3.396 \(\int \frac{x^3 \tanh ^{-1}(a x)^2}{(1-a^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=186 \[ \frac{2 i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a^4}-\frac{2 i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a^4}+\frac{2}{a^4 \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{a^4}+\frac{\tanh ^{-1}(a x)^2}{a^4 \sqrt{1-a^2 x^2}}-\frac{2 x \tanh ^{-1}(a x)}{a^3 \sqrt{1-a^2 x^2}}+\frac{4 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{a^4} \]

[Out]

2/(a^4*Sqrt[1 - a^2*x^2]) - (2*x*ArcTanh[a*x])/(a^3*Sqrt[1 - a^2*x^2]) + (4*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]
]*ArcTanh[a*x])/a^4 + ArcTanh[a*x]^2/(a^4*Sqrt[1 - a^2*x^2]) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/a^4 + ((2*I)
*PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a^4 - ((2*I)*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a^4

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Rubi [A]  time = 0.310389, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6028, 5994, 5950, 5958} \[ \frac{2 i \text{PolyLog}\left (2,-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a^4}-\frac{2 i \text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a^4}+\frac{2}{a^4 \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{a^4}+\frac{\tanh ^{-1}(a x)^2}{a^4 \sqrt{1-a^2 x^2}}-\frac{2 x \tanh ^{-1}(a x)}{a^3 \sqrt{1-a^2 x^2}}+\frac{4 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{a^4} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*ArcTanh[a*x]^2)/(1 - a^2*x^2)^(3/2),x]

[Out]

2/(a^4*Sqrt[1 - a^2*x^2]) - (2*x*ArcTanh[a*x])/(a^3*Sqrt[1 - a^2*x^2]) + (4*ArcTan[Sqrt[1 - a*x]/Sqrt[1 + a*x]
]*ArcTanh[a*x])/a^4 + ArcTanh[a*x]^2/(a^4*Sqrt[1 - a^2*x^2]) + (Sqrt[1 - a^2*x^2]*ArcTanh[a*x]^2)/a^4 + ((2*I)
*PolyLog[2, ((-I)*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a^4 - ((2*I)*PolyLog[2, (I*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a^4

Rule 6028

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/e, Int
[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x])^p, x], x] - Dist[d/e, Int[x^(m - 2)*(d + e*x^2)^q*(a + b*A
rcTanh[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegersQ[p, 2*q] && LtQ[q, -1] &&
IGtQ[m, 1] && NeQ[p, -1]

Rule 5994

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

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]

Rule 5958

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[b/(c*d*Sqrt[d + e*x^2]
), x] + Simp[(x*(a + b*ArcTanh[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
]

Rubi steps

\begin{align*} \int \frac{x^3 \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac{\int \frac{x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^2}-\frac{\int \frac{x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{a^2}\\ &=\frac{\tanh ^{-1}(a x)^2}{a^4 \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{a^4}-\frac{2 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^3}-\frac{2 \int \frac{\tanh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{a^3}\\ &=\frac{2}{a^4 \sqrt{1-a^2 x^2}}-\frac{2 x \tanh ^{-1}(a x)}{a^3 \sqrt{1-a^2 x^2}}+\frac{4 \tan ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right ) \tanh ^{-1}(a x)}{a^4}+\frac{\tanh ^{-1}(a x)^2}{a^4 \sqrt{1-a^2 x^2}}+\frac{\sqrt{1-a^2 x^2} \tanh ^{-1}(a x)^2}{a^4}+\frac{2 i \text{Li}_2\left (-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{a^4}-\frac{2 i \text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.371595, size = 165, normalized size = 0.89 \[ \frac{\frac{-2 i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+\left (2-a^2 x^2\right ) \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x) \left (-i \sqrt{1-a^2 x^2} \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )+i \sqrt{1-a^2 x^2} \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+a x\right )+2}{\sqrt{1-a^2 x^2}}+2 i \text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )}{a^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^3*ArcTanh[a*x]^2)/(1 - a^2*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.273, size = 230, normalized size = 1.2 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-2\,{\it Artanh} \left ( ax \right ) +2}{2\,{a}^{4} \left ( ax-1 \right ) }\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+2\,{\it Artanh} \left ( ax \right ) +2}{2\,{a}^{4} \left ( ax+1 \right ) }\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}{{a}^{4}}\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{2\,i{\it Artanh} \left ( ax \right ) }{{a}^{4}}\ln \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{2\,i{\it Artanh} \left ( ax \right ) }{{a}^{4}}\ln \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }+{\frac{2\,i}{{a}^{4}}{\it dilog} \left ( 1+{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) }-{\frac{2\,i}{{a}^{4}}{\it dilog} \left ( 1-{i \left ( ax+1 \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arctanh(a*x)^2/(-a^2*x^2+1)^(3/2),x)

[Out]

-1/2*(arctanh(a*x)^2-2*arctanh(a*x)+2)*(-(a*x-1)*(a*x+1))^(1/2)/a^4/(a*x-1)+1/2*(arctanh(a*x)^2+2*arctanh(a*x)
+2)*(-(a*x-1)*(a*x+1))^(1/2)/a^4/(a*x+1)+arctanh(a*x)^2*(-(a*x-1)*(a*x+1))^(1/2)/a^4+2*I*ln(1+I*(a*x+1)/(-a^2*
x^2+1)^(1/2))*arctanh(a*x)/a^4-2*I*ln(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))*arctanh(a*x)/a^4+2*I*dilog(1+I*(a*x+1)/(
-a^2*x^2+1)^(1/2))/a^4-2*I*dilog(1-I*(a*x+1)/(-a^2*x^2+1)^(1/2))/a^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \operatorname{artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arctanh(a*x)^2/(-a^2*x^2+1)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^3*arctanh(a*x)^2/(-a^2*x^2 + 1)^(3/2), 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} x^{3} \operatorname{artanh}\left (a x\right )^{2}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arctanh(a*x)^2/(-a^2*x^2+1)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*x^2 + 1)*x^3*arctanh(a*x)^2/(a^4*x^4 - 2*a^2*x^2 + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \operatorname{atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*atanh(a*x)**2/(-a**2*x**2+1)**(3/2),x)

[Out]

Integral(x**3*atanh(a*x)**2/(-(a*x - 1)*(a*x + 1))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \operatorname{artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arctanh(a*x)^2/(-a^2*x^2+1)^(3/2),x, algorithm="giac")

[Out]

integrate(x^3*arctanh(a*x)^2/(-a^2*x^2 + 1)^(3/2), x)