Optimal. Leaf size=120 \[ -\frac{\cosh ^{-1}(a x)^4}{4 a^2}-\frac{3 \cosh ^{-1}(a x)^2}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^4+\frac{3}{2} x^2 \cosh ^{-1}(a x)^2-\frac{x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{a}-\frac{3 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{2 a}+\frac{3 x^2}{4} \]
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Rubi [A] time = 0.602508, antiderivative size = 120, 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 = {5662, 5759, 5676, 30} \[ -\frac{\cosh ^{-1}(a x)^4}{4 a^2}-\frac{3 \cosh ^{-1}(a x)^2}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^4+\frac{3}{2} x^2 \cosh ^{-1}(a x)^2-\frac{x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{a}-\frac{3 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{2 a}+\frac{3 x^2}{4} \]
Antiderivative was successfully verified.
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Rule 5662
Rule 5759
Rule 5676
Rule 30
Rubi steps
\begin{align*} \int x \cosh ^{-1}(a x)^4 \, dx &=\frac{1}{2} x^2 \cosh ^{-1}(a x)^4-(2 a) \int \frac{x^2 \cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^4+3 \int x \cosh ^{-1}(a x)^2 \, dx-\frac{\int \frac{\cosh ^{-1}(a x)^3}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{a}\\ &=\frac{3}{2} x^2 \cosh ^{-1}(a x)^2-\frac{x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a}-\frac{\cosh ^{-1}(a x)^4}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^4-(3 a) \int \frac{x^2 \cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{2 a}+\frac{3}{2} x^2 \cosh ^{-1}(a x)^2-\frac{x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a}-\frac{\cosh ^{-1}(a x)^4}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^4+\frac{3 \int x \, dx}{2}-\frac{3 \int \frac{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{2 a}\\ &=\frac{3 x^2}{4}-\frac{3 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)}{2 a}-\frac{3 \cosh ^{-1}(a x)^2}{4 a^2}+\frac{3}{2} x^2 \cosh ^{-1}(a x)^2-\frac{x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a}-\frac{\cosh ^{-1}(a x)^4}{4 a^2}+\frac{1}{2} x^2 \cosh ^{-1}(a x)^4\\ \end{align*}
Mathematica [A] time = 0.0700765, size = 104, normalized size = 0.87 \[ \frac{3 a^2 x^2+\left (2 a^2 x^2-1\right ) \cosh ^{-1}(a x)^4+\left (6 a^2 x^2-3\right ) \cosh ^{-1}(a x)^2-4 a x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3-6 a x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{4 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}{a}^{2}{x}^{2}}{2}}- \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}\sqrt{ax-1}\sqrt{ax+1}ax-{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}}{4}}+{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{2}{x}^{2}}{2}}-{\frac{3\,ax{\rm arccosh} \left (ax\right )}{2}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{4}}+{\frac{3\,{a}^{2}{x}^{2}}{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 x + 1} \sqrt{a x - 1}\right )^{4} - \int \frac{2 \,{\left (a^{3} x^{4} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} x^{3} - a x^{2}\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{3}}{a^{3} x^{3} +{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} \sqrt{a x - 1} - a x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.60556, size = 316, normalized size = 2.63 \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 = 2.61227, size = 110, normalized size = 0.92 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{acosh}^{4}{\left (a x \right )}}{2} + \frac{3 x^{2} \operatorname{acosh}^{2}{\left (a x \right )}}{2} + \frac{3 x^{2}}{4} - \frac{x \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{3}{\left (a x \right )}}{a} - \frac{3 x \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}{\left (a x \right )}}{2 a} - \frac{\operatorname{acosh}^{4}{\left (a x \right )}}{4 a^{2}} - \frac{3 \operatorname{acosh}^{2}{\left (a x \right )}}{4 a^{2}} & \text{for}\: a \neq 0 \\\frac{\pi ^{4} x^{2}}{32} & \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{arcosh}\left (a x\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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