Optimal. Leaf size=98 \[ \frac {x \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{a^2 \sqrt {a^2-x^2}}+\frac {\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \text {Int}\left (\frac {x}{\left (1-\frac {x^2}{a^2}\right ) \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}},x\right )}{2 a^3 \sqrt {a^2-x^2}} \]
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Rubi [A]
time = 0.06, 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 = {}
\begin {gather*} \int \frac {\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx \end {gather*}
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 = 0.70, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 4.25, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {\operatorname {acosh}{\left (\frac {x}{a} \right )}}}{\left (- \left (- a + x\right ) \left (a + x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {\mathrm {acosh}\left (\frac {x}{a}\right )}}{{\left (a^2-x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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