Optimal. Leaf size=28 \[ -\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a^2} \]
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Rubi [A] time = 0.0402895, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4636, 4406, 12, 3305, 3351} \[ -\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 4636
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{\cos ^{-1}(a x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^2}\\ &=-\frac{\operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{a^2}\\ &=-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0230338, size = 28, normalized size = 1. \[ -\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 21, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{\pi }}{2\,{a}^{2}}{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \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 \frac{x}{\sqrt{\operatorname{acos}{\left (a x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31227, size = 69, normalized size = 2.46 \begin{align*} \frac{\sqrt{\pi } i \operatorname{erf}\left ({\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{4 \, a^{2}{\left (i - 1\right )}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{4 \, a^{2}{\left (i - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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