Optimal. Leaf size=126 \[ \frac {4 \sqrt {2 \pi } S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{3 a^4}-\frac {4 \sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^4}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}} \]
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Rubi [A] time = 0.34, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4633, 4719, 4635, 4406, 3305, 3351, 12} \[ \frac {4 \sqrt {2 \pi } S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{3 a^4}-\frac {4 \sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^4}-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3305
Rule 3351
Rule 4406
Rule 4633
Rule 4635
Rule 4719
Rubi steps
\begin {align*} \int \frac {x^3}{\sin ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac {2 \int \frac {x^2}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx}{a}-\frac {1}{3} (8 a) \int \frac {x^4}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}}-\frac {64}{3} \int \frac {x^3}{\sqrt {\sin ^{-1}(a x)}} \, dx+\frac {8 \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx}{a^2}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}}+\frac {8 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^4}-\frac {64 \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^3(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}}+\frac {8 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^4}-\frac {64 \operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}-\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}}+\frac {8 \operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^4}+\frac {4 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^4}-\frac {16 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}}+\frac {16 \operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{3 a^4}+\frac {8 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{a^4}-\frac {32 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{3 a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\sin ^{-1}(a x)}}+\frac {4 \sqrt {2 \pi } S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{3 a^4}-\frac {4 \sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^4}\\ \end {align*}
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Mathematica [C] time = 0.38, size = 200, normalized size = 1.59 \[ \frac {-2 \sin \left (2 \sin ^{-1}(a x)\right )+\sin \left (4 \sin ^{-1}(a x)\right )-4 \sin ^{-1}(a x) \left (e^{-2 i \sin ^{-1}(a x)}+e^{2 i \sin ^{-1}(a x)}-\sqrt {2} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \sin ^{-1}(a x)\right )-\sqrt {2} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \sin ^{-1}(a x)\right )\right )+4 \sin ^{-1}(a x) \left (e^{-4 i \sin ^{-1}(a x)}+e^{4 i \sin ^{-1}(a x)}-2 \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \sin ^{-1}(a x)\right )-2 \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \sin ^{-1}(a x)\right )\right )}{12 a^4 \sin ^{-1}(a x)^{3/2}} \]
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(-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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maple [A] time = 0.08, size = 109, normalized size = 0.87 \[ -\frac {-16 \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+16 \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+8 \arcsin \left (a x \right ) \cos \left (2 \arcsin \left (a x \right )\right )-8 \arcsin \left (a x \right ) \cos \left (4 \arcsin \left (a x \right )\right )+2 \sin \left (2 \arcsin \left (a x \right )\right )-\sin \left (4 \arcsin \left (a x \right )\right )}{12 a^{4} \arcsin \left (a x \right )^{\frac {3}{2}}} \]
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 \[ \int \frac {x^3}{{\mathrm {asin}\left (a\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{3}}{\operatorname {asin}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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