Optimal. Leaf size=223 \[ \frac{\sqrt{\frac{\pi }{2}} \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{6}} \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c^3} \]
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Rubi [A] time = 0.370696, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {4636, 4406, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{\frac{\pi }{2}} \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{6}} \sin \left (\frac{3 a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c^3} \]
Antiderivative was successfully verified.
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Rule 4636
Rule 4406
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{a+b \cos ^{-1}(c x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{c^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 \sqrt{a+b x}}+\frac{\sin (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{c^3}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{4 c^3}-\frac{\operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{4 c^3}\\ &=-\frac{\cos \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{4 c^3}-\frac{\cos \left (\frac{3 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{4 c^3}+\frac{\sin \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{a}{b}+x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{4 c^3}+\frac{\sin \left (\frac{3 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{3 a}{b}+3 x\right )}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{4 c^3}\\ &=-\frac{\cos \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \sin \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{2 b c^3}-\frac{\cos \left (\frac{3 a}{b}\right ) \operatorname{Subst}\left (\int \sin \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{2 b c^3}+\frac{\sin \left (\frac{a}{b}\right ) \operatorname{Subst}\left (\int \cos \left (\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{2 b c^3}+\frac{\sin \left (\frac{3 a}{b}\right ) \operatorname{Subst}\left (\int \cos \left (\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{2 b c^3}\\ &=-\frac{\sqrt{\frac{\pi }{2}} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{6}} \cos \left (\frac{3 a}{b}\right ) S\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{2}} C\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{a}{b}\right )}{2 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{6}} C\left (\frac{\sqrt{\frac{6}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right ) \sin \left (\frac{3 a}{b}\right )}{2 \sqrt{b} c^3}\\ \end{align*}
Mathematica [C] time = 0.421384, size = 225, normalized size = 1.01 \[ \frac{e^{-\frac{3 i a}{b}} \left (3 e^{\frac{2 i a}{b}} \sqrt{-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )+3 e^{\frac{4 i a}{b}} \sqrt{\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )+\sqrt{3} \left (\sqrt{-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )+e^{\frac{6 i a}{b}} \sqrt{\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}} \text{Gamma}\left (\frac{1}{2},\frac{3 i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )\right )\right )}{24 c^3 \sqrt{a+b \cos ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.085, size = 167, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}\sqrt{\pi }}{12\,{c}^{3}}\sqrt{{b}^{-1}} \left ( \sqrt{3}\cos \left ( 3\,{\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}}{\sqrt{\pi }b}\sqrt{a+b\arccos \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) -\sqrt{3}\sin \left ( 3\,{\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}}{\sqrt{\pi }b}\sqrt{a+b\arccos \left ( cx \right ) }{\frac{1}{\sqrt{{b}^{-1}}}}} \right ) +3\,\cos \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{a+b\arccos \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) -3\,\sin \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{a+b\arccos \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^{2}}{\sqrt{b \arccos \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^{2}}{\sqrt{a + b \operatorname{acos}{\left (c x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.086, size = 446, normalized size = 2. \begin{align*} \frac{\sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{6} \sqrt{b \arccos \left (c x\right ) + a} \sqrt{b} i}{2 \,{\left | b \right |}} - \frac{\sqrt{6} \sqrt{b \arccos \left (c x\right ) + a}}{2 \, \sqrt{b}}\right ) e^{\left (\frac{3 \, a i}{b}\right )}}{4 \,{\left (\frac{\sqrt{6} b^{\frac{3}{2}} i}{{\left | b \right |}} + \sqrt{6} \sqrt{b}\right )} c^{3}} + \frac{\sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{2} \sqrt{b \arccos \left (c x\right ) + a} i}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arccos \left (c x\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (\frac{a i}{b}\right )}}{4 \,{\left (\frac{\sqrt{2} b i}{\sqrt{{\left | b \right |}}} + \sqrt{2} \sqrt{{\left | b \right |}}\right )} c^{3}} + \frac{\sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{b \arccos \left (c x\right ) + a} i}{2 \, \sqrt{{\left | b \right |}}} - \frac{\sqrt{2} \sqrt{b \arccos \left (c x\right ) + a} \sqrt{{\left | b \right |}}}{2 \, b}\right ) e^{\left (-\frac{a i}{b}\right )}}{4 \,{\left (\frac{\sqrt{2} b i}{\sqrt{{\left | b \right |}}} - \sqrt{2} \sqrt{{\left | b \right |}}\right )} c^{3}} + \frac{\sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{6} \sqrt{b \arccos \left (c x\right ) + a} \sqrt{b} i}{2 \,{\left | b \right |}} - \frac{\sqrt{6} \sqrt{b \arccos \left (c x\right ) + a}}{2 \, \sqrt{b}}\right ) e^{\left (-\frac{3 \, a i}{b}\right )}}{4 \,{\left (\frac{\sqrt{6} b^{\frac{3}{2}} i}{{\left | b \right |}} - \sqrt{6} \sqrt{b}\right )} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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