Optimal. Leaf size=71 \[ \frac {\sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a^3}-\frac {\sqrt {\frac {\pi }{6}} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a^3} \]
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Rubi [A] time = 0.09, antiderivative size = 71, 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, 3304, 3352} \[ \frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a^3}-\frac {\sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3352
Rule 4406
Rule 4635
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {\sin ^{-1}(a x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {x}}-\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^3}-\frac {\operatorname {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^3}\\ &=\frac {\operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{2 a^3}-\frac {\operatorname {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{2 a^3}\\ &=\frac {\sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a^3}-\frac {\sqrt {\frac {\pi }{6}} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a^3}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 128, normalized size = 1.80 \[ -\frac {i \left (3 \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-i \sin ^{-1}(a x)\right )-3 \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},i \sin ^{-1}(a x)\right )+\sqrt {3} \left (\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},3 i \sin ^{-1}(a x)\right )-\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 i \sin ^{-1}(a x)\right )\right )\right )}{24 a^3 \sqrt {\sin ^{-1}(a x)}} \]
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 = 0.37, size = 93, normalized size = 1.31 \[ \frac {\left (i + 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{48 \, a^{3}} - \frac {\left (i - 1\right ) \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {6} \sqrt {\arcsin \left (a x\right )}\right )}{48 \, a^{3}} - \frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{16 \, a^{3}} + \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \left (a x\right )}\right )}{16 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 51, normalized size = 0.72 \[ \frac {\sqrt {2}\, \sqrt {\pi }\, \left (-\sqrt {3}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )+3 \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right )\right )}{12 a^{3}} \]
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 \[ \int \frac {x^2}{\sqrt {\mathrm {asin}\left (a\,x\right )}} \,d x \]
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 \[ \int \frac {x^{2}}{\sqrt {\operatorname {asin}{\left (a x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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