Optimal. Leaf size=89 \[ \frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{8 a}-\frac{\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)^{3/2} \]
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Rubi [A] time = 0.17861, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {4630, 4708, 4642, 4636, 4406, 12, 3305, 3351} \[ \frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{8 a}-\frac{\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 4630
Rule 4708
Rule 4642
Rule 4636
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int x \cos ^{-1}(a x)^{3/2} \, dx &=\frac{1}{2} x^2 \cos ^{-1}(a x)^{3/2}+\frac{1}{4} (3 a) \int \frac{x^2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{8 a}+\frac{1}{2} x^2 \cos ^{-1}(a x)^{3/2}-\frac{3}{16} \int \frac{x}{\sqrt{\cos ^{-1}(a x)}} \, dx+\frac{3 \int \frac{\sqrt{\cos ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{8 a}\\ &=-\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{8 a}-\frac{\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^2}\\ &=-\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{8 a}-\frac{\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^2}\\ &=-\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{8 a}-\frac{\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^2}\\ &=-\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{8 a}-\frac{\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{16 a^2}\\ &=-\frac{3 x \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{8 a}-\frac{\cos ^{-1}(a x)^{3/2}}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x)^{3/2}+\frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{32 a^2}\\ \end{align*}
Mathematica [A] time = 0.0657384, size = 64, normalized size = 0.72 \[ \frac{3 \sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )-2 \sqrt{\cos ^{-1}(a x)} \left (3 \sin \left (2 \cos ^{-1}(a x)\right )-4 \cos ^{-1}(a x) \cos \left (2 \cos ^{-1}(a x)\right )\right )}{32 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 64, normalized size = 0.7 \begin{align*}{\frac{1}{32\,{a}^{2}} \left ( 8\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\cos \left ( 2\,\arccos \left ( ax \right ) \right ) +3\,\sqrt{\pi }\sqrt{\arccos \left ( ax \right ) }{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -6\,\arccos \left ( ax \right ) \sin \left ( 2\,\arccos \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arccos \left ( ax \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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 x \operatorname{acos}^{\frac{3}{2}}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28056, size = 174, normalized size = 1.96 \begin{align*} \frac{3 \, i \sqrt{\arccos \left (a x\right )} e^{\left (2 \, i \arccos \left (a x\right )\right )}}{32 \, a^{2}} + \frac{\arccos \left (a x\right )^{\frac{3}{2}} e^{\left (2 \, i \arccos \left (a x\right )\right )}}{8 \, a^{2}} - \frac{3 \, i \sqrt{\arccos \left (a x\right )} e^{\left (-2 \, i \arccos \left (a x\right )\right )}}{32 \, a^{2}} + \frac{\arccos \left (a x\right )^{\frac{3}{2}} e^{\left (-2 \, i \arccos \left (a x\right )\right )}}{8 \, a^{2}} - \frac{3 \, \sqrt{\pi } i \operatorname{erf}\left ({\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{64 \, a^{2}{\left (i - 1\right )}} + \frac{3 \, \sqrt{\pi } \operatorname{erf}\left (-{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{64 \, a^{2}{\left (i - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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