Integrate2 is like Integrate, but Integrate2[a_Plus, b__] := Map[Integrate2[#, b]&, a] ( more linear algebra and partial fraction decomposition is done)
Integrate2[f[x] DeltaFunction[x], x] -> f[0]
Integrate2[f[x] DeltaFunction[x0-x], x] -> f[x0]
Integrate2[f[x] DeltaFunction[a + b x], x] -> Integrate[f[x] (1/Abs[b]) DeltaFunction[a/b + x], x], where Abs[b] -> b, if b is a symbol, and if b = -c, then Abs[-c] -> c, i.e., the variable contained in b is supposed to be positive.
\pi ^2 is replaced by 6 Zeta2.
Integrate2[1/(1-y),{y,x,1}] is interpreted as distribution, i.e. as Integrate2[-1/(1-y)],{y, 0, x}] -> Log[1-y].
Integrate2[1/(1-x),{x,0,1}] -> 0
Since Integrate2 does do a reordering and partial fraction decomposition before calling the integral table of Integrate3, it will in general be slower compared to Integrate3 for sums of integrals. I.e., if the integrand has already an expanded form and if partial fraction decomposition is not necessary it is more effective to use Integrate3.
Overview, DeltaFunction, Integrate3, Integrate5, SumS, SumT.
Integrate2[Log[1 + x] Log[x]/(1 - x), {x, 0, 1}] // Timing\left\{0.057955,\zeta (3)-\frac{3}{2} \zeta (2) \log (2)\right\}
Since Integrate2 uses table-look-up methods it is much faster than Mathematica’s Integrate.
Integrate2[PolyLog[2, x^2], {x, 0, 1}]\zeta (2)-4+4 \log (2)
Integrate2[PolyLog[3, -x], {x, 0, 1}]\frac{\zeta (2)}{2}-\frac{3 \zeta (3)}{4}+1-2 \log (2)
Integrate2[PolyLog[3, 1/(1 + x)], {x, 0, 1}]\zeta (2) (-\log (2))+\frac{3 \zeta (3)}{4}+\frac{\log ^3(2)}{3}-\log ^2(2)+2 \log (2)
Integrate2[DeltaFunction[1 - x] f[x], {x, 0, 1}]f(1)
Integrate2 does integration in a Hadamard sense, i.e., \int _0^1 \, f(x) \, d x means actually expanding the result of \int _{\delta }^{1-\delta} \, f(x) \, dx up to \mathcal{O}(\delta ) and neglecting all \delta-dependent terms. E.g. \int_{\delta }^{1-\delta} \frac{1}{1-x} \, d x = - \log (1-x) \biggl |_{\delta }^{1-\delta } = -\log (\delta )+log (1) \Rightarrow 0
Integrate2[1/(1 - x), {x, 0, 1}]0
In the physics literature sometimes the “+” notation is used. In FeynCalc the \left(frac{1}{1-x} \right)_{+} is represented by PlusDistribution}[1/(1-x)] or just 1/(1-x)
Integrate2[PlusDistribution[1/(1 - x)], {x, 0, 1}]0
Integrate2[PolyLog[2, 1 - x]/(1 - x)^2, {x, 0, 1}]2-\zeta (2)
Integrate2[(Log[x] Log[1 + x])/(1 + x), {x, 0, 1}]-\frac{\zeta (3)}{8}
Integrate2[Log[x]^2/(1 - x), {x, 0, 1}]2 \zeta (3)
Integrate2[PolyLog[2, -x]/(1 + x), {x, 0, 1}]\frac{\zeta (3)}{4}-\frac{1}{2} \zeta (2) \log (2)
Integrate2[Log[x] PolyLog[2, x], {x, 0, 1}]3-2 \zeta (2)
Integrate2[x PolyLog[3, x], {x, 0, 1}]-\frac{\zeta (2)}{4}+\frac{\zeta (3)}{2}+\frac{3}{16}
Integrate2[(Log[x]^2 Log[1 - x])/(1 + x), {x, 0, 1}]\zeta (4)+\zeta (2) \log ^2(2)-4 \;\text{Li}_4\left(\frac{1}{2}\right)-\frac{\log ^4(2)}{6}
Integrate2[PolyLog[2, ((x (1 - z) + z) (1 - x + x z))/z]/(1 - x + x z), {x, 0, 1}]\frac{2 i \pi \;\text{Li}_2(-z)}{1-z}-\frac{4 \;\text{Li}_3\left(\frac{1-z}{2}\right)}{1-z}+\frac{4 \;\text{Li}_3(1-z)}{1-z}+\frac{2 \;\text{Li}_3(-z)}{1-z}+\frac{4 \;\text{Li}_3\left(\frac{1}{z+1}\right)}{1-z}-\frac{4 \;\text{Li}_3\left(\frac{1-z}{z+1}\right)}{1-z}-\frac{4 \;\text{Li}_3\left(\frac{z+1}{2}\right)}{1-z}-\frac{2 \;\text{Li}_2(1-z) \log (z)}{1-z}-\frac{2 \;\text{Li}_2(-z) \log (z)}{1-z}+\frac{4 \;\text{Li}_2(-z) \log (1-z)}{1-z}-\frac{2 S_{12}(1-z)}{1-z}+\frac{i \pi \zeta (2)}{1-z}-\frac{\zeta (2) \log (z)}{1-z}+\frac{2 \zeta (2) \log (1-z)}{1-z}+\frac{6 \zeta (2) \log (z+1)}{1-z}-\frac{4 \zeta (2) \log (2)}{1-z}+\frac{2 \zeta (3)}{1-z}+\frac{\log ^3(z)}{6 (1-z)}+\frac{4 \log ^3(2)}{3 (1-z)}-\frac{\log (1-z) \log ^2(z)}{1-z}-\frac{\log (z+1) \log ^2(z)}{1-z}-\frac{i \pi \log ^2(z)}{2 (1-z)}-\frac{2 \log (1-z) \log ^2(z+1)}{1-z}-\frac{2 \log ^2(2) \log (1-z)}{1-z}-\frac{2 \log ^2(2) \log (z+1)}{1-z}+\frac{4 \log (1-z) \log (z+1) \log (z)}{1-z}+\frac{2 i \pi \log (z+1) \log (z)}{1-z}+\frac{4 \log (2) \log (1-z) \log (z+1)}{1-z}
Apart[Integrate2[x^(OPEm - 1) PolyLog[3, 1 - x], {x, 0, 1}], OPEm]-\frac{\zeta (2)}{m^2}-\frac{\zeta (2)}{m-1}+\frac{\zeta (2)+\zeta (2) \left(-S_1(m-2)\right)+S_{12}(m)+\zeta (3)}{m}
Integrate2[x^(OPEm - 1) Log[1 - x] Log[x] Log[1 + x]/(1 + x), {x, 0, 1}] // Simplify
% /. OPEm -> 2
N[%]\frac{1}{24} (-1)^m \left(48 \zeta (4)+30 \zeta (2) \log ^2(2)+6 \zeta (2) S_{-1}^2(m-1)+18 \zeta (2) S_2(m-1)-24 \zeta (2) S_{1-1}(m-1)-12 S_{-2}(m-1) \left(\zeta (2)-\log (4) S_{-1}(m-1)-\log ^2(2)\right)-36 \zeta (2) \log (2) S_1(m-1)+12 S_{-1}(m-1) (\zeta (2) \log (8)-2 \zeta (3))+39 \zeta (3) S_1(m-1)+24 S_{-2-1-1}(m-1)+24 S_{-1-2-1}(m-1)+24 S_{-1-1-2}(m-1)+24 S_{1-21}(m-1)+24 S_{1-12}(m-1)+24 S_{2-11}(m-1)-12 \log ^2(2) S_2(m-1)+24 \log (2) S_3(m-1)-24 \log (2) S_{-21}(m-1)-24 \log (2) S_{-12}(m-1)-48 \;\text{Li}_4\left(\frac{1}{2}\right)-63 \zeta (3) \log (2)-2 \log ^4(2)\right)
\frac{1}{24} \left(48 \zeta (2)+48 \zeta (4)+30 \zeta (2) \log ^2(2)+12 \left(\zeta (2)-\log ^2(2)+\log (4)\right)-36 \zeta (2) \log (2)-48 \;\text{Li}_4\left(\frac{1}{2}\right)-12 (\zeta (2) \log (8)-2 \zeta (3))+39 \zeta (3)-63 \zeta (3) \log (2)-144-2 \log ^4(2)-12 \log ^2(2)+72 \log (2)\right)
0.0505138
Integrate2[x^(OPEm - 1) (PolyLog[3, (1 - x)/(1 + x)] - PolyLog[3, -((1 - x)/(1 + x))]), {x, 0, 1}]\frac{3 \zeta (2) (-1)^m \log (2)}{2 m}-\frac{3 \zeta (2) \log (2)}{2 m}+\frac{\zeta (2) (-1)^m S_{-1}(m)}{m}-\frac{\zeta (2) S_{-1}(m)}{2 m}+\frac{\zeta (2) (-1)^m S_1(m)}{2 m}-\frac{\zeta (2) S_1(m)}{m}+\frac{(-1)^m S_{-3}(m)}{m}+\frac{(-1)^m S_{-2}(m) S_1(m)}{m}+\frac{S_1(m) S_2(m)}{m}+\frac{S_3(m)}{m}-\frac{(-1)^m S_{-21}(m)}{m}-\frac{S_{-1-2}(m)}{m}-\frac{(-1)^m S_{-12}(m)}{m}-\frac{S_{21}(m)}{m}-\frac{7 (-1)^m \zeta (3)}{8 m}+\frac{21 \zeta (3)}{8 m}
DataType[OPEm, PositiveInteger]
Integrate2[x^(OPEm - 1) DeltaFunction[1 - x], {x, 0, 1}]\text{True}
1
This is the polarized non-singlet spin splitting function whose first moment vanishes.
t = SplittingFunction[PQQNS] /. FCGV[z_] :> ToExpression[z]-8 C_F \left(C_F-\frac{C_A}{2}\right) \left(\frac{\left(x^2+1\right) \left(-2 \zeta (2)-4 \;\text{Li}_2(-x)+\log ^2(x)-4 \log (x+1) \log (x)\right)}{x+1}+4 (1-x)+2 (x+1) \log (x)\right)+C_A C_F \left(\frac{4 \left(x^2+1\right) \log ^2(x)}{1-x}+8 \zeta (2) (x+1)+\left(\frac{536}{9}-16 \zeta (2)\right) \left(\frac{1}{1-x}\right)_++\delta (1-x) \left(\frac{88 \zeta (2)}{3}-24 \zeta (3)+\frac{17}{3}\right)+\frac{4}{9} (53-187 x)-\frac{4}{3} \left(5 x-\frac{22}{1-x}+5\right) \log (x)\right)+C_F N_f \left(-\frac{8 \left(x^2+1\right) \log (x)}{3 (1-x)}+\left(-\frac{16 \zeta (2)}{3}-\frac{2}{3}\right) \delta (1-x)+\frac{88 x}{9}-\frac{80}{9} \left(\frac{1}{1-x}\right)_+-\frac{8}{9}\right)+C_F^2 \left(-\frac{16 \left(x^2+1\right) \log (1-x) \log (x)}{1-x}+\delta (1-x) (-24 \zeta (2)+48 \zeta (3)+3)-40 (1-x)-4 (x+1) \log ^2(x)-8 \left(2 x+\frac{3}{1-x}\right) \log (x)\right)
t // Expand8 \zeta (2) C_A C_F-\frac{16 x^2 C_A C_F \;\text{Li}_2(-x)}{x+1}-\frac{16 C_A C_F \;\text{Li}_2(-x)}{x+1}-\frac{8 \zeta (2) x^2 C_A C_F}{x+1}+\frac{4 x^2 C_A C_F \log ^2(x)}{1-x}+\frac{4 x^2 C_A C_F \log ^2(x)}{x+1}-\frac{16 x^2 C_A C_F \log (x) \log (x+1)}{x+1}+\frac{88}{3} \zeta (2) C_A C_F \delta (1-x)+\frac{17}{3} C_A C_F \delta (1-x)+8 \zeta (2) x C_A C_F-\frac{8 \zeta (2) C_A C_F}{x+1}-16 \zeta (2) \left(\frac{1}{1-x}\right)_+ C_A C_F-24 \zeta (3) C_A C_F \delta (1-x)-\frac{892}{9} x C_A C_F+\frac{536}{9} \left(\frac{1}{1-x}\right)_+ C_A C_F+\frac{4 C_A C_F \log ^2(x)}{1-x}+\frac{4 C_A C_F \log ^2(x)}{x+1}+\frac{4}{3} C_A C_F \log (x)+\frac{4}{3} x C_A C_F \log (x)+\frac{88 C_A C_F \log (x)}{3 (1-x)}-\frac{16 C_A C_F \log (x) \log (x+1)}{x+1}+\frac{356 C_A C_F}{9}-\frac{8 x^2 C_F N_f \log (x)}{3 (1-x)}-\frac{16}{3} \zeta (2) C_F N_f \delta (1-x)-\frac{2}{3} C_F N_f \delta (1-x)+\frac{88}{9} x C_F N_f-\frac{80}{9} \left(\frac{1}{1-x}\right)_+ C_F N_f-\frac{8 C_F N_f \log (x)}{3 (1-x)}-\frac{8 C_F N_f}{9}+\frac{32 x^2 C_F^2 \;\text{Li}_2(-x)}{x+1}+\frac{32 C_F^2 \;\text{Li}_2(-x)}{x+1}+\frac{16 \zeta (2) x^2 C_F^2}{x+1}-\frac{8 x^2 C_F^2 \log ^2(x)}{x+1}-\frac{16 x^2 C_F^2 \log (1-x) \log (x)}{1-x}+\frac{32 x^2 C_F^2 \log (x) \log (x+1)}{x+1}-24 \zeta (2) C_F^2 \delta (1-x)+3 C_F^2 \delta (1-x)+\frac{16 \zeta (2) C_F^2}{x+1}+48 \zeta (3) C_F^2 \delta (1-x)+72 x C_F^2-4 x C_F^2 \log ^2(x)-\frac{8 C_F^2 \log ^2(x)}{x+1}-4 C_F^2 \log ^2(x)-32 x C_F^2 \log (x)-\frac{16 C_F^2 \log (1-x) \log (x)}{1-x}-\frac{24 C_F^2 \log (x)}{1-x}-16 C_F^2 \log (x)+\frac{32 C_F^2 \log (x) \log (x+1)}{x+1}-72 C_F^2
Integrate2[t, {x, 0, 1}] // Timing\{0.040008,0\}
Expanding t with respect to x yields a form already suitable for Integrate3 and therefore the following is faster:
Integrate3[Expand[t, x], {x, 0, 1}] // Expand // Timing\{0.018181,0\}
Clear[t];
Integrate2[DeltaFunction[1 - x] f[x], {x, 0, 1}]f(1)
Integrate2[x^5 Log[1 + x]^2, {x, 0, 1}]
N[%]\frac{46 \log (2)}{45}-\frac{6959}{10800}
0.0641986
NIntegrate[x^5 Log[1 + x]^2, {x, 0, 1}]0.0641986
Integrate2[x^(OPEm - 1) Log[1 + x]^2, {x, 0, 1}]-\frac{2 (-1)^m S_1^2(m)}{m}+\frac{(-1)^m S_1\left(\frac{m-1}{2}\right) S_1(m)}{m}-\frac{S_1\left(\frac{m-1}{2}\right) S_1(m)}{m}+\frac{(-1)^m S_1\left(\frac{m}{2}\right) S_1(m)}{m}+\frac{S_1\left(\frac{m}{2}\right) S_1(m)}{m}+\frac{(-1)^m S_2\left(\frac{m-1}{2}\right)}{2 m}-\frac{S_2\left(\frac{m-1}{2}\right)}{2 m}+\frac{(-1)^m S_2\left(\frac{m}{2}\right)}{2 m}+\frac{S_2\left(\frac{m}{2}\right)}{2 m}-\frac{2 (-1)^m S_2(m)}{m}-\frac{2 (-1)^m S_{-11}(m)}{m}+\frac{4 (-1)^m \log (2) S_1(m)}{m}-\frac{(-1)^m \log (2) S_1\left(\frac{m-1}{2}\right)}{m}+\frac{\log (2) S_1\left(\frac{m-1}{2}\right)}{m}-\frac{(-1)^m \log (2) S_1\left(\frac{m}{2}\right)}{m}-\frac{\log (2) S_1\left(\frac{m}{2}\right)}{m}-\frac{(-1)^m \log ^2(2)}{m}+\frac{\log ^2(2)}{m}