Optimal. Leaf size=243 \[ \frac {\sqrt {a x-1} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt {1-a x}}+\frac {15 \sqrt {a x-1} \cosh ^{-1}(a x)}{64 a^5 \sqrt {1-a x}}-\frac {15 x \sqrt {1-a x} \sqrt {a x+1}}{64 a^4}-\frac {3 x^2 \sqrt {a x-1} \cosh ^{-1}(a x)}{8 a^3 \sqrt {1-a x}}-\frac {x^3 \sqrt {1-a x} \sqrt {a x+1}}{32 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{4 a^2}-\frac {3 x \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{8 a^4}-\frac {x^4 \sqrt {a x-1} \cosh ^{-1}(a x)}{8 a \sqrt {1-a x}} \]
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Rubi [A] time = 0.79, antiderivative size = 329, normalized size of antiderivative = 1.35, number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5798, 5759, 5676, 5662, 90, 52, 100, 12} \[ -\frac {x^3 (1-a x) (a x+1)}{32 a^2 \sqrt {1-a^2 x^2}}-\frac {15 x (1-a x) (a x+1)}{64 a^4 \sqrt {1-a^2 x^2}}-\frac {x^4 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{8 a \sqrt {1-a^2 x^2}}-\frac {x^3 (1-a x) (a x+1) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt {1-a^2 x^2}}-\frac {3 x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{8 a^3 \sqrt {1-a^2 x^2}}-\frac {3 x (1-a x) (a x+1) \cosh ^{-1}(a x)^2}{8 a^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{64 a^5 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 52
Rule 90
Rule 100
Rule 5662
Rule 5676
Rule 5759
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^4 \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^4 \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{4 a^2 \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)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{4 a^2 \sqrt {1-a^2 x^2}}-\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int x^3 \cosh ^{-1}(a x) \, dx}{2 a \sqrt {1-a^2 x^2}}\\ &=-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{8 a \sqrt {1-a^2 x^2}}-\frac {3 x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{8 a^4 \sqrt {1-a^2 x^2}}-\frac {x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^4}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 \sqrt {1-a^2 x^2}}+\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 a^4 \sqrt {1-a^2 x^2}}-\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int x \cosh ^{-1}(a x) \, dx}{4 a^3 \sqrt {1-a^2 x^2}}\\ &=-\frac {x^3 (1-a x) (1+a x)}{32 a^2 \sqrt {1-a^2 x^2}}-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{8 a^3 \sqrt {1-a^2 x^2}}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{8 a \sqrt {1-a^2 x^2}}-\frac {3 x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{8 a^4 \sqrt {1-a^2 x^2}}-\frac {x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt {1-a^2 x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {3 x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 a^2 \sqrt {1-a^2 x^2}}\\ &=-\frac {3 x (1-a x) (1+a x)}{16 a^4 \sqrt {1-a^2 x^2}}-\frac {x^3 (1-a x) (1+a x)}{32 a^2 \sqrt {1-a^2 x^2}}-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{8 a^3 \sqrt {1-a^2 x^2}}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{8 a \sqrt {1-a^2 x^2}}-\frac {3 x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{8 a^4 \sqrt {1-a^2 x^2}}-\frac {x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt {1-a^2 x^2}}+\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 a^4 \sqrt {1-a^2 x^2}}+\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 a^2 \sqrt {1-a^2 x^2}}\\ &=-\frac {15 x (1-a x) (1+a x)}{64 a^4 \sqrt {1-a^2 x^2}}-\frac {x^3 (1-a x) (1+a x)}{32 a^2 \sqrt {1-a^2 x^2}}+\frac {3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{16 a^5 \sqrt {1-a^2 x^2}}-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{8 a^3 \sqrt {1-a^2 x^2}}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{8 a \sqrt {1-a^2 x^2}}-\frac {3 x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{8 a^4 \sqrt {1-a^2 x^2}}-\frac {x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt {1-a^2 x^2}}+\frac {\left (3 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{64 a^4 \sqrt {1-a^2 x^2}}\\ &=-\frac {15 x (1-a x) (1+a x)}{64 a^4 \sqrt {1-a^2 x^2}}-\frac {x^3 (1-a x) (1+a x)}{32 a^2 \sqrt {1-a^2 x^2}}+\frac {15 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{64 a^5 \sqrt {1-a^2 x^2}}-\frac {3 x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{8 a^3 \sqrt {1-a^2 x^2}}-\frac {x^4 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{8 a \sqrt {1-a^2 x^2}}-\frac {3 x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{8 a^4 \sqrt {1-a^2 x^2}}-\frac {x^3 (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{4 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 116, normalized size = 0.48 \[ \frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1) \left (32 \cosh ^{-1}(a x)^3-4 \left (16 \cosh \left (2 \cosh ^{-1}(a x)\right )+\cosh \left (4 \cosh ^{-1}(a x)\right )\right ) \cosh ^{-1}(a x)+8 \cosh ^{-1}(a x)^2 \left (8 \sinh \left (2 \cosh ^{-1}(a x)\right )+\sinh \left (4 \cosh ^{-1}(a x)\right )\right )+32 \sinh \left (2 \cosh ^{-1}(a x)\right )+\sinh \left (4 \cosh ^{-1}(a x)\right )\right )}{256 a^5 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} x^{4} \operatorname {arcosh}\left (a x\right )^{2}}{a^{2} x^{2} - 1}, x\right ) \]
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 \[ \int \frac {x^{4} \operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.77, size = 488, normalized size = 2.01 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{3}}{8 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (8 x^{5} a^{5}-12 x^{3} a^{3}+8 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}+4 a x -8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (8 \mathrm {arccosh}\left (a x \right )^{2}-4 \,\mathrm {arccosh}\left (a x \right )+1\right )}{512 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 x^{3} a^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (2 \mathrm {arccosh}\left (a x \right )^{2}-2 \,\mathrm {arccosh}\left (a x \right )+1\right )}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 x^{3} a^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (2 \mathrm {arccosh}\left (a x \right )^{2}+2 \,\mathrm {arccosh}\left (a x \right )+1\right )}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (8 x^{5} a^{5}-12 x^{3} a^{3}-8 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}+4 a x +8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (8 \mathrm {arccosh}\left (a x \right )^{2}+4 \,\mathrm {arccosh}\left (a x \right )+1\right )}{512 a^{5} \left (a^{2} x^{2}-1\right )} \]
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 = -0.00 \[ \int \frac {x^4\,{\mathrm {acosh}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{4} \operatorname {acosh}^{2}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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