Optimal. Leaf size=55 \[ \frac{15 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a}+\frac{3 \text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a}+\frac{\text{Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a}+\frac{5 \log \left (\tanh ^{-1}(a x)\right )}{16 a} \]
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Rubi [A] time = 0.103471, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {5968, 3312, 3301} \[ \frac{15 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a}+\frac{3 \text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a}+\frac{\text{Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a}+\frac{5 \log \left (\tanh ^{-1}(a x)\right )}{16 a} \]
Antiderivative was successfully verified.
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Rule 5968
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh ^6(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{5}{16 x}+\frac{15 \cosh (2 x)}{32 x}+\frac{3 \cosh (4 x)}{16 x}+\frac{\cosh (6 x)}{32 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac{5 \log \left (\tanh ^{-1}(a x)\right )}{16 a}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (6 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac{15 \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}\\ &=\frac{15 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a}+\frac{3 \text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a}+\frac{\text{Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a}+\frac{5 \log \left (\tanh ^{-1}(a x)\right )}{16 a}\\ \end{align*}
Mathematica [A] time = 0.160401, size = 40, normalized size = 0.73 \[ \frac{15 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )+6 \text{Chi}\left (4 \tanh ^{-1}(a x)\right )+\text{Chi}\left (6 \tanh ^{-1}(a x)\right )+10 \log \left (\tanh ^{-1}(a x)\right )}{32 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 48, normalized size = 0.9 \begin{align*}{\frac{15\,{\it Chi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{32\,a}}+{\frac{3\,{\it Chi} \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{16\,a}}+{\frac{{\it Chi} \left ( 6\,{\it Artanh} \left ( ax \right ) \right ) }{32\,a}}+{\frac{5\,\ln \left ({\it Artanh} \left ( ax \right ) \right ) }{16\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} x^{2} - 1\right )}^{4} \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 [B] time = 2.02222, size = 564, normalized size = 10.25 \begin{align*} \frac{20 \, \log \left (\log \left (-\frac{a x + 1}{a x - 1}\right )\right ) + \logintegral \left (-\frac{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + \logintegral \left (-\frac{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}\right ) + 6 \, \logintegral \left (\frac{a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 6 \, \logintegral \left (\frac{a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + 15 \, \logintegral \left (-\frac{a x + 1}{a x - 1}\right ) + 15 \, \logintegral \left (-\frac{a x - 1}{a x + 1}\right )}{64 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4} \operatorname{atanh}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname{artanh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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