3.369 \(\int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\)

Optimal. Leaf size=95 \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{a} \]

[Out]

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

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Rubi [A]  time = 0.0264132, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {5950} \[ -\frac{i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a}+\frac{i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]/Sqrt[1 - a^2*x^2],x]

[Out]

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

Rule 5950

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-2*(a + b*ArcTanh[c*x])*
ArcTan[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(c*Sqrt[d]), x] + (-Simp[(I*b*PolyLog[2, -((I*Sqrt[1 - c*x])/Sqrt[1 + c*x
])])/(c*Sqrt[d]), x] + Simp[(I*b*PolyLog[2, (I*Sqrt[1 - c*x])/Sqrt[1 + c*x]])/(c*Sqrt[d]), x]) /; FreeQ[{a, b,
c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0]

Rubi steps

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

Mathematica [A]  time = 0.0733041, size = 76, normalized size = 0.8 \[ -\frac{i \left (\text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x) \left (\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )\right )}{a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTanh[a*x]/Sqrt[1 - a^2*x^2],x]

[Out]

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

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Maple [B]  time = 0.247, size = 366, normalized size = 3.9 \begin{align*}{\frac{{\frac{i}{2}}{\it Artanh} \left ( ax \right ) }{a}\ln \left ({-i{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{iax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{i{\it Artanh} \left ( ax \right ) }{a}\ln \left ( \left ( 1-i \right ) \cosh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) + \left ( 1+i \right ) \sinh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) \right ) }-{\frac{{\frac{i}{2}}{\it Artanh} \left ( ax \right ) }{a}\ln \left ({i{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{iax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{i{\it Artanh} \left ( ax \right ) }{a}\ln \left ( \left ( 1+i \right ) \cosh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) + \left ( 1-i \right ) \sinh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) \right ) }+{\frac{i}{a}\ln \left ( \left ( 1-i \right ) \cosh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) + \left ( 1+i \right ) \sinh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) \right ) \ln \left ({-i{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{iax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{i}{a}\ln \left ( \left ( 1+i \right ) \cosh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) + \left ( 1-i \right ) \sinh \left ({\frac{{\it Artanh} \left ( ax \right ) }{2}} \right ) \right ) \ln \left ({i{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{iax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{i}{a}{\it dilog} \left ({-i{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{iax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{i}{a}{\it dilog} \left ({i{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{iax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I/a*arctanh(a*x)*ln(-I/(-a^2*x^2+1)^(1/2)-I*a*x/(-a^2*x^2+1)^(1/2))-I/a*ln((1-I)*cosh(1/2*arctanh(a*x))+(1
+I)*sinh(1/2*arctanh(a*x)))*arctanh(a*x)-1/2*I/a*arctanh(a*x)*ln(I/(-a^2*x^2+1)^(1/2)+I*a*x/(-a^2*x^2+1)^(1/2)
)+I/a*ln((1+I)*cosh(1/2*arctanh(a*x))+(1-I)*sinh(1/2*arctanh(a*x)))*arctanh(a*x)+I/a*ln((1-I)*cosh(1/2*arctanh
(a*x))+(1+I)*sinh(1/2*arctanh(a*x)))*ln(-I/(-a^2*x^2+1)^(1/2)-I*a*x/(-a^2*x^2+1)^(1/2))-I/a*ln((1+I)*cosh(1/2*
arctanh(a*x))+(1-I)*sinh(1/2*arctanh(a*x)))*ln(I/(-a^2*x^2+1)^(1/2)+I*a*x/(-a^2*x^2+1)^(1/2))+I/a*dilog(-I/(-a
^2*x^2+1)^(1/2)-I*a*x/(-a^2*x^2+1)^(1/2))-I/a*dilog(I/(-a^2*x^2+1)^(1/2)+I*a*x/(-a^2*x^2+1)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} \operatorname{artanh}\left (a x\right )}{a^{2} x^{2} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-a^2*x^2 + 1)*arctanh(a*x)/(a^2*x^2 - 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}{\left (a x \right )}}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atanh(a*x)/sqrt(-(a*x - 1)*(a*x + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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