Optimal. Leaf size=243 \[ -\frac {15 x \sqrt {1-a x} \sqrt {1+a x}}{64 a^4}-\frac {x^3 \sqrt {1-a x} \sqrt {1+a x}}{32 a^2}+\frac {15 \sqrt {-1+a x} \cosh ^{-1}(a x)}{64 a^5 \sqrt {1-a x}}-\frac {3 x^2 \sqrt {-1+a x} \cosh ^{-1}(a x)}{8 a^3 \sqrt {1-a x}}-\frac {x^4 \sqrt {-1+a x} \cosh ^{-1}(a x)}{8 a \sqrt {1-a x}}-\frac {3 x \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{8 a^4}-\frac {x^3 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{4 a^2}+\frac {\sqrt {-1+a x} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt {1-a x}} \]
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Rubi [A]
time = 0.22, antiderivative size = 243, normalized size of antiderivative = 1.00, 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 = {5938, 5892,
5883, 92, 54, 102, 12} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 54
Rule 92
Rule 102
Rule 5883
Rule 5892
Rule 5938
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 = 0.21, size = 116, normalized size = 0.48 \begin {gather*} \frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (32 \cosh ^{-1}(a x)^3-4 \cosh ^{-1}(a x) \left (16 \cosh \left (2 \cosh ^{-1}(a x)\right )+\cosh \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 )+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 )\right )}{256 a^5 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs.
\(2(199)=398\).
time = 4.58, size = 488, normalized size = 2.01
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{3}}{8 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}+8 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{4} x^{4}+4 a x -8 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (8 \mathrm {arccosh}\left (a x \right )^{2}-4 \,\mathrm {arccosh}\left (a x \right )+1\right )}{512 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x +2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (2 \mathrm {arccosh}\left (a x \right )^{2}-2 \,\mathrm {arccosh}\left (a x \right )+1\right )}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x -2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (2 \mathrm {arccosh}\left (a x \right )^{2}+2 \,\mathrm {arccosh}\left (a x \right )+1\right )}{16 a^{5} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}-8 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{4} x^{4}+4 a x +8 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (8 \mathrm {arccosh}\left (a x \right )^{2}+4 \,\mathrm {arccosh}\left (a x \right )+1\right )}{512 a^{5} \left (a^{2} x^{2}-1\right )}\) | \(488\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{4} \operatorname {acosh}^{2}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^4\,{\mathrm {acosh}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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