Optimal. Leaf size=188 \[ \frac {\sqrt {a x-1} \cosh ^{-1}(a x)^4}{8 a^3 \sqrt {1-a x}}+\frac {3 \sqrt {a x-1} \cosh ^{-1}(a x)^2}{8 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^3}{2 a^2}-\frac {3 x \sqrt {1-a x} \sqrt {a x+1} \cosh ^{-1}(a x)}{4 a^2}-\frac {3 x^2 \sqrt {a x-1}}{8 a \sqrt {1-a x}}-\frac {3 x^2 \sqrt {a x-1} \cosh ^{-1}(a x)^2}{4 a \sqrt {1-a x}} \]
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Rubi [A] time = 0.77, antiderivative size = 257, normalized size of antiderivative = 1.37, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5798, 5759, 5676, 5662, 30} \[ -\frac {3 x^2 \sqrt {a x-1} \sqrt {a x+1}}{8 a \sqrt {1-a^2 x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^4}{8 a^3 \sqrt {1-a^2 x^2}}-\frac {x (1-a x) (a x+1) \cosh ^{-1}(a x)^3}{2 a^2 \sqrt {1-a^2 x^2}}-\frac {3 x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{4 a \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{8 a^3 \sqrt {1-a^2 x^2}}-\frac {3 x (1-a x) (a x+1) \cosh ^{-1}(a x)}{4 a^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 5662
Rule 5676
Rule 5759
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^2 \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^2 \cosh ^{-1}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {x (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{2 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 a^2 \sqrt {1-a^2 x^2}}-\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int x \cosh ^{-1}(a x)^2 \, dx}{2 a \sqrt {1-a^2 x^2}}\\ &=-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 a \sqrt {1-a^2 x^2}}-\frac {x (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{2 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^4}{8 a^3 \sqrt {1-a^2 x^2}}+\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^2 \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a^2 x^2}}\\ &=-\frac {3 x (1-a x) (1+a x) \cosh ^{-1}(a x)}{4 a^2 \sqrt {1-a^2 x^2}}-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 a \sqrt {1-a^2 x^2}}-\frac {x (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{2 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^4}{8 a^3 \sqrt {1-a^2 x^2}}+\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{4 a^2 \sqrt {1-a^2 x^2}}-\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int x \, dx}{4 a \sqrt {1-a^2 x^2}}\\ &=-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x}}{8 a \sqrt {1-a^2 x^2}}-\frac {3 x (1-a x) (1+a x) \cosh ^{-1}(a x)}{4 a^2 \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{8 a^3 \sqrt {1-a^2 x^2}}-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 a \sqrt {1-a^2 x^2}}-\frac {x (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{2 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^4}{8 a^3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 98, normalized size = 0.52 \[ -\frac {\sqrt {-((a x-1) (a x+1))} \left (2 \cosh ^{-1}(a x) \left (\cosh ^{-1}(a x)^3+\left (2 \cosh ^{-1}(a x)^2+3\right ) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )-3 \left (2 \cosh ^{-1}(a x)^2+1\right ) \cosh \left (2 \cosh ^{-1}(a x)\right )\right )}{16 a^3 \sqrt {\frac {a x-1}{a x+1}} (a x+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {arcosh}\left (a x\right )^{3}}{a^{2} x^{2} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 255, normalized size = 1.36 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{4}}{8 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 x^{3} a^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \mathrm {arccosh}\left (a x \right )^{3}-6 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )-3\right )}{32 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 x^{3} a^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \mathrm {arccosh}\left (a x \right )^{3}+6 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+3\right )}{32 a^{3} \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {x^2\,{\mathrm {acosh}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{2} \operatorname {acosh}^{3}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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