Optimal. Leaf size=75 \[ \frac {3 \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{2 a}-\frac {3 \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a}+x \sin ^{-1}(a x)^{3/2} \]
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Rubi [A] time = 0.10, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4619, 4677, 4623, 3304, 3352} \[ \frac {3 \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{2 a}-\frac {3 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a}+x \sin ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3352
Rule 4619
Rule 4623
Rule 4677
Rubi steps
\begin {align*} \int \sin ^{-1}(a x)^{3/2} \, dx &=x \sin ^{-1}(a x)^{3/2}-\frac {1}{2} (3 a) \int \frac {x \sqrt {\sin ^{-1}(a x)}}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{2 a}+x \sin ^{-1}(a x)^{3/2}-\frac {3}{4} \int \frac {1}{\sqrt {\sin ^{-1}(a x)}} \, dx\\ &=\frac {3 \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{2 a}+x \sin ^{-1}(a x)^{3/2}-\frac {3 \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{4 a}\\ &=\frac {3 \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{2 a}+x \sin ^{-1}(a x)^{3/2}-\frac {3 \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{2 a}\\ &=\frac {3 \sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}}{2 a}+x \sin ^{-1}(a x)^{3/2}-\frac {3 \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 76, normalized size = 1.01 \[ \frac {\sqrt {\sin ^{-1}(a x)} \left (\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},-i \sin ^{-1}(a x)\right )+\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {5}{2},i \sin ^{-1}(a x)\right )\right )}{2 a \sqrt {\sin ^{-1}(a x)^2}} \]
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 [C] time = 0.33, size = 119, normalized size = 1.59 \[ -\frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (i \, \arcsin \left (a x\right )\right )}}{2 \, a} + \frac {i \, \arcsin \left (a x\right )^{\frac {3}{2}} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{2 \, a} + \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{16 \, a} - \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{16 \, a} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (i \, \arcsin \left (a x\right )\right )}}{4 \, a} + \frac {3 \, \sqrt {\arcsin \left (a x\right )} e^{\left (-i \, \arcsin \left (a x\right )\right )}}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 72, normalized size = 0.96 \[ \frac {\sqrt {2}\, \left (2 \arcsin \left (a x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\pi }\, x a +3 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}-3 \pi \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{4 a \sqrt {\pi }} \]
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 = -0.01 \[ \int {\mathrm {asin}\left (a\,x\right )}^{3/2} \,d x \]
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 \[ \int \operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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