Optimal. Leaf size=67 \[ -\frac{4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{a}-\frac{24 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a}+x \sinh ^{-1}(a x)^4+12 x \sinh ^{-1}(a x)^2+24 x \]
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Rubi [A] time = 0.125115, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5653, 5717, 8} \[ -\frac{4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{a}-\frac{24 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a}+x \sinh ^{-1}(a x)^4+12 x \sinh ^{-1}(a x)^2+24 x \]
Antiderivative was successfully verified.
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Rule 5653
Rule 5717
Rule 8
Rubi steps
\begin{align*} \int \sinh ^{-1}(a x)^4 \, dx &=x \sinh ^{-1}(a x)^4-(4 a) \int \frac{x \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+x \sinh ^{-1}(a x)^4+12 \int \sinh ^{-1}(a x)^2 \, dx\\ &=12 x \sinh ^{-1}(a x)^2-\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+x \sinh ^{-1}(a x)^4-(24 a) \int \frac{x \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{24 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{a}+12 x \sinh ^{-1}(a x)^2-\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+x \sinh ^{-1}(a x)^4+24 \int 1 \, dx\\ &=24 x-\frac{24 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)}{a}+12 x \sinh ^{-1}(a x)^2-\frac{4 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3}{a}+x \sinh ^{-1}(a x)^4\\ \end{align*}
Mathematica [A] time = 0.018935, size = 67, normalized size = 1. \[ -\frac{4 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{a}-\frac{24 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a}+x \sinh ^{-1}(a x)^4+12 x \sinh ^{-1}(a x)^2+24 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 65, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{4}ax-4\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{3}\sqrt{{a}^{2}{x}^{2}+1}+12\, \left ({\it Arcsinh} \left ( ax \right ) \right ) ^{2}ax-24\,{\it Arcsinh} \left ( ax \right ) \sqrt{{a}^{2}{x}^{2}+1}+24\,ax \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13815, size = 99, normalized size = 1.48 \begin{align*} x \operatorname{arsinh}\left (a x\right )^{4} - \frac{4 \, \sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )^{3}}{a} + 12 \,{\left (\frac{x \operatorname{arsinh}\left (a x\right )^{2}}{a} + \frac{2 \,{\left (x - \frac{\sqrt{a^{2} x^{2} + 1} \operatorname{arsinh}\left (a x\right )}{a}\right )}}{a}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05502, size = 262, normalized size = 3.91 \begin{align*} \frac{a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{4} + 12 \, a x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} - 4 \, \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3} + 24 \, a x - 24 \, \sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.03254, size = 65, normalized size = 0.97 \begin{align*} \begin{cases} x \operatorname{asinh}^{4}{\left (a x \right )} + 12 x \operatorname{asinh}^{2}{\left (a x \right )} + 24 x - \frac{4 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}^{3}{\left (a x \right )}}{a} - \frac{24 \sqrt{a^{2} x^{2} + 1} \operatorname{asinh}{\left (a x \right )}}{a} & \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 = 1.58945, size = 169, normalized size = 2.52 \begin{align*} x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{4} - 4 \,{\left (\frac{\sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{3}}{a^{2}} - \frac{3 \,{\left (x \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )^{2} + 2 \, a{\left (\frac{x}{a} - \frac{\sqrt{a^{2} x^{2} + 1} \log \left (a x + \sqrt{a^{2} x^{2} + 1}\right )}{a^{2}}\right )}\right )}}{a}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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