Optimal. Leaf size=41 \[ \frac{\text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{2 a^3}-\frac{x^2}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \]
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Rubi [A] time = 0.287269, antiderivative size = 60, normalized size of antiderivative = 1.46, number of steps used = 12, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6028, 5966, 6034, 5448, 12, 3298} \[ \frac{\text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{2 a^3}+\frac{1}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{1}{a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 6028
Rule 5966
Rule 6034
Rule 5448
Rule 12
Rule 3298
Rubi steps
\begin{align*} \int \frac{x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx &=\frac{\int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx}{a^2}-\frac{\int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx}{a^2}\\ &=-\frac{1}{a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{1}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{2 \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx}{a}+\frac{4 \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx}{a}\\ &=-\frac{1}{a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{1}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{2 \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}+\frac{4 \operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{1}{a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{1}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{2 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}+\frac{4 \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^3}\\ &=-\frac{1}{a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{1}{a^3 \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 )}{2 a^3}\\ &=-\frac{1}{a^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{1}{a^3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.202416, size = 56, normalized size = 1.37 \[ \frac{\left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x) \text{Shi}\left (4 \tanh ^{-1}(a x)\right )-2 a^2 x^2}{2 a^3 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 38, normalized size = 0.9 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{1}{8\,{\it Artanh} \left ( ax \right ) }}-{\frac{\cosh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{8\,{\it Artanh} \left ( ax \right ) }}+{\frac{{\it Shi} \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{2}} \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 \, x^{2}}{{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right ) -{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-a x + 1\right )} + \int -\frac{4 \,{\left (a^{2} x^{3} + x\right )}}{{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) -{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\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 = 2.33203, size = 370, normalized size = 9.02 \begin{align*} -\frac{8 \, 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 )}{4 \,{\left (a^{7} x^{4} - 2 \, a^{5} x^{2} + a^{3}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )} \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}^{2}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname{atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname{atanh}^{2}{\left (a x \right )} - \operatorname{atanh}^{2}{\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 )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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