Optimal. Leaf size=177 \[ -\frac{2 x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{x}{15 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{a^2 x^2+1}{90 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{a^2 x^2+1}{60 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{1}{6 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}+\frac{2 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{45 a} \]
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Rubi [A] time = 0.344156, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {5966, 5996, 6032, 6034, 3312, 3301, 5968} \[ -\frac{2 x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{x}{15 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{a^2 x^2+1}{90 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{a^2 x^2+1}{60 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{1}{6 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}+\frac{2 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{45 a} \]
Antiderivative was successfully verified.
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Rule 5966
Rule 5996
Rule 6032
Rule 6034
Rule 3312
Rule 3301
Rule 5968
Rubi steps
\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^7} \, dx &=-\frac{1}{6 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}+\frac{1}{3} a \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^6} \, dx\\ &=-\frac{1}{6 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{x}{15 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{1+a^2 x^2}{60 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}+\frac{1}{15} a \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4} \, dx\\ &=-\frac{1}{6 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{x}{15 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{1+a^2 x^2}{60 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{1+a^2 x^2}{90 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}+\frac{1}{45} (2 a) \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac{1}{6 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{x}{15 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{1+a^2 x^2}{60 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{1+a^2 x^2}{90 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{2}{45} \int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+\frac{1}{45} \left (2 a^2\right ) \int \frac{x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{6 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{x}{15 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{1+a^2 x^2}{60 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{1+a^2 x^2}{90 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{45 a}+\frac{2 \operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{45 a}\\ &=-\frac{1}{6 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{x}{15 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{1+a^2 x^2}{60 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{1+a^2 x^2}{90 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{2 \operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{45 a}+\frac{2 \operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{45 a}\\ &=-\frac{1}{6 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{x}{15 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{1+a^2 x^2}{60 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{1+a^2 x^2}{90 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{45 a}\\ &=-\frac{1}{6 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^6}-\frac{x}{15 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^5}-\frac{1+a^2 x^2}{60 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^4}-\frac{x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac{1+a^2 x^2}{90 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 x}{45 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{2 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{45 a}\\ \end{align*}
Mathematica [A] time = 0.0950506, size = 112, normalized size = 0.63 \[ \frac{8 \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^6 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )+2 \left (a^2 x^2+1\right ) \tanh ^{-1}(a x)^4+3 \left (a^2 x^2+1\right ) \tanh ^{-1}(a x)^2+8 a x \tanh ^{-1}(a x)^5+4 a x \tanh ^{-1}(a x)^3+12 a x \tanh ^{-1}(a x)+30}{180 a \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 113, normalized size = 0.6 \begin{align*}{\frac{1}{a} \left ( -{\frac{1}{12\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{6}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{12\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{6}}}-{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{30\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{5}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{60\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}}}-{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{90\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{90\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}-{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{45\,{\it Artanh} \left ( ax \right ) }}+{\frac{2\,{\it Chi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{45}} \right ) } \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*} \frac{2 \,{\left (2 \, a x \log \left (a x + 1\right )^{5} - 2 \, a x \log \left (-a x + 1\right )^{5} + 4 \, a x \log \left (a x + 1\right )^{3} +{\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{4} +{\left (a^{2} x^{2} + 10 \, a x \log \left (a x + 1\right ) + 1\right )} \log \left (-a x + 1\right )^{4} - 4 \,{\left (5 \, a x \log \left (a x + 1\right )^{2} + a x +{\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{3} + 48 \, a x \log \left (a x + 1\right ) + 6 \,{\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 2 \,{\left (10 \, a x \log \left (a x + 1\right )^{3} + 3 \, a^{2} x^{2} + 6 \, a x \log \left (a x + 1\right ) + 3 \,{\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 3\right )} \log \left (-a x + 1\right )^{2} - 2 \,{\left (5 \, a x \log \left (a x + 1\right )^{4} + 6 \, a x \log \left (a x + 1\right )^{2} + 2 \,{\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{3} + 24 \, a x + 6 \,{\left (a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right ) + 240\right )}}{45 \,{\left ({\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{6} - 6 \,{\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{5} \log \left (-a x + 1\right ) + 15 \,{\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{4} \log \left (-a x + 1\right )^{2} - 20 \,{\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{3} \log \left (-a x + 1\right )^{3} + 15 \,{\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right )^{4} - 6 \,{\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right )^{5} +{\left (a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )^{6}\right )}} - \int -\frac{4 \,{\left (a^{2} x^{2} + 1\right )}}{45 \,{\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) -{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-a x + 1\right )\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07757, size = 524, normalized size = 2.96 \begin{align*} \frac{4 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right )^{5} +{\left ({\left (a^{2} x^{2} - 1\right )} \logintegral \left (-\frac{a x + 1}{a x - 1}\right ) +{\left (a^{2} x^{2} - 1\right )} \logintegral \left (-\frac{a x - 1}{a x + 1}\right )\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{6} + 8 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} + 2 \,{\left (a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{4} + 96 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right ) + 12 \,{\left (a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} + 480}{45 \,{\left (a^{3} x^{2} - a\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{6}} \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 )^{2} \left (a x + 1\right )^{2} \operatorname{atanh}^{7}{\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 )}^{2} \operatorname{artanh}\left (a x\right )^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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