Optimal. Leaf size=50 \[ \frac {2 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {a^2-x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5892}
\begin {gather*} \frac {2 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {a^2-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 5892
Rubi steps
\begin {align*} \int \frac {\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{\sqrt {a^2-x^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 )}}{\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 )^{3/2}}{3 \sqrt {a^2-x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 50, normalized size = 1.00 \begin {gather*} \frac {2 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{3 \sqrt {a^2-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.69, size = 44, normalized size = 0.88
method | result | size |
default | \(\frac {2 \mathrm {arccosh}\left (\frac {x}{a}\right )^{\frac {3}{2}} a \sqrt {-\frac {a -x}{a}}\, \sqrt {\frac {a +x}{a}}}{3 \sqrt {\left (a -x \right ) \left (a +x \right )}}\) | \(44\) |
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 \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\operatorname {acosh}{\left (\frac {x}{a} \right )}}}{\sqrt {- \left (- a + x\right ) \left (a + x\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 = -0.02 \begin {gather*} \int \frac {\sqrt {\mathrm {acosh}\left (\frac {x}{a}\right )}}{\sqrt {a^2-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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