Optimal. Leaf size=98 \[ \frac {\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \text {Int}\left (\frac {x}{\left (1-\frac {x^2}{a^2}\right ) \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}},x\right )}{2 a^3 \sqrt {a^2-x^2}}+\frac {x \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{a^2 \sqrt {a^2-x^2}} \]
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Rubi [A] time = 0.26, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}\right ) \int \frac {\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{\left (-1+\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx}{a^2 \sqrt {a^2-x^2}}\\ &=\frac {x \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{a^2 \sqrt {a^2-x^2}}+\frac {\left (\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}\right ) \int \frac {x}{\left (1-\frac {x^2}{a^2}\right ) \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}} \, dx}{2 a^3 \sqrt {a^2-x^2}}\\ \end {align*}
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Mathematica [A] time = 0.96, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [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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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {arcosh}\left (\frac {x}{a}\right )}}{{\left (a^{2} - x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\mathrm {arccosh}\left (\frac {x}{a}\right )}}{\left (a^{2}-x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {arcosh}\left (\frac {x}{a}\right )}}{{\left (a^{2} - x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {\mathrm {acosh}\left (\frac {x}{a}\right )}}{{\left (a^2-x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\operatorname {acosh}{\left (\frac {x}{a} \right )}}}{\left (- \left (- a + x\right ) \left (a + x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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