Optimal. Leaf size=77 \[ x \cosh ^{-1}(a x)^4-\frac{4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{a}+12 x \cosh ^{-1}(a x)^2-\frac{24 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{a}+24 x \]
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Rubi [A] time = 0.298058, antiderivative size = 77, 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 = {5654, 5718, 8} \[ x \cosh ^{-1}(a x)^4-\frac{4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{a}+12 x \cosh ^{-1}(a x)^2-\frac{24 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{a}+24 x \]
Antiderivative was successfully verified.
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Rule 5654
Rule 5718
Rule 8
Rubi steps
\begin{align*} \int \cosh ^{-1}(a x)^4 \, dx &=x \cosh ^{-1}(a x)^4-(4 a) \int \frac{x \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a}+x \cosh ^{-1}(a x)^4+12 \int \cosh ^{-1}(a x)^2 \, dx\\ &=12 x \cosh ^{-1}(a x)^2-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a}+x \cosh ^{-1}(a x)^4-(24 a) \int \frac{x \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{24 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{a}+12 x \cosh ^{-1}(a x)^2-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a}+x \cosh ^{-1}(a x)^4+24 \int 1 \, dx\\ &=24 x-\frac{24 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{a}+12 x \cosh ^{-1}(a x)^2-\frac{4 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a}+x \cosh ^{-1}(a x)^4\\ \end{align*}
Mathematica [A] time = 0.0284939, size = 77, normalized size = 1. \[ x \cosh ^{-1}(a x)^4-\frac{4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{a}+12 x \cosh ^{-1}(a x)^2-\frac{24 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{a}+24 x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 71, normalized size = 0.9 \begin{align*}{\frac{1}{a} \left ( \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}ax-4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}\sqrt{ax-1}\sqrt{ax+1}+12\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}ax-24\,{\rm arccosh} \left (ax\right )\sqrt{ax-1}\sqrt{ax+1}+24\,ax \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15713, size = 99, normalized size = 1.29 \begin{align*} x \operatorname{arcosh}\left (a x\right )^{4} - \frac{4 \, \sqrt{a^{2} x^{2} - 1} \operatorname{arcosh}\left (a x\right )^{3}}{a} + 12 \,{\left (\frac{x \operatorname{arcosh}\left (a x\right )^{2}}{a} + \frac{2 \,{\left (x - \frac{\sqrt{a^{2} x^{2} - 1} \operatorname{arcosh}\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 = 2.31198, size = 262, normalized size = 3.4 \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 = 1.18142, size = 70, normalized size = 0.91 \begin{align*} \begin{cases} x \operatorname{acosh}^{4}{\left (a x \right )} + 12 x \operatorname{acosh}^{2}{\left (a x \right )} + 24 x - \frac{4 \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{3}{\left (a x \right )}}{a} - \frac{24 \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\\frac{\pi ^{4} x}{16} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.46721, size = 169, normalized size = 2.19 \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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