Optimal. Leaf size=56 \[ -\frac{\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{2 a^4}-\frac{\text{CosIntegral}\left (4 \cos ^{-1}(a x)\right )}{2 a^4}+\frac{x^3 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)} \]
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Rubi [A] time = 0.0502264, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4632, 3302} \[ -\frac{\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{2 a^4}-\frac{\text{CosIntegral}\left (4 \cos ^{-1}(a x)\right )}{2 a^4}+\frac{x^3 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4632
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^3}{\cos ^{-1}(a x)^2} \, dx &=\frac{x^3 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \left (-\frac{\cos (2 x)}{2 x}-\frac{\cos (4 x)}{2 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=\frac{x^3 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^4}-\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^4}\\ &=\frac{x^3 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{Ci}\left (2 \cos ^{-1}(a x)\right )}{2 a^4}-\frac{\text{Ci}\left (4 \cos ^{-1}(a x)\right )}{2 a^4}\\ \end{align*}
Mathematica [A] time = 0.138831, size = 50, normalized size = 0.89 \[ -\frac{-\frac{2 a^3 x^3 \sqrt{1-a^2 x^2}}{\cos ^{-1}(a x)}+\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )+\text{CosIntegral}\left (4 \cos ^{-1}(a x)\right )}{2 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 54, normalized size = 1. \begin{align*}{\frac{1}{{a}^{4}} \left ({\frac{\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{4\,\arccos \left ( ax \right ) }}-{\frac{{\it Ci} \left ( 2\,\arccos \left ( ax \right ) \right ) }{2}}+{\frac{\sin \left ( 4\,\arccos \left ( ax \right ) \right ) }{8\,\arccos \left ( ax \right ) }}-{\frac{{\it Ci} \left ( 4\,\arccos \left ( ax \right ) \right ) }{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{\arccos \left (a x\right )^{2}}, x\right ) \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}^{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 = 1.18811, size = 68, normalized size = 1.21 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} x^{3}}{a \arccos \left (a x\right )} - \frac{\operatorname{Ci}\left (4 \, \arccos \left (a x\right )\right )}{2 \, a^{4}} - \frac{\operatorname{Ci}\left (2 \, \arccos \left (a x\right )\right )}{2 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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