Optimal. Leaf size=65 \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^4}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{4 a^4} \]
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Rubi [A] time = 0.0777967, antiderivative size = 65, 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} \[ -\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^4}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{4 a^4} \]
Antiderivative was successfully verified.
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Rule 4636
Rule 4406
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{\cos ^{-1}(a x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\cos ^3(x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{\sin (2 x)}{4 \sqrt{x}}+\frac{\sin (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}-\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{\operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{4 a^4}-\frac{\operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{2 a^4}\\ &=-\frac{\sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{8 a^4}-\frac{\sqrt{\pi } S\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{4 a^4}\\ \end{align*}
Mathematica [C] time = 0.0678617, size = 130, normalized size = 2. \[ -\frac{-2 \sqrt{2} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \cos ^{-1}(a x)\right )-2 \sqrt{2} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \cos ^{-1}(a x)\right )-\sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \cos ^{-1}(a x)\right )-\sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \cos ^{-1}(a x)\right )}{32 a^4 \sqrt{\cos ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.07, size = 43, normalized size = 0.7 \begin{align*} -{\frac{\sqrt{\pi }}{16\,{a}^{4}} \left ( \sqrt{2}{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +4\,{\it FresnelS} \left ( 2\,{\frac{\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^{3}}{\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 = 1.3581, size = 153, normalized size = 2.35 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (\sqrt{2}{\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{32 \, a^{4}{\left (i - 1\right )}} + \frac{\sqrt{\pi } i \operatorname{erf}\left ({\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{8 \, a^{4}{\left (i - 1\right )}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\sqrt{2}{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{32 \, a^{4}{\left (i - 1\right )}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{8 \, a^{4}{\left (i - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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