Optimal. Leaf size=50 \[ \frac {2 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]
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Rubi [A] time = 0.17, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5713, 5676} \[ \frac {2 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]
Antiderivative was successfully verified.
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Rule 5676
Rule 5713
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx &=\frac {\left (\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}\right ) \int \frac {\cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx}{\sqrt {a^2-x^2}}\\ &=\frac {2 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 50, normalized size = 1.00 \[ \frac {2 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]
Antiderivative was successfully verified.
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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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{\sqrt {a^{2} - x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 44, normalized size = 0.88 \[ \frac {2 \mathrm {arccosh}\left (\frac {x}{a}\right )^{\frac {5}{2}} a \sqrt {\frac {-a +x}{a}}\, \sqrt {\frac {a +x}{a}}}{5 \sqrt {-\left (-a +x \right ) \left (a +x \right )}} \]
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 \[ \int \frac {\operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{\sqrt {a^{2} - x^{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {acosh}\left (\frac {x}{a}\right )}^{3/2}}{\sqrt {a^2-x^2}} \,d x \]
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 \[ \int \frac {\operatorname {acosh}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}}{\sqrt {- \left (- a + x\right ) \left (a + x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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