Optimal. Leaf size=107 \[ -\frac {\cosh ^{-1}(a x)^3}{4 a^2}-\frac {3 \cosh ^{-1}(a x)}{8 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^3+\frac {3}{4} x^2 \cosh ^{-1}(a x)-\frac {3 x \sqrt {a x-1} \sqrt {a x+1}}{8 a}-\frac {3 x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{4 a} \]
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Rubi [A] time = 0.38, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5662, 5759, 5676, 90, 52} \[ -\frac {\cosh ^{-1}(a x)^3}{4 a^2}-\frac {3 \cosh ^{-1}(a x)}{8 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^3+\frac {3}{4} x^2 \cosh ^{-1}(a x)-\frac {3 x \sqrt {a x-1} \sqrt {a x+1}}{8 a}-\frac {3 x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{4 a} \]
Antiderivative was successfully verified.
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Rule 52
Rule 90
Rule 5662
Rule 5676
Rule 5759
Rubi steps
\begin {align*} \int x \cosh ^{-1}(a x)^3 \, dx &=\frac {1}{2} x^2 \cosh ^{-1}(a x)^3-\frac {1}{2} (3 a) \int \frac {x^2 \cosh ^{-1}(a x)^2}{\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}{4 a}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^3+\frac {3}{2} \int x \cosh ^{-1}(a x) \, dx-\frac {3 \int \frac {\cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{4 a}\\ &=\frac {3}{4} x^2 \cosh ^{-1}(a x)-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 a}-\frac {\cosh ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^3-\frac {1}{4} (3 a) \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x}}{8 a}+\frac {3}{4} x^2 \cosh ^{-1}(a x)-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 a}-\frac {\cosh ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^3-\frac {3 \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 a}\\ &=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x}}{8 a}-\frac {3 \cosh ^{-1}(a x)}{8 a^2}+\frac {3}{4} x^2 \cosh ^{-1}(a x)-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 a}-\frac {\cosh ^{-1}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \cosh ^{-1}(a x)^3\\ \end {align*}
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Mathematica [A] time = 0.09, size = 113, normalized size = 1.06 \[ \frac {\left (4 a^2 x^2-2\right ) \cosh ^{-1}(a x)^3+6 a^2 x^2 \cosh ^{-1}(a x)-3 \left (a x \sqrt {a x-1} \sqrt {a x+1}+\log \left (a x+\sqrt {a x-1} \sqrt {a x+1}\right )\right )-6 a x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{8 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 112, normalized size = 1.05 \[ -\frac {6 \, \sqrt {a^{2} x^{2} - 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 2 \, {\left (2 \, a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} + 3 \, \sqrt {a^{2} x^{2} - 1} a x - 3 \, {\left (2 \, a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 88, normalized size = 0.82 \[ \frac {\frac {a^{2} x^{2} \mathrm {arccosh}\left (a x \right )^{3}}{2}-\frac {3 \mathrm {arccosh}\left (a x \right )^{2} a x \sqrt {a x -1}\, \sqrt {a x +1}}{4}-\frac {\mathrm {arccosh}\left (a x \right )^{3}}{4}+\frac {3 a^{2} x^{2} \mathrm {arccosh}\left (a x \right )}{4}-\frac {3 \sqrt {a x +1}\, \sqrt {a x -1}\, a x}{8}-\frac {3 \,\mathrm {arccosh}\left (a x \right )}{8}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, x^{2} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{3} - \int \frac {3 \, {\left (a^{3} x^{4} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x^{3} - a x^{2}\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}}{2 \, {\left (a^{3} x^{3} + {\left (a^{2} x^{2} - 1\right )} \sqrt {a x + 1} \sqrt {a x - 1} - a x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {acosh}\left (a\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.98, size = 102, normalized size = 0.95 \[ \begin {cases} \frac {x^{2} \operatorname {acosh}^{3}{\left (a x \right )}}{2} + \frac {3 x^{2} \operatorname {acosh}{\left (a x \right )}}{4} - \frac {3 x \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (a x \right )}}{4 a} - \frac {3 x \sqrt {a^{2} x^{2} - 1}}{8 a} - \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{4 a^{2}} - \frac {3 \operatorname {acosh}{\left (a x \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\- \frac {i \pi ^{3} x^{2}}{16} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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