Optimal. Leaf size=49 \[ -\text {Int}\left (\frac {1}{x \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)},x\right )+\frac {3}{4} \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+\frac {1}{8} \text {Shi}\left (4 \tanh ^{-1}(a x)\right ) \]
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Rubi [A] time = 0.06, 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 = {} \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx \]
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 )^3 \tanh ^{-1}(a x)} \, dx &=\int \frac {1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx\\ \end {align*}
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Mathematica [A] time = 1.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {1}{{\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )} \operatorname {artanh}\left (a x\right )}, x\right ) \]
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 \[ \int -\frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname {artanh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (-a^{2} x^{2}+1\right )^{3} \arctanh \left (a x \right )}\, dx \]
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 \[ -\int \frac {1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname {artanh}\left (a x\right )}\,{d x} \]
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 \[ -\int \frac {1}{x\,\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^3} \,d x \]
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 \[ - \int \frac {1}{a^{6} x^{7} \operatorname {atanh}{\left (a x \right )} - 3 a^{4} x^{5} \operatorname {atanh}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atanh}{\left (a x \right )} - x \operatorname {atanh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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