Optimal. Leaf size=243 \[ \frac{\sqrt{\frac{\pi }{2}} e^2 \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{6}} e^2 \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} e^2 \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{6}} e^2 \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d} \]
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Rubi [A] time = 0.566976, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {4805, 12, 4635, 4406, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} e^2 \cos \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{6}} e^2 \cos \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} e^2 \sin \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{6}} e^2 \sin \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 12
Rule 4635
Rule 4406
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{(c e+d e x)^2}{\sqrt{a+b \sin ^{-1}(c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e^2 x^2}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 \sqrt{a+b x}}-\frac{\cos (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{d}\\ &=\frac{e^2 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d}-\frac{e^2 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d}\\ &=\frac{\left (e^2 \cos \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d}-\frac{\left (e^2 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d}+\frac{\left (e^2 \sin \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d}-\frac{\left (e^2 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4 d}\\ &=\frac{\left (e^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 )}{2 b d}-\frac{\left (e^2 \cos \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 b d}+\frac{\left (e^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 )}{2 b d}-\frac{\left (e^2 \sin \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sin ^{-1}(c+d x)}\right )}{2 b d}\\ &=\frac{e^2 \sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d}-\frac{e^2 \sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d}+\frac{e^2 \sqrt{\frac{\pi }{2}} S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{2 \sqrt{b} d}-\frac{e^2 \sqrt{\frac{\pi }{6}} S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \sin ^{-1}(c+d x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{2 \sqrt{b} d}\\ \end{align*}
Mathematica [C] time = 0.253284, size = 249, normalized size = 1.02 \[ -\frac{i e^2 e^{-\frac{3 i a}{b}} \left (3 e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-3 e^{\frac{4 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+\sqrt{3} \left (e^{\frac{6 i a}{b}} \sqrt{\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-\sqrt{-\frac{i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )\right )}{24 d \sqrt{a+b \sin ^{-1}(c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.061, size = 179, normalized size = 0.7 \begin{align*}{\frac{{e}^{2}\sqrt{2}\sqrt{\pi }}{12\,d}\sqrt{{b}^{-1}} \left ( -\sqrt{3}\cos \left ( 3\,{\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( dx+c \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) -\sqrt{3}\sin \left ( 3\,{\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}}{\sqrt{\pi }b}\sqrt{a+b\arcsin \left ( dx+c \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) +3\,\cos \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) +3\,\sin \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{a+b\arcsin \left ( dx+c \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{{\left (d e x + c e\right )}^{2}}{\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*} e^{2} \left (\int \frac{c^{2}}{\sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}}\, dx + \int \frac{d^{2} x^{2}}{\sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}}\, dx + \int \frac{2 c d x}{\sqrt{a + b \operatorname{asin}{\left (c + d x \right )}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.35676, size = 473, normalized size = 1.95 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{6} \sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{b} i}{2 \,{\left | b \right |}} - \frac{\sqrt{6} \sqrt{b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt{b}}\right ) e^{\left (\frac{3 \, a i}{b} + 2\right )}}{4 \,{\left (\frac{\sqrt{6} b^{\frac{3}{2}} i}{{\left | b \right |}} + \sqrt{6} \sqrt{b}\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} + 2\right )}}{4 \,{\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} + 2\right )}}{4 \,{\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{6} \sqrt{b \arcsin \left (d x + c\right ) + a} \sqrt{b} i}{2 \,{\left | b \right |}} - \frac{\sqrt{6} \sqrt{b \arcsin \left (d x + c\right ) + a}}{2 \, \sqrt{b}}\right ) e^{\left (-\frac{3 \, a i}{b} + 2\right )}}{4 \,{\left (\frac{\sqrt{6} b^{\frac{3}{2}} i}{{\left | b \right |}} - \sqrt{6} \sqrt{b}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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