Optimal. Leaf size=101 \[ -\frac{\text{Unintegrable}\left (\frac{1}{\tanh ^{-1}(a x)^2},x\right )}{2 a^3}+\frac{\text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{a^4}-\frac{x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{a^2 x^2+1}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{x}{2 a^3 \tanh ^{-1}(a x)^2} \]
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Rubi [A] time = 0.22487, 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 = {} \[ \int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx &=\frac{\int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx}{a^2}-\frac{\int \frac{x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx}{a^2}\\ &=\frac{x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac{x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{\int \frac{1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}+\frac{2 \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx}{a^2}\\ &=\frac{x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac{x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}-\frac{\int \frac{1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}\\ &=\frac{x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac{x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}-\frac{\int \frac{1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}\\ &=\frac{x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac{x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\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)^2} \, dx}{2 a^3}\\ &=\frac{x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac{x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{1+a^2 x^2}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{a^4}-\frac{\int \frac{1}{\tanh ^{-1}(a x)^2} \, dx}{2 a^3}\\ \end{align*}
Mathematica [A] time = 10.8905, size = 0, normalized size = 0. \[ \int \frac{x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.227, 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 ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, a x^{3} -{\left (a^{2} x^{4} - 3 \, x^{2}\right )} \log \left (a x + 1\right ) +{\left (a^{2} x^{4} - 3 \, x^{2}\right )} \log \left (-a x + 1\right )}{{\left (a^{4} x^{2} - a^{2}\right )} \log \left (a x + 1\right )^{2} - 2 \,{\left (a^{4} x^{2} - a^{2}\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right ) +{\left (a^{4} x^{2} - a^{2}\right )} \log \left (-a x + 1\right )^{2}} - \int -\frac{2 \,{\left (a^{4} x^{5} - 2 \, a^{2} x^{3} + 3 \, x\right )}}{{\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (a x + 1\right ) -{\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (-a x + 1\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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 )^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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}^{3}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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 )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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