Optimal. Leaf size=125 \[ -\frac{2 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{8}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac{1}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{2 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac{4 \text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{3 a} \]
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Rubi [A] time = 0.499948, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {5966, 6032, 6028, 6034, 5448, 12, 3298} \[ -\frac{2 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{8}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac{1}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{2 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac{4 \text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{3 a} \]
Antiderivative was successfully verified.
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Rule 5966
Rule 6032
Rule 6028
Rule 6034
Rule 5448
Rule 12
Rule 3298
Rubi steps
\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^4} \, dx &=-\frac{1}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{1}{3} (4 a) \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac{1}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{2}{3} \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx+\left (2 a^2\right ) \int \frac{x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac{1}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+2 \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx-2 \int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx+\frac{1}{3} (8 a) \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{8}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{8 \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{3 a}-(4 a) \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+(8 a) \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{8}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{8 \operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 x}+\frac{\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{3 a}-\frac{4 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac{8 \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{1}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{8}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{3 a}+\frac{2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{3 a}-\frac{4 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac{8 \operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 x}+\frac{\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{1}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{8}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{2 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac{\text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{3 a}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{1}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{8}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{2}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{2 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac{4 \text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{3 a}\\ \end{align*}
Mathematica [A] time = 0.247077, size = 108, normalized size = 0.86 \[ -\frac{-2 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^3 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )-4 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^3 \text{Shi}\left (4 \tanh ^{-1}(a x)\right )+6 a^2 x^2 \tanh ^{-1}(a x)^2+2 \tanh ^{-1}(a x)^2+2 a x \tanh ^{-1}(a x)+1}{3 a \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 122, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( -{\frac{1}{8\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{6\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}}-{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{6\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{3\,{\it Artanh} \left ( ax \right ) }}+{\frac{2\,{\it Shi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{3}}-{\frac{\cosh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{24\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}}-{\frac{\sinh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{12\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}-{\frac{\cosh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{3\,{\it Artanh} \left ( ax \right ) }}+{\frac{4\,{\it Shi} \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{3}} \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{4 \,{\left (2 \, a x \log \left (a x + 1\right ) +{\left (3 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} +{\left (3 \, a^{2} x^{2} + 1\right )} \log \left (-a x + 1\right )^{2} - 2 \,{\left (a x +{\left (3 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right ) + 2\right )}}{3 \,{\left ({\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right )^{3} - 3 \,{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right ) + 3 \,{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right )^{2} -{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-a x + 1\right )^{3}\right )}} + \int -\frac{8 \,{\left (3 \, a^{3} x^{3} + 5 \, a x\right )}}{3 \,{\left ({\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) -{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, 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 [B] time = 2.05897, size = 647, normalized size = 5.18 \begin{align*} \frac{{\left (2 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \logintegral \left (\frac{a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 2 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \logintegral \left (\frac{a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) +{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \logintegral \left (-\frac{a x + 1}{a x - 1}\right ) -{\left (a^{4} x^{4} - 2 \, 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 )^{3} - 8 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right ) - 4 \,{\left (3 \, a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 8}{3 \,{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{3}} \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}{a^{6} x^{6} \operatorname{atanh}^{4}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname{atanh}^{4}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname{atanh}^{4}{\left (a x \right )} - \operatorname{atanh}^{4}{\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 )}^{3} \operatorname{artanh}\left (a x\right )^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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