Optimal. Leaf size=65 \[ \frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{8 a^4} \]
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Rubi [A] time = 0.08, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4635, 4406, 3305, 3351} \[ \frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{4 a^4}-\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{8 a^4} \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3351
Rule 4406
Rule 4635
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {\sin ^{-1}(a x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cos (x) \sin ^3(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=\frac {\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 )}{a^4}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{8 a^4}+\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {\operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{4 a^4}+\frac {\operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{2 a^4}\\ &=-\frac {\sqrt {\frac {\pi }{2}} S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{8 a^4}+\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{4 a^4}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 128, normalized size = 1.97 \[ \frac {-2 \sqrt {2} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \sin ^{-1}(a x)\right )-2 \sqrt {2} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \sin ^{-1}(a x)\right )+\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \sin ^{-1}(a x)\right )+\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \sin ^{-1}(a x)\right )}{32 a^4 \sqrt {\sin ^{-1}(a x)}} \]
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 = 0.31, size = 81, normalized size = 1.25 \[ -\frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{64 \, a^{4}} + \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{64 \, a^{4}} + \frac {\left (i - 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{16 \, a^{4}} - \frac {\left (i + 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{16 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 44, normalized size = 0.68 \[ \frac {\sqrt {\pi }\, \left (-\sqrt {2}\, \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+4 \,\mathrm {S}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{16 a^{4}} \]
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 = -0.02 \[ \int \frac {x^3}{\sqrt {\mathrm {asin}\left (a\,x\right )}} \,d x \]
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 \[ \int \frac {x^{3}}{\sqrt {\operatorname {asin}{\left (a x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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