Optimal. Leaf size=54 \[ -\frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^3}+\frac{3 \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^3}-\frac{x^2 \sqrt{a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.0474095, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5665, 3298} \[ -\frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^3}+\frac{3 \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^3}-\frac{x^2 \sqrt{a^2 x^2+1}}{a \sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 5665
Rule 3298
Rubi steps
\begin{align*} \int \frac{x^2}{\sinh ^{-1}(a x)^2} \, dx &=-\frac{x^2 \sqrt{1+a^2 x^2}}{a \sinh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{4 x}+\frac{3 \sinh (3 x)}{4 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{x^2 \sqrt{1+a^2 x^2}}{a \sinh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac{x^2 \sqrt{1+a^2 x^2}}{a \sinh ^{-1}(a x)}-\frac{\text{Shi}\left (\sinh ^{-1}(a x)\right )}{4 a^3}+\frac{3 \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^3}\\ \end{align*}
Mathematica [A] time = 0.165811, size = 49, normalized size = 0.91 \[ -\frac{\frac{4 a^2 x^2 \sqrt{a^2 x^2+1}}{\sinh ^{-1}(a x)}+\text{Shi}\left (\sinh ^{-1}(a x)\right )-3 \text{Shi}\left (3 \sinh ^{-1}(a x)\right )}{4 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 56, normalized size = 1. \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{1}{4\,{\it Arcsinh} \left ( ax \right ) }\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{{\it Shi} \left ({\it Arcsinh} \left ( ax \right ) \right ) }{4}}-{\frac{\cosh \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{4\,{\it Arcsinh} \left ( ax \right ) }}+{\frac{3\,{\it Shi} \left ( 3\,{\it Arcsinh} \left ( ax \right ) \right ) }{4}} \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{a^{3} x^{5} + a x^{3} +{\left (a^{2} x^{4} + x^{2}\right )} \sqrt{a^{2} x^{2} + 1}}{{\left (a^{3} x^{2} + \sqrt{a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )} + \int \frac{3 \, a^{5} x^{6} + 6 \, a^{3} x^{4} + 3 \, a x^{2} +{\left (3 \, a^{3} x^{4} + a x^{2}\right )}{\left (a^{2} x^{2} + 1\right )} +{\left (6 \, a^{4} x^{5} + 7 \, a^{2} x^{3} + 2 \, x\right )} \sqrt{a^{2} x^{2} + 1}}{{\left (a^{5} x^{4} +{\left (a^{2} x^{2} + 1\right )} a^{3} x^{2} + 2 \, a^{3} x^{2} + 2 \,{\left (a^{4} x^{3} + a^{2} x\right )} \sqrt{a^{2} x^{2} + 1} + a\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2}}{\operatorname{arsinh}\left (a x\right )^{2}}, x\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}}{\operatorname{asinh}^{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}}{\operatorname{arsinh}\left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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