Optimal. Leaf size=28 \[ \frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{2 a^2} \]
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Rubi [A] time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4635, 4406, 12, 3305, 3351} \[ \frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3305
Rule 3351
Rule 4406
Rule 4635
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {\sin ^{-1}(a x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{2 a^2}\\ &=\frac {\operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{a^2}\\ &=\frac {\sqrt {\pi } S\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{2 a^2}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 71, normalized size = 2.54 \[ -\frac {\sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \sin ^{-1}(a x)\right )+\sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \sin ^{-1}(a x)\right )}{4 \sqrt {2} a^2 \sqrt {\sin ^{-1}(a x)}} \]
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 = 35, normalized size = 1.25 \[ \frac {\left (i - 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{8 \, a^{2}} - \frac {\left (i + 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arcsin \left (a x\right )}\right )}{8 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 21, normalized size = 0.75 \[ \frac {\mathrm {S}\left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\pi }}{2 a^{2}} \]
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 = -0.04 \[ \int \frac {x}{\sqrt {\mathrm {asin}\left (a\,x\right )}} \,d x \]
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 \[ \int \frac {x}{\sqrt {\operatorname {asin}{\left (a x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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