Optimal. Leaf size=33 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \tanh ^{-1}(a x)},x\right )}{a}-\frac {1}{a x \tanh ^{-1}(a x)} \]
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Rubi [A] time = 0.08, 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 = {} \[ \int \frac {1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx \]
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 ) \tanh ^{-1}(a x)^2} \, dx &=-\frac {1}{a x \tanh ^{-1}(a x)}-\frac {\int \frac {1}{x^2 \tanh ^{-1}(a x)} \, dx}{a}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{{\left (a^{2} x^{3} - x\right )} \operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]
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 \[ \int -\frac {1}{{\left (a^{2} x^{2} - 1\right )} x \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (-a^{2} x^{2}+1\right ) \arctanh \left (a x \right )^{2}}\, dx \]
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 \[ -\frac {2}{a x \log \left (a x + 1\right ) - a x \log \left (-a x + 1\right )} + 2 \, \int -\frac {1}{a x^{2} \log \left (a x + 1\right ) - a x^{2} \log \left (-a x + 1\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ -\int \frac {1}{x\,{\mathrm {atanh}\left (a\,x\right )}^2\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {1}{a^{2} x^{3} \operatorname {atanh}^{2}{\left (a x \right )} - x \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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