Optimal. Leaf size=61 \[ -\frac {\text {Int}\left (\frac {1}{\tanh ^{-1}(a x)},x\right )}{a^3}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^4}+\frac {x}{a^3 \tanh ^{-1}(a x)}-\frac {x}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)} \]
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Rubi [A] time = 0.32, 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 = {} \[ \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \]
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)^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*}
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Mathematica [A] time = 3.40, size = 0, normalized size = 0.00 \[ \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]
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 \[ \int \frac {x^{3}}{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]
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 \[ \int \frac {x^{3}}{\left (-a^{2} x^{2}+1\right )^{2} \arctanh \left (a x \right )^{2}}\, dx \]
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 \[ \frac {2 \, x^{3}}{{\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) - {\left (a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )} + \int -\frac {2 \, {\left (a^{2} x^{4} - 3 \, x^{2}\right )}}{{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right ) - {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-a x + 1\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^3}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2} \,d x \]
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 \[ \int \frac {x^{3}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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