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

Optimal. Leaf size=182 \[ -\frac{i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a \sqrt{c-a^2 c x^2}}+\frac{i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{1-a^2 x^2} \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{a \sqrt{c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0622075, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {5954, 5950} \[ -\frac{i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a \sqrt{c-a^2 c x^2}}+\frac{i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{1-a^2 x^2} \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{a \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]/Sqrt[c - a^2*c*x^2],x]

[Out]

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

Rule 5954

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[1 - c^2*x^2]/S
qrt[d + e*x^2], Int[(a + b*ArcTanh[c*x])^p/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d
+ e, 0] && IGtQ[p, 0] &&  !GtQ[d, 0]

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

Mathematica [A]  time = 0.118544, size = 109, normalized size = 0.6 \[ -\frac{i \sqrt{c \left (1-a^2 x^2\right )} \left (\text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x) \left (\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )\right )}{a c \sqrt{1-a^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*Sqrt[c*(1 - a^2*x^2)]*(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*c*Sqrt[1 - a^2*x^2])

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Maple [A]  time = 0.362, size = 290, normalized size = 1.6 \begin{align*}{\frac{i{\it Artanh} \left ( ax \right ) }{ \left ({a}^{2}{x}^{2}-1 \right ) ac}\ln \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) \sqrt{-{a}^{2}{x}^{2}+1}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}}-{\frac{i{\it Artanh} \left ( ax \right ) }{ \left ({a}^{2}{x}^{2}-1 \right ) ac}\ln \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) \sqrt{-{a}^{2}{x}^{2}+1}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}}+{\frac{i}{ \left ({a}^{2}{x}^{2}-1 \right ) ac}{\it dilog} \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) \sqrt{-{a}^{2}{x}^{2}+1}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}}-{\frac{i}{ \left ({a}^{2}{x}^{2}-1 \right ) ac}{\it dilog} \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) \sqrt{-{a}^{2}{x}^{2}+1}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*ln(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)*(-a^2*x^2+1)^(1/2)*(-(a*x-1)*(a*x+1)*c)^(1/2)/(a^2*x^2-1)/a/
c-I*ln(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)*(-a^2*x^2+1)^(1/2)*(-(a*x-1)*(a*x+1)*c)^(1/2)/(a^2*x^2-1)/
a/c+I*dilog(1+I*(a*x+1)/(-a^2*x^2+1)^(1/2))*(-a^2*x^2+1)^(1/2)*(-(a*x-1)*(a*x+1)*c)^(1/2)/(a^2*x^2-1)/a/c-I*di
log(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))*(-a^2*x^2+1)^(1/2)*(-(a*x-1)*(a*x+1)*c)^(1/2)/(a^2*x^2-1)/a/c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Integral(atanh(a*x)/sqrt(-c*(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} c x^{2} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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