Optimal. Leaf size=99 \[ -\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\text {Shi}\left (2 \tanh ^{-1}(a x)\right )-\frac {\text {Int}\left (\frac {1}{x^2 \tanh ^{-1}(a x)^2},x\right )}{2 a} \]
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Rubi [A]
time = 0.18, 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 {1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx &=a^2 \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx+\int \frac {1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}+\left (2 a^2\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}+\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac {1}{2 a x \tanh ^{-1}(a x)^2}-\frac {a x}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\text {Shi}\left (2 \tanh ^{-1}(a x)\right )-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)^2} \, dx}{2 a}\\ \end {align*}
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Mathematica [A]
time = 2.57, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 20.35, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{2} \arctanh \left (a x \right )^{3}}\, dx\]
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 \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 {1}{x \left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{3}{\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 = -0.01 \begin {gather*} \int \frac {1}{x\,{\mathrm {atanh}\left (a\,x\right )}^3\,{\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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