Optimal. Leaf size=101 \[ \frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b} c}+\frac{\sqrt{2 \pi } \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b} c} \]
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Rubi [A] time = 0.0950421, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4623, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b} c}+\frac{\sqrt{2 \pi } \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b} c} \]
Antiderivative was successfully verified.
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Rule 4623
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b \sin ^{-1}(c x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b c}\\ &=\frac{\cos \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b c}+\frac{\sin \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b c}\\ &=\frac{\left (2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b c}+\frac{\left (2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c x)}\right )}{b c}\\ &=\frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right )}{\sqrt{b} c}+\frac{\sqrt{2 \pi } S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{\sqrt{b} c}\\ \end{align*}
Mathematica [C] time = 0.0964359, size = 121, normalized size = 1.2 \[ \frac{i e^{-\frac{i a}{b}} \left (e^{\frac{2 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt{-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )\right )}{2 c \sqrt{a+b \sin ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0., size = 83, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{c}\sqrt{{b}^{-1}} \left ( \cos \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) +\sin \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \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{1}{\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{1}{\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 = 1.79073, size = 215, normalized size = 2.13 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{i \, \sqrt{2} \sqrt{b \arcsin \left (c x\right ) + a}}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arcsin \left (c x\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac{i \, a}{b}\right )}}{c{\left (\frac{i \, \sqrt{2} b}{\sqrt{{\left | b \right |}}} + \sqrt{2} \sqrt{{\left | b \right |}}\right )}} - \frac{\sqrt{\pi } \operatorname{erf}\left (\frac{i \, \sqrt{2} \sqrt{b \arcsin \left (c x\right ) + a}}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arcsin \left (c x\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac{i \, a}{b}\right )}}{c{\left (-\frac{i \, \sqrt{2} b}{\sqrt{{\left | b \right |}}} + \sqrt{2} \sqrt{{\left | b \right |}}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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