Optimal. Leaf size=54 \[ -\frac{\text{CosIntegral}\left (\cos ^{-1}(a x)\right )}{4 a^3}-\frac{3 \text{CosIntegral}\left (3 \cos ^{-1}(a x)\right )}{4 a^3}+\frac{x^2 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)} \]
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Rubi [A] time = 0.0461439, antiderivative size = 54, 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} \[ -\frac{\text{CosIntegral}\left (\cos ^{-1}(a x)\right )}{4 a^3}-\frac{3 \text{CosIntegral}\left (3 \cos ^{-1}(a x)\right )}{4 a^3}+\frac{x^2 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4632
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^2}{\cos ^{-1}(a x)^2} \, dx &=\frac{x^2 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \left (-\frac{\cos (x)}{4 x}-\frac{3 \cos (3 x)}{4 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=\frac{x^2 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}\\ &=\frac{x^2 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{\text{Ci}\left (\cos ^{-1}(a x)\right )}{4 a^3}-\frac{3 \text{Ci}\left (3 \cos ^{-1}(a x)\right )}{4 a^3}\\ \end{align*}
Mathematica [A] time = 0.122994, size = 50, normalized size = 0.93 \[ -\frac{-\frac{4 a^2 x^2 \sqrt{1-a^2 x^2}}{\cos ^{-1}(a x)}+\text{CosIntegral}\left (\cos ^{-1}(a x)\right )+3 \text{CosIntegral}\left (3 \cos ^{-1}(a x)\right )}{4 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 57, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{\sin \left ( 3\,\arccos \left ( ax \right ) \right ) }{4\,\arccos \left ( ax \right ) }}-{\frac{3\,{\it Ci} \left ( 3\,\arccos \left ( ax \right ) \right ) }{4}}+{\frac{1}{4\,\arccos \left ( ax \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{\it Ci} \left ( \arccos \left ( ax \right ) \right ) }{4}} \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^{2}}{\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^{2}}{\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 = 1.17569, size = 65, normalized size = 1.2 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{a \arccos \left (a x\right )} - \frac{3 \, \operatorname{Ci}\left (3 \, \arccos \left (a x\right )\right )}{4 \, a^{3}} - \frac{\operatorname{Ci}\left (\arccos \left (a x\right )\right )}{4 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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