Optimal. Leaf size=25 \[ \frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 \sqrt{\pi } b c} \]
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Rubi [A] time = 0.0502889, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {5675} \[ \frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 \sqrt{\pi } b c} \]
Antiderivative was successfully verified.
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Rule 5675
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{\pi +c^2 \pi x^2}} \, dx &=\frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.0240984, size = 25, normalized size = 1. \[ \frac{\left (a+b \sinh ^{-1}(c x)\right )^3}{3 \sqrt{\pi } b c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.063, size = 72, normalized size = 2.9 \begin{align*}{{a}^{2}\ln \left ({\pi \,{c}^{2}x{\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+\sqrt{\pi \,{c}^{2}{x}^{2}+\pi } \right ){\frac{1}{\sqrt{\pi \,{c}^{2}}}}}+{\frac{{b}^{2} \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{3}}{3\,c\sqrt{\pi }}}+{\frac{ab \left ({\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{c\sqrt{\pi }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.363, size = 230, normalized size = 9.2 \begin{align*} \frac{b^{2} \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right ) \operatorname{arsinh}\left (c x\right )^{2}}{\sqrt{\pi c^{2}}} + \frac{2 \, a b \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right ) \operatorname{arsinh}\left (c x\right )}{\sqrt{\pi c^{2}}} + \frac{1}{3} \,{\left (\frac{\operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )^{3}}{\sqrt{\pi c^{2}}} - \frac{3 \, \sqrt{c^{2}} \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )^{2} \operatorname{arsinh}\left (c x\right )}{\sqrt{\pi c^{2}} c}\right )} b^{2} - \frac{a b \sqrt{c^{2}} \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )^{2}}{\sqrt{\pi c^{2}} c} + \frac{a^{2} \operatorname{arsinh}\left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{\pi c^{2}}} \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{b^{2} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname{arsinh}\left (c x\right ) + a^{2}}{\sqrt{\pi + \pi c^{2} x^{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.12885, size = 88, normalized size = 3.52 \begin{align*} \begin{cases} a^{2} \left (\begin{cases} \frac{\sqrt{- \frac{1}{c^{2}}} \operatorname{asin}{\left (x \sqrt{- c^{2}} \right )}}{\sqrt{\pi }} & \text{for}\: \pi c^{2} < 0 \\\frac{\sqrt{\frac{1}{c^{2}}} \operatorname{asinh}{\left (x \sqrt{c^{2}} \right )}}{\sqrt{\pi }} & \text{for}\: \pi c^{2} > 0 \end{cases}\right ) & \text{for}\: b = 0 \\\frac{a^{2} x}{\sqrt{\pi }} & \text{for}\: c = 0 \\\frac{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{3}}{3 \sqrt{\pi } b c} & \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{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt{\pi + \pi c^{2} x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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