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

Optimal. Leaf size=60 \[ -\frac{\text{Unintegrable}\left (\frac{1}{\tanh ^{-1}(a x)},x\right )}{a^3}+\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^4}-\frac{x}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{x}{a^3 \tanh ^{-1}(a x)} \]

[Out]

x/(a^3*ArcTanh[a*x]) - x/(a^3*(1 - a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[2*ArcTanh[a*x]]/a^4 - Unintegrable[Ar
cTanh[a*x]^(-1), x]/a^3

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Rubi [A]  time = 0.31626, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

x/(a^3*ArcTanh[a*x]) - x/(a^3*(1 - a^2*x^2)*ArcTanh[a*x]) + CoshIntegral[2*ArcTanh[a*x]]/a^4 - Defer[Int][ArcT
anh[a*x]^(-1), x]/a^3

Rubi steps

\begin{align*} \int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx &=\frac{\int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx}{a^2}-\frac{\int \frac{x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx}{a^2}\\ &=\frac{x}{a^3 \tanh ^{-1}(a x)}-\frac{x}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{\int \frac{1}{\tanh ^{-1}(a x)} \, dx}{a^3}+\frac{\int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx}{a^3}+\frac{\int \frac{x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx}{a}\\ &=\frac{x}{a^3 \tanh ^{-1}(a x)}-\frac{x}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}+\frac{\operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}-\frac{\int \frac{1}{\tanh ^{-1}(a x)} \, dx}{a^3}\\ &=\frac{x}{a^3 \tanh ^{-1}(a x)}-\frac{x}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}-\frac{\int \frac{1}{\tanh ^{-1}(a x)} \, dx}{a^3}\\ &=\frac{x}{a^3 \tanh ^{-1}(a x)}-\frac{x}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^4}-\frac{\int \frac{1}{\tanh ^{-1}(a x)} \, dx}{a^3}\\ &=\frac{x}{a^3 \tanh ^{-1}(a x)}-\frac{x}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^4}-\frac{\int \frac{1}{\tanh ^{-1}(a x)} \, dx}{a^3}\\ \end{align*}

Mathematica [A]  time = 3.33926, size = 0, normalized size = 0. \[ \int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.209, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ \left ( -{a}^{2}{x}^{2}+1 \right ) ^{2} \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, x^{3}}{{\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) -{\left (a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )} + \int -\frac{2 \,{\left (a^{2} x^{4} - 3 \, x^{2}\right )}}{{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right ) -{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-a x + 1\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^3/((a^3*x^2 - a)*log(a*x + 1) - (a^3*x^2 - a)*log(-a*x + 1)) + integrate(-2*(a^2*x^4 - 3*x^2)/((a^5*x^4 -
2*a^3*x^2 + a)*log(a*x + 1) - (a^5*x^4 - 2*a^3*x^2 + a)*log(-a*x + 1)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname{artanh}\left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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