Optimal. Leaf size=102 \[ -\frac {\text {Int}\left (\frac {1}{\tanh ^{-1}(a x)^2},x\right )}{2 a^3}+\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{a^4}+\frac {x}{2 a^3 \tanh ^{-1}(a x)^2}-\frac {a^2 x^2+1}{2 a^4 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {x}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \]
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Rubi [A] time = 0.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}
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Mathematica [A] time = 9.43, size = 0, normalized size = 0.00 \[ \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (-a^{2} x^{2}+1\right )^{2} \arctanh \left (a x \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, a x^{3} - {\left (a^{2} x^{4} - 3 \, x^{2}\right )} \log \left (a x + 1\right ) + {\left (a^{2} x^{4} - 3 \, x^{2}\right )} \log \left (-a x + 1\right )}{{\left (a^{4} x^{2} - a^{2}\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{4} x^{2} - a^{2}\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right ) + {\left (a^{4} x^{2} - a^{2}\right )} \log \left (-a x + 1\right )^{2}} - \int -\frac {2 \, {\left (a^{4} x^{5} - 2 \, a^{2} x^{3} + 3 \, x\right )}}{{\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (a x + 1\right ) - {\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (-a x + 1\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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