Optimal. Leaf size=65 \[ \frac {\sqrt {a x-1} \text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^3 \sqrt {1-a x}}+\frac {\sqrt {a x-1} \log \left (\cosh ^{-1}(a x)\right )}{2 a^3 \sqrt {1-a x}} \]
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Rubi [A] time = 0.43, antiderivative size = 91, normalized size of antiderivative = 1.40, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5798, 5781, 3312, 3301} \[ \frac {\sqrt {a x-1} \sqrt {a x+1} \text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^3 \sqrt {1-a^2 x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \log \left (\cosh ^{-1}(a x)\right )}{2 a^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 3312
Rule 5781
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {\cosh ^2(x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^3 \sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^3 \sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (\cosh ^{-1}(a x)\right )}{2 a^3 \sqrt {1-a^2 x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^3 \sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^3 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \log \left (\cosh ^{-1}(a x)\right )}{2 a^3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 60, normalized size = 0.92 \[ -\frac {\sqrt {-((a x-1) (a x+1))} \left (\text {Chi}\left (2 \cosh ^{-1}(a x)\right )+\log \left (\cosh ^{-1}(a x)\right )\right )}{2 a^3 \sqrt {\frac {a x-1}{a x+1}} (a x+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{{\left (a^{2} x^{2} - 1\right )} \operatorname {arcosh}\left (a x\right )}, x\right ) \]
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 \[ \int \frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.47, size = 149, normalized size = 2.29 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \Ei \left (1, 2 \,\mathrm {arccosh}\left (a x \right )\right )}{4 a^{3} \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \Ei \left (1, -2 \,\mathrm {arccosh}\left (a x \right )\right )}{4 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (\mathrm {arccosh}\left (a x \right )\right )}{2 a^{3} \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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