Optimal. Leaf size=368 \[ \frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \]
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Rubi [A]
time = 0.36, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5897, 5895,
5893, 5887, 5556, 12, 3389, 2211, 2235, 2236, 5912, 5952} \begin {gather*} \frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {\sqrt {\pi } a^3 \sqrt {a^2-x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {\sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2-x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {\sqrt {\pi } a^3 \sqrt {a^2-x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2-x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5887
Rule 5893
Rule 5895
Rule 5897
Rule 5912
Rule 5952
Rubi steps
\begin {align*} \int \left (a^2-x^2\right )^{3/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \, dx &=-\frac {\left (a^2 \sqrt {a^2-x^2}\right ) \int \left (-1+\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{3/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \, dx}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}+\frac {\left (a \sqrt {a^2-x^2}\right ) \int \frac {x \left (-1+\frac {x^2}{a^2}\right )}{\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}} \, dx}{8 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^2 \sqrt {a^2-x^2}\right ) \int \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \, dx}{4 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {\left (3 a \sqrt {a^2-x^2}\right ) \int \frac {x}{\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}} \, dx}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^2 \sqrt {a^2-x^2}\right ) \int \frac {\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx}{8 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{8 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{8 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{128 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{128 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}-\frac {a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 165, normalized size = 0.45 \begin {gather*} -\frac {a^4 \sqrt {a^2-x^2} \left (-\sqrt {-\cosh ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-4 \cosh ^{-1}\left (\frac {x}{a}\right )\right )+8 \sqrt {2} \sqrt {-\cosh ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-2 \cosh ^{-1}\left (\frac {x}{a}\right )\right )+\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \left (32 \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+8 \sqrt {2} \Gamma \left (\frac {3}{2},2 \cosh ^{-1}\left (\frac {x}{a}\right )\right )-\Gamma \left (\frac {3}{2},4 \cosh ^{-1}\left (\frac {x}{a}\right )\right )\right )\right )}{128 \sqrt {\frac {-a+x}{a+x}} (a+x) \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a^{2}-x^{2}\right )^{\frac {3}{2}} \sqrt {\mathrm {arccosh}\left (\frac {x}{a}\right )}\, dx\]
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 \left (- \left (- a + x\right ) \left (a + x\right )\right )^{\frac {3}{2}} \sqrt {\operatorname {acosh}{\left (\frac {x}{a} \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 = -0.00 \begin {gather*} \int \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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