Optimal. Leaf size=170 \[ \frac{x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac{x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}+\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac{4 \text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{3 a} \]
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Rubi [A] time = 0.963358, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {5966, 6032, 6028, 6034, 3312, 3301, 5968, 5448} \[ \frac{x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}-\frac{x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}-\frac{1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}+\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac{4 \text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{3 a} \]
Antiderivative was successfully verified.
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Rule 5966
Rule 6032
Rule 6028
Rule 6034
Rule 3312
Rule 3301
Rule 5968
Rule 5448
Rubi steps
\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^5} \, dx &=-\frac{1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}+a \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^4} \, dx\\ &=-\frac{1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}+\frac{1}{3} \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx+a^2 \int \frac{x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac{1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{1}{6 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{3} (2 a) \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx+\int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx-\int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac{1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{2}{3} \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx-a \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx+(2 a) \int \frac{x}{\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)} \, dx\\ &=-\frac{1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx+\frac{2 \operatorname{Subst}\left (\int \frac{\cosh ^4(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{3 a}+\frac{2 \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}-a^2 \int \frac{x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+\left (6 a^2\right ) \int \frac{x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx-\int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \left (\frac{3}{8 x}+\frac{\cosh (2 x)}{2 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{3 a}-\frac{\operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}-\frac{\operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac{2 \operatorname{Subst}\left (\int \frac{\cosh ^4(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{1}{8 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac{6 \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{12 a}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{3 a}+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac{2 \operatorname{Subst}\left (\int \left (\frac{3}{8 x}+\frac{\cosh (2 x)}{2 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac{6 \operatorname{Subst}\left (\int \left (-\frac{1}{8 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{3 a}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}-2 \frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac{1}{4 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4}-\frac{x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3}-\frac{2}{3 a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{8 x}{3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}+\frac{4 \text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{3 a}\\ \end{align*}
Mathematica [A] time = 0.164388, size = 132, normalized size = 0.78 \[ -\frac{-4 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^4 \text{Chi}\left (2 \tanh ^{-1}(a x)\right )-16 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^4 \text{Chi}\left (4 \tanh ^{-1}(a x)\right )+12 a^3 x^3 \tanh ^{-1}(a x)^3+6 a^2 x^2 \tanh ^{-1}(a x)^2+20 a x \tanh ^{-1}(a x)^3+2 \tanh ^{-1}(a x)^2+4 a x \tanh ^{-1}(a x)+3}{12 a \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 152, normalized size = 0.9 \begin{align*}{\frac{1}{a} \left ( -{\frac{3}{32\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{8\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}}}-{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{12\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}}-{\frac{\cosh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{12\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}-{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{6\,{\it Artanh} \left ( ax \right ) }}+{\frac{{\it Chi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{3}}-{\frac{\cosh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{32\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{4}}}-{\frac{\sinh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{24\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{3}}}-{\frac{\cosh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{12\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}-{\frac{\sinh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{3\,{\it Artanh} \left ( ax \right ) }}+{\frac{4\,{\it Chi} \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{2 \,{\left ({\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (a x + 1\right )^{3} -{\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (-a x + 1\right )^{3} + 4 \, 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} + 3 \,{\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (a x + 1\right ) + 1\right )} \log \left (-a x + 1\right )^{2} -{\left (3 \,{\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (a x + 1\right )^{2} + 4 \, a x + 2 \,{\left (3 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right ) + 6\right )}}{3 \,{\left ({\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right )^{4} - 4 \,{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right )^{3} \log \left (-a x + 1\right ) + 6 \,{\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 )^{2} - 4 \,{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right ) \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 )^{4}\right )}} + \int -\frac{2 \,{\left (3 \, a^{4} x^{4} + 24 \, a^{2} x^{2} + 5\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 = 2.03757, size = 720, normalized size = 4.24 \begin{align*} \frac{{\left (4 \,{\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 ) + 4 \,{\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 )^{4} - 4 \,{\left (3 \, a^{3} x^{3} + 5 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{3} - 16 \, 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} - 24}{6 \,{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{4}} \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}^{5}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname{atanh}^{5}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname{atanh}^{5}{\left (a x \right )} - \operatorname{atanh}^{5}{\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 )^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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