Optimal. Leaf size=359 \[ -\frac{3 \sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{512 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3 \sqrt{\pi } a^3 \sqrt{a^2-x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3 a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1-\frac{x^2}{a^2}}}+\frac{27 a^3 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{9 a x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 \left (a^2-x^2\right )^{5/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1-\frac{x^2}{a^2}}} \]
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Rubi [A] time = 0.421075, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4649, 4647, 4641, 4629, 4723, 3312, 3304, 3352, 4677, 4661} \[ -\frac{3 \sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{512 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3 \sqrt{\pi } a^3 \sqrt{a^2-x^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3 a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1-\frac{x^2}{a^2}}}+\frac{27 a^3 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{9 a x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 \left (a^2-x^2\right )^{5/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1-\frac{x^2}{a^2}}} \]
Antiderivative was successfully verified.
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Rule 4649
Rule 4647
Rule 4641
Rule 4629
Rule 4723
Rule 3312
Rule 3304
Rule 3352
Rule 4677
Rule 4661
Rubi steps
\begin{align*} \int \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2} \, dx &=\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} \left (3 a^2\right ) \int \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2} \, dx-\frac{\left (3 a \sqrt{a^2-x^2}\right ) \int 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 \left (a^2-x^2\right )^{5/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}-\frac{\left (9 a \sqrt{a^2-x^2}\right ) \int 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{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}} \, dx}{64 \sqrt{1-\frac{x^2}{a^2}}}+\frac{\left (3 a^2 \sqrt{a^2-x^2}\right ) \int \frac{\sin ^{-1}\left (\frac{x}{a}\right )^{3/2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx}{8 \sqrt{1-\frac{x^2}{a^2}}}\\ &=-\frac{9 a x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3 \left (a^2-x^2\right )^{5/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1-\frac{x^2}{a^2}}}+\frac{\left (9 \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}{64 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos ^4(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1-\frac{x^2}{a^2}}}\\ &=-\frac{9 a x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3 \left (a^2-x^2\right )^{5/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}+\frac{\cos (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}\left (\frac{x}{a}\right )\right )}{64 \sqrt{1-\frac{x^2}{a^2}}}+\frac{\left (9 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 )}{64 \sqrt{1-\frac{x^2}{a^2}}}\\ &=-\frac{9 a^3 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{1-\frac{x^2}{a^2}}}-\frac{9 a x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3 \left (a^2-x^2\right )^{5/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}\left (\frac{x}{a}\right )\right )}{512 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \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 )}{128 \sqrt{1-\frac{x^2}{a^2}}}+\frac{\left (9 a^3 \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 )}{64 \sqrt{1-\frac{x^2}{a^2}}}\\ &=\frac{27 a^3 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{1-\frac{x^2}{a^2}}}-\frac{9 a x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3 \left (a^2-x^2\right )^{5/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \sqrt{a^2-x^2}\right ) \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{256 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (3 a^3 \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 )}{64 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (9 a^3 \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 )}{128 \sqrt{1-\frac{x^2}{a^2}}}\\ &=\frac{27 a^3 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{1-\frac{x^2}{a^2}}}-\frac{9 a x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3 \left (a^2-x^2\right )^{5/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3 a^3 \sqrt{\frac{\pi }{2}} \sqrt{a^2-x^2} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{512 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3 a^3 \sqrt{\pi } \sqrt{a^2-x^2} C\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )}{128 \sqrt{1-\frac{x^2}{a^2}}}-\frac{\left (9 a^3 \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 )}{64 \sqrt{1-\frac{x^2}{a^2}}}\\ &=\frac{27 a^3 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{256 \sqrt{1-\frac{x^2}{a^2}}}-\frac{9 a x^2 \sqrt{a^2-x^2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 \sqrt{1-\frac{x^2}{a^2}}}+\frac{3 \left (a^2-x^2\right )^{5/2} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{32 a \sqrt{1-\frac{x^2}{a^2}}}+\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{1}{4} x \left (a^2-x^2\right )^{3/2} \sin ^{-1}\left (\frac{x}{a}\right )^{3/2}+\frac{3 a^3 \sqrt{a^2-x^2} \sin ^{-1}\left (\frac{x}{a}\right )^{5/2}}{20 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3 a^3 \sqrt{\frac{\pi }{2}} \sqrt{a^2-x^2} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}\right )}{512 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3 a^3 \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*}
Mathematica [C] time = 0.434541, size = 209, normalized size = 0.58 \[ \frac{a^3 \sqrt{a^2-x^2} \left (-240 \sqrt{\pi } \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )^2} \text{FresnelC}\left (\frac{2 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )}}{\sqrt{\pi }}\right )+\sqrt{\sin ^{-1}\left (\frac{x}{a}\right )} \left (5 \sqrt{i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{5}{2},-4 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+5 \sqrt{-i \sin ^{-1}\left (\frac{x}{a}\right )} \text{Gamma}\left (\frac{5}{2},4 i \sin ^{-1}\left (\frac{x}{a}\right )\right )+32 \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )^2} \left (12 \sin ^{-1}\left (\frac{x}{a}\right )^2+20 \sin \left (2 \sin ^{-1}\left (\frac{x}{a}\right )\right ) \sin ^{-1}\left (\frac{x}{a}\right )+15 \cos \left (2 \sin ^{-1}\left (\frac{x}{a}\right )\right )\right )\right )\right )}{2560 \sqrt{1-\frac{x^2}{a^2}} \sqrt{\sin ^{-1}\left (\frac{x}{a}\right )^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.184, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}-{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( \arcsin \left ({\frac{x}{a}} \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} - x^{2}\right )}^{\frac{3}{2}} \arcsin \left (\frac{x}{a}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} - x^{2}\right )}^{\frac{3}{2}} \arcsin \left (\frac{x}{a}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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