Optimal. Leaf size=183 \[ \frac{9 x^2 \cosh ^{-1}(a x)}{32 a^2}-\frac{45 x \sqrt{a x-1} \sqrt{a x+1}}{256 a^3}-\frac{9 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{32 a^3}-\frac{3 \cosh ^{-1}(a x)^3}{32 a^4}-\frac{45 \cosh ^{-1}(a x)}{256 a^4}-\frac{3 x^3 \sqrt{a x-1} \sqrt{a x+1}}{128 a}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^3+\frac{3}{32} x^4 \cosh ^{-1}(a x)-\frac{3 x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{16 a} \]
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Rubi [A] time = 0.666215, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5662, 5759, 5676, 90, 52, 100, 12} \[ \frac{9 x^2 \cosh ^{-1}(a x)}{32 a^2}-\frac{45 x \sqrt{a x-1} \sqrt{a x+1}}{256 a^3}-\frac{9 x \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{32 a^3}-\frac{3 \cosh ^{-1}(a x)^3}{32 a^4}-\frac{45 \cosh ^{-1}(a x)}{256 a^4}-\frac{3 x^3 \sqrt{a x-1} \sqrt{a x+1}}{128 a}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^3+\frac{3}{32} x^4 \cosh ^{-1}(a x)-\frac{3 x^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{16 a} \]
Antiderivative was successfully verified.
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Rule 5662
Rule 5759
Rule 5676
Rule 90
Rule 52
Rule 100
Rule 12
Rubi steps
\begin{align*} \int x^3 \cosh ^{-1}(a x)^3 \, dx &=\frac{1}{4} x^4 \cosh ^{-1}(a x)^3-\frac{1}{4} (3 a) \int \frac{x^4 \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{16 a}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^3+\frac{3}{8} \int x^3 \cosh ^{-1}(a x) \, dx-\frac{9 \int \frac{x^2 \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{16 a}\\ &=\frac{3}{32} x^4 \cosh ^{-1}(a x)-\frac{9 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{32 a^3}-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{16 a}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^3-\frac{9 \int \frac{\cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{32 a^3}+\frac{9 \int x \cosh ^{-1}(a x) \, dx}{16 a^2}-\frac{1}{32} (3 a) \int \frac{x^4}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{128 a}+\frac{9 x^2 \cosh ^{-1}(a x)}{32 a^2}+\frac{3}{32} x^4 \cosh ^{-1}(a x)-\frac{9 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{32 a^3}-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{16 a}-\frac{3 \cosh ^{-1}(a x)^3}{32 a^4}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^3-\frac{3 \int \frac{3 x^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{128 a}-\frac{9 \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{32 a}\\ &=-\frac{9 x \sqrt{-1+a x} \sqrt{1+a x}}{64 a^3}-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{128 a}+\frac{9 x^2 \cosh ^{-1}(a x)}{32 a^2}+\frac{3}{32} x^4 \cosh ^{-1}(a x)-\frac{9 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{32 a^3}-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{16 a}-\frac{3 \cosh ^{-1}(a x)^3}{32 a^4}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^3-\frac{9 \int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{64 a^3}-\frac{9 \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{128 a}\\ &=-\frac{45 x \sqrt{-1+a x} \sqrt{1+a x}}{256 a^3}-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{128 a}-\frac{9 \cosh ^{-1}(a x)}{64 a^4}+\frac{9 x^2 \cosh ^{-1}(a x)}{32 a^2}+\frac{3}{32} x^4 \cosh ^{-1}(a x)-\frac{9 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{32 a^3}-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{16 a}-\frac{3 \cosh ^{-1}(a x)^3}{32 a^4}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^3-\frac{9 \int \frac{1}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{256 a^3}\\ &=-\frac{45 x \sqrt{-1+a x} \sqrt{1+a x}}{256 a^3}-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x}}{128 a}-\frac{45 \cosh ^{-1}(a x)}{256 a^4}+\frac{9 x^2 \cosh ^{-1}(a x)}{32 a^2}+\frac{3}{32} x^4 \cosh ^{-1}(a x)-\frac{9 x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{32 a^3}-\frac{3 x^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{16 a}-\frac{3 \cosh ^{-1}(a x)^3}{32 a^4}+\frac{1}{4} x^4 \cosh ^{-1}(a x)^3\\ \end{align*}
Mathematica [A] time = 0.131483, size = 143, normalized size = 0.78 \[ \frac{-3 a x \sqrt{a x-1} \sqrt{a x+1} \left (2 a^2 x^2+15\right )+8 \left (8 a^4 x^4-3\right ) \cosh ^{-1}(a x)^3-24 a x \sqrt{a x-1} \sqrt{a x+1} \left (2 a^2 x^2+3\right ) \cosh ^{-1}(a x)^2+24 a^2 x^2 \left (a^2 x^2+3\right ) \cosh ^{-1}(a x)-45 \log \left (a x+\sqrt{a x-1} \sqrt{a x+1}\right )}{256 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 184, normalized size = 1. \begin{align*}{\frac{1}{{a}^{4}} \left ({\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3} \left ( ax-1 \right ) \left ( ax+1 \right ){a}^{2}{x}^{2}}{4}}+{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{2}{x}^{2}}{4}}-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{a}^{3}{x}^{3}}{16}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{9\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}ax}{32}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{32}}+{\frac{3\,{\rm arccosh} \left (ax\right ) \left ( ax-1 \right ) \left ( ax+1 \right ){a}^{2}{x}^{2}}{32}}-{\frac{3\,{x}^{3}{a}^{3}}{128}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{45\,ax}{256}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{45\,{\rm arccosh} \left (ax\right )}{256}}+{\frac{3\,{a}^{2}{x}^{2}{\rm arccosh} \left (ax\right )}{8}} \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}{4} \, x^{4} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{3} - \int \frac{3 \,{\left (a^{3} x^{6} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} x^{5} - a x^{4}\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{2}}{4 \,{\left (a^{3} x^{3} +{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} \sqrt{a x - 1} - a x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52813, size = 327, normalized size = 1.79 \begin{align*} \frac{8 \,{\left (8 \, a^{4} x^{4} - 3\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} - 24 \,{\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 3 \,{\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - 3 \,{\left (2 \, a^{3} x^{3} + 15 \, a x\right )} \sqrt{a^{2} x^{2} - 1}}{256 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.03371, size = 170, normalized size = 0.93 \begin{align*} \begin{cases} \frac{x^{4} \operatorname{acosh}^{3}{\left (a x \right )}}{4} + \frac{3 x^{4} \operatorname{acosh}{\left (a x \right )}}{32} - \frac{3 x^{3} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{16 a} - \frac{3 x^{3} \sqrt{a^{2} x^{2} - 1}}{128 a} + \frac{9 x^{2} \operatorname{acosh}{\left (a x \right )}}{32 a^{2}} - \frac{9 x \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{32 a^{3}} - \frac{45 x \sqrt{a^{2} x^{2} - 1}}{256 a^{3}} - \frac{3 \operatorname{acosh}^{3}{\left (a x \right )}}{32 a^{4}} - \frac{45 \operatorname{acosh}{\left (a x \right )}}{256 a^{4}} & \text{for}\: a \neq 0 \\- \frac{i \pi ^{3} x^{4}}{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^{3} \operatorname{arcosh}\left (a x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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