### 3.280 $$\int \frac{\log (1+\frac{i \sqrt{1-a x}}{\sqrt{1+a x}})}{1-a^2 x^2} \, dx$$

Optimal. Leaf size=29 $\frac{\text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a}$

[Out]

PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/a

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Rubi [A]  time = 0.0335913, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.026, Rules used = {2518} $\frac{\text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a}$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[1 + (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(1 - a^2*x^2),x]

[Out]

PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/a

Rule 2518

Int[Log[v_]*(u_), x_Symbol] :> With[{w = DerivativeDivides[v, u*(1 - v), x]}, Simp[w*PolyLog[2, 1 - v], x] /;
!FalseQ[w]]

Rubi steps

\begin{align*} \int \frac{\log \left (1+\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{1-a^2 x^2} \, dx &=\frac{\text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{a}\\ \end{align*}

Mathematica [B]  time = 0.616548, size = 134, normalized size = 4.62 $\frac{\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )-2 \left (-\text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )+\text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x) \left (\log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )-\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )+\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )\right )+4 \log \left (1+\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{4 a}$

Warning: Unable to verify antiderivative.

[In]

Integrate[Log[1 + (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(1 - a^2*x^2),x]

[Out]

(4*ArcTanh[a*x]*Log[1 + (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]] + PolyLog[2, -E^(-2*ArcTanh[a*x])] - 2*(ArcTanh[a*x]*
(Log[1 + E^(-2*ArcTanh[a*x])] - Log[1 - I/E^ArcTanh[a*x]] + Log[1 + I/E^ArcTanh[a*x]]) - PolyLog[2, (-I)/E^Arc
Tanh[a*x]] + PolyLog[2, I/E^ArcTanh[a*x]]))/(4*a)

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{a}^{2}{x}^{2}+1}\ln \left ( 1+{i\sqrt{-ax+1}{\frac{1}{\sqrt{ax+1}}}} \right ) }\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(1+I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x)

[Out]

int(ln(1+I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \,{\left (\log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} + 2 \, \log \left (a x + 1\right ) \log \left (-a x + 1\right ) - \log \left (-a x + 1\right )^{2} - 4 \,{\left (\log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )} \log \left (\sqrt{a x + 1} + i \, \sqrt{-a x + 1}\right )}{8 \, a} - \int -\frac{\sqrt{a x + 1}{\left (\log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )}}{2 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} +{\left (2 i \, a^{2} x^{2} - 2 i\right )} \sqrt{-a x + 1}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1+I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x, algorithm="maxima")

[Out]

-1/8*(2*(log(a*x + 1) - log(-a*x + 1))*log(a*x + 1) - log(a*x + 1)^2 + 2*log(a*x + 1)*log(-a*x + 1) - log(-a*x
+ 1)^2 - 4*(log(a*x + 1) - log(-a*x + 1))*log(sqrt(a*x + 1) + I*sqrt(-a*x + 1)))/a - integrate(-sqrt(a*x + 1)
*(log(a*x + 1) - log(-a*x + 1))/(2*(a^2*x^2 - 1)*sqrt(a*x + 1) + (2*I*a^2*x^2 - 2*I)*sqrt(-a*x + 1)), x)

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Fricas [A]  time = 2.19388, size = 92, normalized size = 3.17 \begin{align*} \frac{{\rm Li}_2\left (-\frac{a x + i \, \sqrt{a x + 1} \sqrt{-a x + 1} + 1}{a x + 1} + 1\right )}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1+I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x, algorithm="fricas")

[Out]

dilog(-(a*x + I*sqrt(a*x + 1)*sqrt(-a*x + 1) + 1)/(a*x + 1) + 1)/a

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(1+I*(-a*x+1)**(1/2)/(a*x+1)**(1/2))/(-a**2*x**2+1),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\log \left (\frac{i \, \sqrt{-a x + 1}}{\sqrt{a x + 1}} + 1\right )}{a^{2} x^{2} - 1}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(1+I*(-a*x+1)^(1/2)/(a*x+1)^(1/2))/(-a^2*x^2+1),x, algorithm="giac")

[Out]

integrate(-log(I*sqrt(-a*x + 1)/sqrt(a*x + 1) + 1)/(a^2*x^2 - 1), x)