Optimal. Leaf size=525 \[ \frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {3 \sqrt {\pi } a^3 \sqrt {a^2-x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3 \sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {3 \sqrt {\pi } a^3 \sqrt {a^2-x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3 \sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}} \]
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Rubi [A] time = 1.28, antiderivative size = 533, normalized size of antiderivative = 1.02, number of steps used = 27, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {5713, 5685, 5683, 5676, 5664, 5781, 3312, 3307, 2180, 2204, 2205, 5716, 5701} \[ -\frac {3 \sqrt {\pi } a^3 \sqrt {a^2-x^2} \text {Erf}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3 \sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2-x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {3 \sqrt {\pi } a^3 \sqrt {a^2-x^2} \text {Erfi}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3 \sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2-x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x (a-x) (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5664
Rule 5676
Rule 5683
Rule 5685
Rule 5701
Rule 5713
Rule 5716
Rule 5781
Rubi steps
\begin {align*} \int \left (a^2-x^2\right )^{3/2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2} \, 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} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2} \, dx}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {\left (3 a \sqrt {a^2-x^2}\right ) \int 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}} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2} \, dx}{4 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {\left (9 a \sqrt {a^2-x^2}\right ) \int 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 {\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}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^2 \sqrt {a^2-x^2}\right ) \int \frac {\cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx}{8 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 \sqrt {a^2-x^2}\right ) \int \frac {x^2}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}} \, dx}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {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 {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {3}{8 \sqrt {x}}-\frac {\cosh (2 x)}{2 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh ^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 {9 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{512 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (2 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 (9 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{1024 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{1024 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (2 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 {27 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{512 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{512 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{128 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{128 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}\left (\frac {x}{a}\right )\right )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 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 )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 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 )}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{128 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (9 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{128 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} (a-x) x (a+x) \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erf}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 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 {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erfi}\left (2 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 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}}}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 219, normalized size = 0.42 \[ \frac {a^4 \sqrt {a^2-x^2} \left (60 \sqrt {2 \pi } \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )+60 \sqrt {2 \pi } \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )-384 \cosh ^{-1}\left (\frac {x}{a}\right )^3-480 \cosh \left (2 \cosh ^{-1}\left (\frac {x}{a}\right )\right ) \cosh ^{-1}\left (\frac {x}{a}\right )+640 \cosh ^{-1}\left (\frac {x}{a}\right )^2 \sinh \left (2 \cosh ^{-1}\left (\frac {x}{a}\right )\right )+5 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {5}{2},4 \cosh ^{-1}\left (\frac {x}{a}\right )\right )-5 \sqrt {-\cosh ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {5}{2},-4 \cosh ^{-1}\left (\frac {x}{a}\right )\right )\right )}{2560 \sqrt {\frac {x-a}{a+x}} (a+x) \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}} \]
Warning: Unable to verify antiderivative.
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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 \[ \int {\left (a^{2} - x^{2}\right )}^{\frac {3}{2}} \operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \left (a^{2}-x^{2}\right )^{\frac {3}{2}} \mathrm {arccosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\left (a^{2} - x^{2}\right )}^{\frac {3}{2}} \operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {acosh}\left (\frac {x}{a}\right )}^{3/2}\,{\left (a^2-x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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