Optimal. Leaf size=48 \[ -\text{Unintegrable}\left (\frac{1}{x \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)},x\right )+\frac{3}{4} \text{Shi}\left (2 \tanh ^{-1}(a x)\right )+\frac{1}{8} \text{Shi}\left (4 \tanh ^{-1}(a x)\right ) \]
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Rubi [A] time = 0.0636374, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [A] time = 1.23197, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.185, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( -{a}^{2}{x}^{2}+1 \right ) ^{3}{\it Artanh} \left ( ax \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname{artanh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} - \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (a^{2} x^{2} - 1\right )}^{3} x \operatorname{artanh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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