Optimal. Leaf size=130 \[ -\frac{2 \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{b^{3/2} c^2}-\frac{2 \sqrt{\pi } \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{b^{3/2} c^2}+\frac{2 x \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \cos ^{-1}(c x)}} \]
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Rubi [A] time = 0.153552, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4632, 3306, 3305, 3351, 3304, 3352} \[ -\frac{2 \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi } \sqrt{b}}\right )}{b^{3/2} c^2}-\frac{2 \sqrt{\pi } \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{b^{3/2} c^2}+\frac{2 x \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \cos ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 4632
Rule 3306
Rule 3305
Rule 3351
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b \cos ^{-1}(c x)\right )^{3/2}} \, dx &=\frac{2 x \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \cos ^{-1}(c x)}}-\frac{2 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{b c^2}\\ &=\frac{2 x \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \cos ^{-1}(c x)}}-\frac{\left (2 \cos \left (\frac{2 a}{b}\right )\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 )}{b c^2}-\frac{\left (2 \sin \left (\frac{2 a}{b}\right )\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 )}{b c^2}\\ &=\frac{2 x \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \cos ^{-1}(c x)}}-\frac{\left (4 \cos \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{b^2 c^2}-\frac{\left (4 \sin \left (\frac{2 a}{b}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \cos ^{-1}(c x)}\right )}{b^2 c^2}\\ &=\frac{2 x \sqrt{1-c^2 x^2}}{b c \sqrt{a+b \cos ^{-1}(c x)}}-\frac{2 \sqrt{\pi } \cos \left (\frac{2 a}{b}\right ) C\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right )}{b^{3/2} c^2}-\frac{2 \sqrt{\pi } S\left (\frac{2 \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b} \sqrt{\pi }}\right ) \sin \left (\frac{2 a}{b}\right )}{b^{3/2} c^2}\\ \end{align*}
Mathematica [A] time = 0.329857, size = 124, normalized size = 0.95 \[ \frac{-2 \sqrt{\pi } \left (\frac{1}{b}\right )^{3/2} \cos \left (\frac{2 a}{b}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right )-2 \sqrt{\pi } \left (\frac{1}{b}\right )^{3/2} \sin \left (\frac{2 a}{b}\right ) S\left (\frac{2 \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{\pi }}\right )+\frac{\sin \left (2 \cos ^{-1}(c x)\right )}{b \sqrt{a+b \cos ^{-1}(c x)}}}{c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.102, size = 142, normalized size = 1.1 \begin{align*}{\frac{1}{b{c}^{2}} \left ( -2\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arccos \left ( cx \right ) }\cos \left ( 2\,{\frac{a}{b}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{a+b\arccos \left ( cx \right ) }}{\sqrt{{b}^{-1}}\sqrt{\pi }b}} \right ) -2\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{a+b\arccos \left ( cx \right ) }\sin \left ( 2\,{\frac{a}{b}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{a+b\arccos \left ( cx \right ) }}{\sqrt{{b}^{-1}}\sqrt{\pi }b}} \right ) +\sin \left ( 2\,{\frac{a+b\arccos \left ( cx \right ) }{b}}-2\,{\frac{a}{b}} \right ) \right ){\frac{1}{\sqrt{a+b\arccos \left ( cx \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}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac{3}{2}}}\,{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}{\left (a + b \operatorname{acos}{\left (c x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (b \arccos \left (c x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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