Optimal. Leaf size=38 \[ \frac{x \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{a^2} \]
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Rubi [A] time = 0.0244747, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4632, 3302} \[ \frac{x \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 4632
Rule 3302
Rubi steps
\begin{align*} \int \frac{x}{\cos ^{-1}(a x)^2} \, dx &=\frac{x \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 )}{a^2}\\ &=\frac{x \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{Ci}\left (2 \cos ^{-1}(a x)\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0854411, size = 37, normalized size = 0.97 \[ \frac{\frac{a x \sqrt{1-a^2 x^2}}{\cos ^{-1}(a x)}-\text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 30, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{2\,\arccos \left ( ax \right ) }}-{\it Ci} \left ( 2\,\arccos \left ( ax \right ) \right ) \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}{\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}{\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 = 1.2191, size = 49, normalized size = 1.29 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} x}{a \arccos \left (a x\right )} - \frac{\operatorname{Ci}\left (2 \, \arccos \left (a x\right )\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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