Optimal. Leaf size=91 \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^4}-\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^4}+\frac{2 x^3 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}} \]
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Rubi [A] time = 0.0651208, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4632, 3304, 3352} \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^4}-\frac{\sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^4}+\frac{2 x^3 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4632
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{x^3}{\cos ^{-1}(a x)^{3/2}} \, dx &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{\cos (2 x)}{2 \sqrt{x}}-\frac{\cos (4 x)}{2 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}-\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}-\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}-\frac{2 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{a^4}-\frac{2 \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{a^4}\\ &=\frac{2 x^3 \sqrt{1-a^2 x^2}}{a \sqrt{\cos ^{-1}(a x)}}-\frac{\sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^4}-\frac{\sqrt{\pi } C\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{a^4}\\ \end{align*}
Mathematica [C] time = 0.39551, size = 154, normalized size = 1.69 \[ \frac{i \sqrt{2} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 i \cos ^{-1}(a x)\right )-i \sqrt{2} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},2 i \cos ^{-1}(a x)\right )+i \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 i \cos ^{-1}(a x)\right )-i \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},4 i \cos ^{-1}(a x)\right )+2 \sin \left (2 \cos ^{-1}(a x)\right )+\sin \left (4 \cos ^{-1}(a x)\right )}{4 a^4 \sqrt{\cos ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.074, size = 81, normalized size = 0.9 \begin{align*}{\frac{1}{4\,{a}^{4}} \left ( -2\,\sqrt{2}\sqrt{\pi }\sqrt{\arccos \left ( ax \right ) }{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -4\,\sqrt{\pi }\sqrt{\arccos \left ( ax \right ) }{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +2\,\sin \left ( 2\,\arccos \left ( ax \right ) \right ) +\sin \left ( 4\,\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 \frac{x^{3}}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\arccos \left (a x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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