Optimal. Leaf size=226 \[ \frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}-\frac {\sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2-x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\sqrt {\pi } a^3 \sqrt {a^2-x^2} S\left (\frac {2 \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}+\frac {a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1-\frac {x^2}{a^2}}} \]
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Rubi [A] time = 0.24, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4649, 4647, 4641, 4635, 4406, 12, 3305, 3351, 4723} \[ -\frac {\sqrt {\frac {\pi }{2}} a^3 \sqrt {a^2-x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\sqrt {\pi } a^3 \sqrt {a^2-x^2} S\left (\frac {2 \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}+\frac {a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3305
Rule 3351
Rule 4406
Rule 4635
Rule 4641
Rule 4647
Rule 4649
Rule 4723
Rubi steps
\begin {align*} \int \left (a^2-x^2\right )^{3/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )} \, dx &=\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} \left (3 a^2\right ) \int \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )} \, dx-\frac {\left (a \sqrt {a^2-x^2}\right ) \int \frac {x \left (1-\frac {x^2}{a^2}\right )}{\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}} \, dx}{8 \sqrt {1-\frac {x^2}{a^2}}}\\ &=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}-\frac {\left (3 a \sqrt {a^2-x^2}\right ) \int \frac {x}{\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}} \, dx}{16 \sqrt {1-\frac {x^2}{a^2}}}+\frac {\left (3 a^2 \sqrt {a^2-x^2}\right ) \int \frac {\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx}{8 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}\\ &=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{16 \sqrt {1-\frac {x^2}{a^2}}}\\ &=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (a^3 \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 )}{32 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\sin ^{-1}\left (\frac {x}{a}\right )\right )}{16 \sqrt {1-\frac {x^2}{a^2}}}\\ &=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}\right )}{32 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \sin \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 {\left (3 a^3 \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 )}{32 \sqrt {1-\frac {x^2}{a^2}}}\\ &=\frac {3}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a^3 \sqrt {\pi } \sqrt {a^2-x^2} S\left (\frac {2 \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{32 \sqrt {1-\frac {x^2}{a^2}}}-\frac {\left (3 a^3 \sqrt {a^2-x^2}\right ) \operatorname {Subst}\left (\int \sin \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}{8} a^2 x \sqrt {a^2-x^2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}+\frac {a^3 \sqrt {a^2-x^2} \sin ^{-1}\left (\frac {x}{a}\right )^{3/2}}{4 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1-\frac {x^2}{a^2}}}-\frac {a^3 \sqrt {\pi } \sqrt {a^2-x^2} S\left (\frac {2 \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}}{\sqrt {\pi }}\right )}{8 \sqrt {1-\frac {x^2}{a^2}}}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 183, normalized size = 0.81 \[ \frac {a^3 \sqrt {a^2-x^2} \left (32 \sin ^{-1}\left (\frac {x}{a}\right )^2+8 \sqrt {2} \sqrt {-i \sin ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-2 i \sin ^{-1}\left (\frac {x}{a}\right )\right )+8 \sqrt {2} \sqrt {i \sin ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},2 i \sin ^{-1}\left (\frac {x}{a}\right )\right )+\sqrt {-i \sin ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},-4 i \sin ^{-1}\left (\frac {x}{a}\right )\right )+\sqrt {i \sin ^{-1}\left (\frac {x}{a}\right )} \Gamma \left (\frac {3}{2},4 i \sin ^{-1}\left (\frac {x}{a}\right )\right )\right )}{128 \sqrt {1-\frac {x^2}{a^2}} \sqrt {\sin ^{-1}\left (\frac {x}{a}\right )}} \]
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 \[ \int {\left (a^{2} - x^{2}\right )}^{\frac {3}{2}} \sqrt {\arcsin \left (\frac {x}{a}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \left (a^{2}-x^{2}\right )^{\frac {3}{2}} \sqrt {\arcsin \left (\frac {x}{a}\right )}\, dx \]
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 \[ \int \sqrt {\mathrm {asin}\left (\frac {x}{a}\right )}\,{\left (a^2-x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- \left (- a + x\right ) \left (a + x\right )\right )^{\frac {3}{2}} \sqrt {\operatorname {asin}{\left (\frac {x}{a} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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