Optimal. Leaf size=86 \[ \frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{a^3}-\frac{x^2}{2 a \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^2}+\frac{x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{2 x}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \]
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Rubi [A] time = 0.585663, antiderivative size = 109, normalized size of antiderivative = 1.27, number of steps used = 22, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6028, 5966, 6032, 6034, 3312, 3301, 5968, 5448} \[ \frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{a^3}+\frac{x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{2 x}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{1}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{1}{2 a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Rule 6028
Rule 5966
Rule 6032
Rule 6034
Rule 3312
Rule 3301
Rule 5968
Rule 5448
Rubi steps
\begin{align*} \int \frac{x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx &=\frac{\int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^3} \, dx}{a^2}-\frac{\int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx}{a^2}\\ &=-\frac{1}{2 a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{\int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx}{a}+\frac{2 \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx}{a}\\ &=-\frac{1}{2 a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 x}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+6 \int \frac{x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx-\frac{\int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx}{a^2}+\frac{2 \int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx}{a^2}-\int \frac{x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac{1}{2 a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 x}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}-\frac{\operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}+\frac{2 \operatorname{Subst}\left (\int \frac{\cosh ^4(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}+\frac{6 \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{1}{2 a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 x}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\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^3}-\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^3}+\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^3}+\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^3}\\ &=-\frac{1}{2 a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 x}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{x}{a^2 \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 )}{4 a^3}-2 \frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^3}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{1}{2 a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2}+\frac{1}{2 a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac{2 x}{a^2 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.154842, size = 56, normalized size = 0.65 \[ \frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{a^3}-\frac{x \left (2 \left (a^2 x^2+1\right ) \tanh ^{-1}(a x)+a x\right )}{2 a^2 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 51, normalized size = 0.6 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{1}{16\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}-{\frac{\cosh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{16\, \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}}}-{\frac{\sinh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{4\,{\it Artanh} \left ( ax \right ) }}+{\it Chi} \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) \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 (a x^{2} +{\left (a^{2} x^{3} + x\right )} \log \left (a x + 1\right ) -{\left (a^{2} x^{3} + x\right )} \log \left (-a x + 1\right )\right )}}{{\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (a x + 1\right )^{2} - 2 \,{\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right ) +{\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (-a x + 1\right )^{2}} + \int -\frac{2 \,{\left (a^{4} x^{4} + 6 \, a^{2} x^{2} + 1\right )}}{{\left (a^{8} x^{6} - 3 \, a^{6} x^{4} + 3 \, a^{4} x^{2} - a^{2}\right )} \log \left (a x + 1\right ) -{\left (a^{8} x^{6} - 3 \, a^{6} x^{4} + 3 \, a^{4} x^{2} - a^{2}\right )} \log \left (-a x + 1\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98297, size = 437, normalized size = 5.08 \begin{align*} -\frac{4 \, a^{2} x^{2} -{\left ({\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^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right )\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} + 4 \,{\left (a^{3} x^{3} + a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{2 \,{\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2}} \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{x^{2}}{a^{6} x^{6} \operatorname{atanh}^{3}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname{atanh}^{3}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname{atanh}^{3}{\left (a x \right )} - \operatorname{atanh}^{3}{\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{x^{2}}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname{artanh}\left (a x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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