Optimal. Leaf size=99 \[ \frac{\sqrt{\pi } \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{2 \sqrt{b} c^2}-\frac{\sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{2 \sqrt{b} c^2} \]
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Rubi [A] time = 0.17077, 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 = {4636, 4406, 12, 3306, 3305, 3351, 3304, 3352} \[ \frac{\sqrt{\pi } \sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{2 \sqrt{b} c^2}-\frac{\sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{2 \sqrt{b} c^2} \]
Antiderivative was successfully verified.
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Rule 4636
Rule 4406
Rule 12
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{a+b \cos ^{-1}(c x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{c^2}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{c^2}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-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,\cos ^{-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,\cos ^{-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 \cos ^{-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 \cos ^{-1}(c x)}\right )}{b c^2}\\ &=-\frac{\sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{2 \sqrt{b} c^2}+\frac{\sqrt{\pi } C\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{2 \sqrt{b} c^2}\\ \end{align*}
Mathematica [A] time = 0.159688, size = 91, normalized size = 0.92 \[ -\frac{\sqrt{\pi } \sqrt{\frac{1}{b}} \left (\cos \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right )-\sin \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right )\right )}{2 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 80, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{\pi }}{2\,{c}^{2}}\sqrt{{b}^{-1}} \left ( \cos \left ( 2\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{a+b\arccos \left ( cx \right ) }}{\sqrt{\pi }\sqrt{{b}^{-1}}b}} \right ) -\sin \left ( 2\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\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}{\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}{\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 = 1.82085, size = 188, normalized size = 1.9 \begin{align*} \frac{\sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{b \arccos \left (c x\right ) + a} \sqrt{b} i}{{\left | b \right |}} - \frac{\sqrt{b \arccos \left (c x\right ) + a}}{\sqrt{b}}\right ) e^{\left (-\frac{2 \, a i}{b}\right )}}{4 \,{\left (\frac{b^{\frac{3}{2}} i}{{\left | b \right |}} - \sqrt{b}\right )} c^{2}} + \frac{\sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{b \arccos \left (c x\right ) + a} \sqrt{b} i}{{\left | b \right |}} - \frac{\sqrt{b \arccos \left (c x\right ) + a}}{\sqrt{b}}\right ) e^{\left (\frac{2 \, a i}{b}\right )}}{4 \, \sqrt{b} c^{2}{\left (\frac{b i}{{\left | b \right |}} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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