Optimal. Leaf size=110 \[ -\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^3}{a^2}-\frac {6 \sqrt {1-a x} \sqrt {a x+1} \cosh ^{-1}(a x)}{a^2}-\frac {6 x \sqrt {a x-1}}{a \sqrt {1-a x}}-\frac {3 x \sqrt {a x-1} \cosh ^{-1}(a x)^2}{a \sqrt {1-a x}} \]
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Rubi [A] time = 0.39, antiderivative size = 153, normalized size of antiderivative = 1.39, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5798, 5718, 5654, 8} \[ -\frac {6 x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {1-a^2 x^2}}-\frac {(1-a x) (a x+1) \cosh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}}-\frac {6 (1-a x) (a x+1) \cosh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 5654
Rule 5718
Rule 5798
Rubi steps
\begin {align*} \int \frac {x \cosh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \cosh ^{-1}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \cosh ^{-1}(a x)^2 \, dx}{a \sqrt {1-a^2 x^2}}\\ &=-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}+\frac {\left (6 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {6 (1-a x) (1+a x) \cosh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}-\frac {\left (6 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int 1 \, dx}{a \sqrt {1-a^2 x^2}}\\ &=-\frac {6 x \sqrt {-1+a x} \sqrt {1+a x}}{a \sqrt {1-a^2 x^2}}-\frac {6 (1-a x) (1+a x) \cosh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {3 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a \sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^3}{a^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 101, normalized size = 0.92 \[ \frac {\sqrt {1-a^2 x^2} \left (6 a x-\sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3+3 a x \cosh ^{-1}(a x)^2-6 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)\right )}{a^2 \sqrt {a x-1} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 159, normalized size = 1.45 \[ \frac {3 \, \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} + 6 \, \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} a x - 6 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a^{4} x^{2} - a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.68, size = 103, normalized size = 0.94 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3}}{a^{2}} - \frac {3 i \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 2 \, a {\left (\frac {x}{a} - \frac {\sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a^{2}}\right )}\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 155, normalized size = 1.41 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x +1}\, \sqrt {a x -1}\, a x +a^{2} x^{2}-1\right ) \left (\mathrm {arccosh}\left (a x \right )^{3}-3 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )-6\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, a x -1\right ) \left (\mathrm {arccosh}\left (a x \right )^{3}+3 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+6\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.33, size = 65, normalized size = 0.59 \[ \frac {3 i \, x \operatorname {arcosh}\left (a x\right )^{2}}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )^{3}}{a^{2}} - \frac {3 \, {\left (-2 i \, x + \frac {2 i \, \sqrt {a^{2} x^{2} - 1} \operatorname {arcosh}\left (a x\right )}{a}\right )}}{a} \]
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 \[ \int \frac {x\,{\mathrm {acosh}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \operatorname {acosh}^{3}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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