Optimal. Leaf size=72 \[ -\frac{\left (a^2 x^2+1\right )^{5/2}}{25 a^5}+\frac{2 \left (a^2 x^2+1\right )^{3/2}}{15 a^5}-\frac{\sqrt{a^2 x^2+1}}{5 a^5}+\frac{1}{5} x^5 \sinh ^{-1}(a x) \]
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Rubi [A] time = 0.0441345, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5661, 266, 43} \[ -\frac{\left (a^2 x^2+1\right )^{5/2}}{25 a^5}+\frac{2 \left (a^2 x^2+1\right )^{3/2}}{15 a^5}-\frac{\sqrt{a^2 x^2+1}}{5 a^5}+\frac{1}{5} x^5 \sinh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 5661
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^4 \sinh ^{-1}(a x) \, dx &=\frac{1}{5} x^5 \sinh ^{-1}(a x)-\frac{1}{5} a \int \frac{x^5}{\sqrt{1+a^2 x^2}} \, dx\\ &=\frac{1}{5} x^5 \sinh ^{-1}(a x)-\frac{1}{10} a \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{5} x^5 \sinh ^{-1}(a x)-\frac{1}{10} a \operatorname{Subst}\left (\int \left (\frac{1}{a^4 \sqrt{1+a^2 x}}-\frac{2 \sqrt{1+a^2 x}}{a^4}+\frac{\left (1+a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1+a^2 x^2}}{5 a^5}+\frac{2 \left (1+a^2 x^2\right )^{3/2}}{15 a^5}-\frac{\left (1+a^2 x^2\right )^{5/2}}{25 a^5}+\frac{1}{5} x^5 \sinh ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0347631, size = 50, normalized size = 0.69 \[ \frac{1}{5} x^5 \sinh ^{-1}(a x)-\frac{\sqrt{a^2 x^2+1} \left (3 a^4 x^4-4 a^2 x^2+8\right )}{75 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 69, normalized size = 1. \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{{a}^{5}{x}^{5}{\it Arcsinh} \left ( ax \right ) }{5}}-{\frac{{a}^{4}{x}^{4}}{25}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{4\,{a}^{2}{x}^{2}}{75}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{8}{75}\sqrt{{a}^{2}{x}^{2}+1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13708, size = 92, normalized size = 1.28 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{arsinh}\left (a x\right ) - \frac{1}{75} \,{\left (\frac{3 \, \sqrt{a^{2} x^{2} + 1} x^{4}}{a^{2}} - \frac{4 \, \sqrt{a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac{8 \, \sqrt{a^{2} x^{2} + 1}}{a^{6}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83797, size = 135, normalized size = 1.88 \begin{align*} \frac{15 \, a^{5} x^{5} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) -{\left (3 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 8\right )} \sqrt{a^{2} x^{2} + 1}}{75 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.16633, size = 70, normalized size = 0.97 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{asinh}{\left (a x \right )}}{5} - \frac{x^{4} \sqrt{a^{2} x^{2} + 1}}{25 a} + \frac{4 x^{2} \sqrt{a^{2} x^{2} + 1}}{75 a^{3}} - \frac{8 \sqrt{a^{2} x^{2} + 1}}{75 a^{5}} & \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 [A] time = 1.30164, size = 90, normalized size = 1.25 \begin{align*} \frac{1}{5} \, x^{5} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - \frac{3 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{5}{2}} - 10 \,{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{a^{2} x^{2} + 1}}{75 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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