Optimal. Leaf size=105 \[ \frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d}+\frac{\sqrt{2 \pi } \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d} \]
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Rubi [A] time = 0.127753, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4803, 4623, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d}+\frac{\sqrt{2 \pi } \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d} \]
Antiderivative was successfully verified.
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Rule 4803
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+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sin ^{-1}(c+d x)\right )}{b d}\\ &=\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+d x)\right )}{b d}+\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+d x)\right )}{b d}\\ &=\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+d x)}\right )}{b d}+\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+d x)}\right )}{b d}\\ &=\frac{\sqrt{2 \pi } \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d}+\frac{\sqrt{2 \pi } S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{\sqrt{b} d}\\ \end{align*}
Mathematica [F] time = 0.0318726, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0., size = 87, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{d}\sqrt{{b}^{-1}} \left ( \cos \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( dx+c \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 ( dx+c \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 (d x + c\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 + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.88713, size = 230, normalized size = 2.19 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac{a i}{b}\right )}}{{\left (\frac{\sqrt{2} b i}{\sqrt{{\left | b \right |}}} + \sqrt{2} \sqrt{{\left | b \right |}}\right )} d} + \frac{\sqrt{\pi } \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a} i}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac{a i}{b}\right )}}{{\left (\frac{\sqrt{2} b i}{\sqrt{{\left | b \right |}}} - \sqrt{2} \sqrt{{\left | b \right |}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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