Optimal. Leaf size=99 \[ \frac{\sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{2 \sqrt{b} c^2}-\frac{\sqrt{\pi } \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{2 \sqrt{b} c^2} \]
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Rubi [A] time = 0.17644, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4635, 4406, 12, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{2 \sqrt{b} c^2}-\frac{\sqrt{\pi } \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{2 \sqrt{b} c^2} \]
Antiderivative was successfully verified.
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Rule 4635
Rule 4406
Rule 12
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 c^2}\\ &=\frac{\cos \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 c^2}-\frac{\sin \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{2 c^2}\\ &=\frac{\cos \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b c^2}-\frac{\sin \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b c^2}\\ &=\frac{\sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{2 \sqrt{b} c^2}-\frac{\sqrt{\pi } C\left (\frac{2 \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{2 \sqrt{b} c^2}\\ \end{align*}
Mathematica [C] time = 0.0610222, size = 123, normalized size = 1.24 \[ -\frac{e^{-\frac{2 i a}{b}} \left (\sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{\frac{4 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{4 \sqrt{2} c^2 \sqrt{a+b \sin ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.038, size = 80, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{\pi }}{2\,{c}^{2}}\sqrt{{b}^{-1}} \left ( \sin \left ( 2\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) -\cos \left ( 2\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{a+b\arcsin \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{b \arcsin \left (c x\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{a + b \operatorname{asin}{\left (c x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.99215, size = 178, normalized size = 1.8 \begin{align*} \frac{i \, \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{b \arcsin \left (c x\right ) + a}}{\sqrt{b}} + \frac{i \, \sqrt{b \arcsin \left (c x\right ) + a} \sqrt{b}}{{\left | b \right |}}\right ) e^{\left (-\frac{2 i \, a}{b}\right )}}{4 \, c^{2}{\left (\sqrt{b} - \frac{i \, b^{\frac{3}{2}}}{{\left | b \right |}}\right )}} - \frac{i \, \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{b \arcsin \left (c x\right ) + a}}{\sqrt{b}} - \frac{i \, \sqrt{b \arcsin \left (c x\right ) + a} \sqrt{b}}{{\left | b \right |}}\right ) e^{\left (\frac{2 i \, a}{b}\right )}}{4 \, \sqrt{b} c^{2}{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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