Optimal. Leaf size=316 \[ \frac {3 \sqrt {\frac {\pi }{2}} a \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3 \sqrt {\frac {\pi }{2}} a \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3 a \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}} \]
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Rubi [A] time = 0.73, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5713, 5683, 5676, 5664, 5781, 3312, 3307, 2180, 2204, 2205} \[ \frac {3 \sqrt {\frac {\pi }{2}} a \sqrt {a^2-x^2} \text {Erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3 \sqrt {\frac {\pi }{2}} a \sqrt {a^2-x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {3 a \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}} \]
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 5713
Rule 5781
Rubi steps
\begin {align*} \int \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2} \, dx &=\frac {\sqrt {a^2-x^2} \int \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2} \, dx}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {1}{2} x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {\sqrt {a^2-x^2} \int \frac {\cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx}{2 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {\left (3 \sqrt {a^2-x^2}\right ) \int x \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \, dx}{4 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 \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}{16 a^2 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \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 )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \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 )}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3 a \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \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 )}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3 a \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \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 )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \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 )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3 a \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \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 )}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {\left (3 a \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 )}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}\\ &=\frac {3 a \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {1}{2} x \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {a \sqrt {a^2-x^2} \cosh ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 a \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 \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 = 0.39, size = 144, normalized size = 0.46 \[ \frac {a^2 \sqrt {a^2-x^2} \left (15 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )+15 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}\right )-8 \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )} \left (16 \cosh ^{-1}\left (\frac {x}{a}\right )^2+15 \cosh \left (2 \cosh ^{-1}\left (\frac {x}{a}\right )\right )-20 \cosh ^{-1}\left (\frac {x}{a}\right ) \sinh \left (2 \cosh ^{-1}\left (\frac {x}{a}\right )\right )\right )\right )}{640 \sqrt {\frac {x-a}{a+x}} (a+x)} \]
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 \[ \int \sqrt {a^{2} - x^{2}} \operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.70, size = 0, normalized size = 0.00 \[ \int \mathrm {arccosh}\left (\frac {x}{a}\right )^{\frac {3}{2}} \sqrt {a^{2}-x^{2}}\, dx \]
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 \[ \int \sqrt {a^{2} - x^{2}} \operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\mathrm {acosh}\left (\frac {x}{a}\right )}^{3/2}\,\sqrt {a^2-x^2} \,d x \]
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 \[ \int \sqrt {- \left (- a + x\right ) \left (a + x\right )} \operatorname {acosh}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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