Optimal. Leaf size=110 \[ -\frac{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{a}-\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{2 a}+\frac{\sinh ^{-1}(a x)^4}{4 a^2}+\frac{3 \sinh ^{-1}(a x)^2}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^4+\frac{3}{2} x^2 \sinh ^{-1}(a x)^2+\frac{3 x^2}{4} \]
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Rubi [A] time = 0.239413, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5661, 5758, 5675, 30} \[ -\frac{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{a}-\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{2 a}+\frac{\sinh ^{-1}(a x)^4}{4 a^2}+\frac{3 \sinh ^{-1}(a x)^2}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^4+\frac{3}{2} x^2 \sinh ^{-1}(a x)^2+\frac{3 x^2}{4} \]
Antiderivative was successfully verified.
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Rule 5661
Rule 5758
Rule 5675
Rule 30
Rubi steps
\begin{align*} \int x \sinh ^{-1}(a x)^4 \, dx &=\frac{1}{2} x^2 \sinh ^{-1}(a x)^4-(2 a) \int \frac{x^2 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^4+3 \int x \sinh ^{-1}(a x)^2 \, dx+\frac{\int \frac{\sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{a}\\ &=\frac{3}{2} x^2 \sinh ^{-1}(a x)^2-\frac{x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac{\sinh ^{-1}(a x)^4}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^4-(3 a) \int \frac{x^2 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{2 a}+\frac{3}{2} x^2 \sinh ^{-1}(a x)^2-\frac{x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac{\sinh ^{-1}(a x)^4}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^4+\frac{3 \int x \, dx}{2}+\frac{3 \int \frac{\sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{2 a}\\ &=\frac{3 x^2}{4}-\frac{3 x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{2 a}+\frac{3 \sinh ^{-1}(a x)^2}{4 a^2}+\frac{3}{2} x^2 \sinh ^{-1}(a x)^2-\frac{x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+\frac{\sinh ^{-1}(a x)^4}{4 a^2}+\frac{1}{2} x^2 \sinh ^{-1}(a x)^4\\ \end{align*}
Mathematica [A] time = 0.0454885, size = 94, normalized size = 0.85 \[ \frac{3 a^2 x^2+\left (2 a^2 x^2+1\right ) \sinh ^{-1}(a x)^4-4 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3+\left (6 a^2 x^2+3\right ) \sinh ^{-1}(a x)^2-6 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{4 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 105, normalized size = 1. \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4} \left ({a}^{2}{x}^{2}+1 \right ) }{2}}- \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}ax\sqrt{{a}^{2}{x}^{2}+1}-{\frac{ \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4}}{4}}+{\frac{3\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2} \left ({a}^{2}{x}^{2}+1 \right ) }{2}}-{\frac{3\,{\it Arcsinh} \left ( ax \right ) ax}{2}\sqrt{{a}^{2}{x}^{2}+1}}-{\frac{3\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}}{4}}+{\frac{3\,{a}^{2}{x}^{2}}{4}}+{\frac{3}{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{1}{2} \, x^{2} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{4} - \int \frac{2 \,{\left (a^{3} x^{4} + \sqrt{a^{2} x^{2} + 1} a^{2} x^{3} + a x^{2}\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3}}{a^{3} x^{3} + a x +{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05094, size = 316, normalized size = 2.87 \begin{align*} -\frac{4 \, \sqrt{a^{2} x^{2} + 1} a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} -{\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{4} - 3 \, a^{2} x^{2} + 6 \, \sqrt{a^{2} x^{2} + 1} a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right ) - 3 \,{\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2}}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.14062, size = 104, normalized size = 0.95 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{asinh}^{4}{\left (a x \right )}}{2} + \frac{3 x^{2} \operatorname{asinh}^{2}{\left (a x \right )}}{2} + \frac{3 x^{2}}{4} - \frac{x \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{3}{\left (a x \right )}}{a} - \frac{3 x \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{2 a} + \frac{\operatorname{asinh}^{4}{\left (a x \right )}}{4 a^{2}} + \frac{3 \operatorname{asinh}^{2}{\left (a x \right )}}{4 a^{2}} & \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 x \operatorname{arsinh}\left (a x\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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