Optimal. Leaf size=49 \[ \frac{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{2 a^2}-\frac{\sinh ^{-1}(a x)^2}{4 a^3}-\frac{x^2}{4 a} \]
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Rubi [A] time = 0.0807102, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5758, 5675, 30} \[ \frac{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{2 a^2}-\frac{\sinh ^{-1}(a x)^2}{4 a^3}-\frac{x^2}{4 a} \]
Antiderivative was successfully verified.
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Rule 5758
Rule 5675
Rule 30
Rubi steps
\begin{align*} \int \frac{x^2 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx &=\frac{x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{2 a^2}-\frac{\int \frac{\sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{2 a^2}-\frac{\int x \, dx}{2 a}\\ &=-\frac{x^2}{4 a}+\frac{x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{2 a^2}-\frac{\sinh ^{-1}(a x)^2}{4 a^3}\\ \end{align*}
Mathematica [A] time = 0.0394678, size = 42, normalized size = 0.86 \[ -\frac{a^2 x^2-2 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+\sinh ^{-1}(a x)^2}{4 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 40, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{a}^{3}} \left ( -2\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}ax+{a}^{2}{x}^{2}+ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}+1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.29806, size = 100, normalized size = 2.04 \begin{align*} -\frac{1}{4} \, a{\left (\frac{x^{2}}{a^{2}} - \frac{\operatorname{arsinh}\left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )^{2}}{a^{4}}\right )} + \frac{1}{2} \,{\left (\frac{\sqrt{a^{2} x^{2} + 1} x}{a^{2}} - \frac{\operatorname{arsinh}\left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{2}}\right )} \operatorname{arsinh}\left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.55848, size = 146, normalized size = 2.98 \begin{align*} -\frac{a^{2} x^{2} - 2 \, \sqrt{a^{2} x^{2} + 1} a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) + \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{4 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.956813, size = 42, normalized size = 0.86 \begin{align*} \begin{cases} - \frac{x^{2}}{4 a} + \frac{x \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{2 a^{2}} - \frac{\operatorname{asinh}^{2}{\left (a x \right )}}{4 a^{3}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \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 )}{\sqrt{a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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