Optimal. Leaf size=151 \[ -\frac {x \sqrt {1-a x} \sqrt {1+a x}}{4 a^2}+\frac {\sqrt {-1+a x} \cosh ^{-1}(a x)}{4 a^3 \sqrt {1-a x}}-\frac {x^2 \sqrt {-1+a x} \cosh ^{-1}(a x)}{2 a \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{2 a^2}+\frac {\sqrt {-1+a x} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt {1-a x}} \]
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Rubi [A]
time = 0.11, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5938, 5892,
5883, 92, 54} \begin {gather*} \frac {\sqrt {a x-1} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt {1-a x}}+\frac {\sqrt {a x-1} \cosh ^{-1}(a x)}{4 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{2 a^2}-\frac {x \sqrt {1-a x} \sqrt {a x+1}}{4 a^2}-\frac {x^2 \sqrt {a x-1} \cosh ^{-1}(a x)}{2 a \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 92
Rule 5883
Rule 5892
Rule 5938
Rubi steps
\begin {align*} \int \frac {x^2 \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^2 \cosh ^{-1}(a x)^2}{\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)^2}{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)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 a^2 \sqrt {1-a^2 x^2}}-\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int x \cosh ^{-1}(a x) \, dx}{a \sqrt {1-a^2 x^2}}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{2 a \sqrt {1-a^2 x^2}}-\frac {x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt {1-a^2 x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a^2 x^2}}\\ &=-\frac {x (1-a x) (1+a x)}{4 a^2 \sqrt {1-a^2 x^2}}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{2 a \sqrt {1-a^2 x^2}}-\frac {x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt {1-a^2 x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{4 a^2 \sqrt {1-a^2 x^2}}\\ &=-\frac {x (1-a x) (1+a x)}{4 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{4 a^3 \sqrt {1-a^2 x^2}}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{2 a \sqrt {1-a^2 x^2}}-\frac {x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 87, normalized size = 0.58 \begin {gather*} -\frac {\sqrt {-((-1+a x) (1+a x))} \left (4 \cosh ^{-1}(a x)^3-6 \cosh ^{-1}(a x) \cosh \left (2 \cosh ^{-1}(a x)\right )+\left (3+6 \cosh ^{-1}(a x)^2\right ) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )}{24 a^3 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.02, size = 239, normalized size = 1.58
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{3}}{6 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x +2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (2 \mathrm {arccosh}\left (a x \right )^{2}-2 \,\mathrm {arccosh}\left (a x \right )+1\right )}{16 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 a^{3} x^{3}-2 a x -2 \sqrt {a x +1}\, \sqrt {a x -1}\, a^{2} x^{2}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (2 \mathrm {arccosh}\left (a x \right )^{2}+2 \,\mathrm {arccosh}\left (a x \right )+1\right )}{16 a^{3} \left (a^{2} x^{2}-1\right )}\) | \(239\) |
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^{2} \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 = -0.01 \begin {gather*} \int \frac {x^2\,{\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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