Optimal. Leaf size=71 \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{2 a^3}-\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{2 a^3} \]
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Rubi [A] time = 0.079498, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4636, 4406, 3305, 3351} \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{2 a^3}-\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 4636
Rule 4406
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{\cos ^{-1}(a x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 \sqrt{x}}+\frac{\sin (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}-\frac{\operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac{\operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{2 a^3}-\frac{\operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{2 a^3}\\ &=-\frac{\sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{2 a^3}-\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{2 a^3}\\ \end{align*}
Mathematica [C] time = 0.0819761, size = 126, normalized size = 1.77 \[ -\frac{-3 \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \cos ^{-1}(a x)\right )-3 \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \cos ^{-1}(a x)\right )-\sqrt{3} \left (\sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \cos ^{-1}(a x)\right )+\sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \cos ^{-1}(a x)\right )\right )}{24 a^3 \sqrt{\cos ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.069, size = 50, normalized size = 0.7 \begin{align*} -{\frac{\sqrt{2}\sqrt{\pi }}{12\,{a}^{3}} \left ( \sqrt{3}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}}{\sqrt{\pi }}\sqrt{\arccos \left ( ax \right ) }} \right ) +3\,{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \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^{2}}{\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 = 1.35129, size = 182, normalized size = 2.56 \begin{align*} \frac{\sqrt{6} \sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{6} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{24 \, a^{3}{\left (i - 1\right )}} + \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{2} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{8 \, a^{3}{\left (i - 1\right )}} - \frac{\sqrt{6} \sqrt{\pi } \operatorname{erf}\left (\frac{\sqrt{6} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{24 \, a^{3}{\left (i - 1\right )}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{8 \, a^{3}{\left (i - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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