Optimal. Leaf size=215 \[ -\frac {3 \sqrt {\pi } a \sqrt {a^2-x^2} C\left (\frac {2 \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{32 \sqrt {1-\frac {x^2}{a^2}}}+\frac {a \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3 a \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {1-\frac {x^2}{a^2}}} \]
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Rubi [A] time = 0.23, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4647, 4641, 4629, 4723, 3312, 3304, 3352} \[ -\frac {3 \sqrt {\pi } a \sqrt {a^2-x^2} \text {FresnelC}\left (\frac {2 \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{32 \sqrt {1-\frac {x^2}{a^2}}}+\frac {a \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {3 a \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {1-\frac {x^2}{a^2}}} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3312
Rule 3352
Rule 4629
Rule 4641
Rule 4647
Rule 4723
Rubi steps
\begin {align*} \int \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2} \, dx &=\frac {1}{2} x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {\sqrt {a^2-x^2} \int \frac {\sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx}{2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 \sqrt {a^2-x^2}\right ) \int x \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )} \, dx}{4 a \sqrt {1-\frac {x^2}{a^2}}}\\ &=-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {\left (3 \sqrt {a^2-x^2}\right ) \int \frac {x^2}{\sqrt {1-\frac {x^2}{a^2}} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}} \, dx}{16 a^2 \sqrt {1-\frac {x^2}{a^2}}}\\ &=-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {\left (3 a \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin ^2(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{16 \sqrt {1-\frac {x^2}{a^2}}}\\ &=-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}+\frac {\left (3 a \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cos (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{16 \sqrt {1-\frac {x^2}{a^2}}}\\ &=\frac {3 a \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 a \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{32 \sqrt {1-\frac {x^2}{a^2}}}\\ &=\frac {3 a \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 a \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}\right )}{16 \sqrt {1-\frac {x^2}{a^2}}}\\ &=\frac {3 a \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{16 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 x^2 \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{8 a \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3 a \sqrt {\pi } \sqrt {a^2-x^2} C\left (\frac {2 \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{32 \sqrt {1-\frac {x^2}{a^2}}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 173, normalized size = 0.80 \[ \frac {\sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )} \left (32 \sin ^{-1}\left (\frac {x}{a}\right ) \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )^2} \left (5 x \sqrt {1-\frac {x^2}{a^2}}+2 a \sin ^{-1}\left (\frac {x}{a}\right )\right )+15 \sqrt {2} a \sqrt {i \sin ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-2 i \sin ^{-1}\left (\frac {x}{a}\right )\right )+15 \sqrt {2} a \sqrt {-i \sin ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},2 i \sin ^{-1}\left (\frac {x}{a}\right )\right )\right )}{320 \sqrt {1-\frac {x^2}{a^2}} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )^2}} \]
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 \[ \int \sqrt {a^{2} - x^{2}} \arcsin \left (\frac {x}{a}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int \sqrt {a^{2}-x^{2}}\, \arcsin \left (\frac {x}{a}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int {\mathrm {asin}\left (\frac {x}{a}\right )}^{3/2}\,\sqrt {a^2-x^2} \,d x \]
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 \[ \int \sqrt {- \left (- a + x\right ) \left (a + x\right )} \operatorname {asin}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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