Optimal. Leaf size=243 \[ -\frac {40 \sqrt {1-a x} \sqrt {a x+1} \cosh ^{-1}(a x)}{9 a^4}-\frac {40 x \sqrt {a x-1}}{9 a^3 \sqrt {1-a x}}-\frac {2 x \sqrt {a x-1} \cosh ^{-1}(a x)^2}{a^3 \sqrt {1-a x}}-\frac {x^2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^3}{3 a^2}-\frac {2 x^2 \sqrt {1-a x} \sqrt {a x+1} \cosh ^{-1}(a x)}{9 a^2}-\frac {2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^3}{3 a^4}-\frac {2 x^3 \sqrt {a x-1}}{27 a \sqrt {1-a x}}-\frac {x^3 \sqrt {a x-1} \cosh ^{-1}(a x)^2}{3 a \sqrt {1-a x}} \]
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Rubi [A] time = 1.03, antiderivative size = 329, normalized size of antiderivative = 1.35, number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5798, 5759, 5718, 5654, 8, 5662, 30} \[ -\frac {2 x^3 \sqrt {a x-1} \sqrt {a x+1}}{27 a \sqrt {1-a^2 x^2}}-\frac {40 x \sqrt {a x-1} \sqrt {a x+1}}{9 a^3 \sqrt {1-a^2 x^2}}-\frac {x^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{3 a \sqrt {1-a^2 x^2}}-\frac {x^2 (1-a x) (a x+1) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt {1-a^2 x^2}}-\frac {2 x^2 (1-a x) (a x+1) \cosh ^{-1}(a x)}{9 a^2 \sqrt {1-a^2 x^2}}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}-\frac {2 (1-a x) (a x+1) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt {1-a^2 x^2}}-\frac {40 (1-a x) (a x+1) \cosh ^{-1}(a x)}{9 a^4 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5654
Rule 5662
Rule 5718
Rule 5759
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^3 \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^3 \cosh ^{-1}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (2 \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}{3 a^2 \sqrt {1-a^2 x^2}}-\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int x^2 \cosh ^{-1}(a x)^2 \, dx}{a \sqrt {1-a^2 x^2}}\\ &=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a \sqrt {1-a^2 x^2}}-\frac {2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt {1-a^2 x^2}}-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^3 \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 \sqrt {1-a^2 x^2}}-\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \cosh ^{-1}(a x)^2 \, dx}{a^3 \sqrt {1-a^2 x^2}}\\ &=-\frac {2 x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^2 \sqrt {1-a^2 x^2}}-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a \sqrt {1-a^2 x^2}}-\frac {2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt {1-a^2 x^2}}-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (4 \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}{9 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (4 \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}{a^2 \sqrt {1-a^2 x^2}}-\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int x^2 \, dx}{9 a \sqrt {1-a^2 x^2}}\\ &=-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{27 a \sqrt {1-a^2 x^2}}-\frac {40 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^4 \sqrt {1-a^2 x^2}}-\frac {2 x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^2 \sqrt {1-a^2 x^2}}-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a \sqrt {1-a^2 x^2}}-\frac {2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt {1-a^2 x^2}}-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt {1-a^2 x^2}}-\frac {\left (4 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int 1 \, dx}{9 a^3 \sqrt {1-a^2 x^2}}-\frac {\left (4 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int 1 \, dx}{a^3 \sqrt {1-a^2 x^2}}\\ &=-\frac {40 x \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3 \sqrt {1-a^2 x^2}}-\frac {2 x^3 \sqrt {-1+a x} \sqrt {1+a x}}{27 a \sqrt {1-a^2 x^2}}-\frac {40 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^4 \sqrt {1-a^2 x^2}}-\frac {2 x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{9 a^2 \sqrt {1-a^2 x^2}}-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{a^3 \sqrt {1-a^2 x^2}}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{3 a \sqrt {1-a^2 x^2}}-\frac {2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^4 \sqrt {1-a^2 x^2}}-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)^3}{3 a^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 140, normalized size = 0.58 \[ \frac {\sqrt {1-a^2 x^2} \left (2 a x \left (a^2 x^2+60\right )-9 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 x^2+2\right ) \cosh ^{-1}(a x)^3+9 a x \left (a^2 x^2+6\right ) \cosh ^{-1}(a x)^2-6 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 x^2+20\right ) \cosh ^{-1}(a x)\right )}{27 a^4 \sqrt {a x-1} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 205, normalized size = 0.84 \[ -\frac {9 \, {\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 9 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 6 \, {\left (a^{4} x^{4} + 19 \, a^{2} x^{2} - 20\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (a^{3} x^{3} + 60 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1}}{27 \, {\left (a^{6} x^{2} - a^{4}\right )}} \]
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 = 0.49, size = 375, normalized size = 1.54 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 x^{4} a^{4}-5 a^{2} x^{2}+4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-3 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +1\right ) \left (9 \mathrm {arccosh}\left (a x \right )^{3}-9 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )-2\right )}{216 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x +1}\, \sqrt {a x -1}\, a x +a^{2} x^{2}-1\right ) \left (\mathrm {arccosh}\left (a x \right )^{3}-3 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )-6\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, a x -1\right ) \left (\mathrm {arccosh}\left (a x \right )^{3}+3 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+6\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 x^{4} a^{4}-5 a^{2} x^{2}-4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}+3 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +1\right ) \left (9 \mathrm {arccosh}\left (a x \right )^{3}+9 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+2\right )}{216 a^{4} \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.71, size = 131, normalized size = 0.54 \[ -\frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arcosh}\left (a x\right )^{3} + \frac {2}{27} \, a {\left (\frac {3 \, {\left (-i \, \sqrt {a^{2} x^{2} - 1} x^{2} - \frac {20 i \, \sqrt {a^{2} x^{2} - 1}}{a^{2}}\right )} \operatorname {arcosh}\left (a x\right )}{a^{3}} + \frac {i \, a^{2} x^{3} + 60 i \, x}{a^{4}}\right )} + \frac {{\left (i \, a^{2} x^{3} + 6 i \, x\right )} \operatorname {arcosh}\left (a x\right )^{2}}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,{\mathrm {acosh}\left (a\,x\right )}^3}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{3} \operatorname {acosh}^{3}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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