Optimal. Leaf size=61 \[ \frac {x}{a^3 \tanh ^{-1}(a x)}-\frac {x}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^4}-\frac {\text {Int}\left (\frac {1}{\tanh ^{-1}(a x)},x\right )}{a^3} \]
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Rubi [A]
time = 0.23, 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 = {}
\begin {gather*} \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \end {gather*}
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 {\text {Subst}\left (\int \frac {\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}+\frac {\text {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 {\text {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 {\text {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 {\text {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 = 2.37, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 32.05, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^3}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (a^2\,x^2-1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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