Optimal. Leaf size=70 \[ -\frac{5 \text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{16 a^6}-\frac{\text{CosIntegral}\left (4 \cos ^{-1}(a x)\right )}{2 a^6}-\frac{3 \text{CosIntegral}\left (6 \cos ^{-1}(a x)\right )}{16 a^6}+\frac{x^5 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)} \]
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Rubi [A] time = 0.0640758, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4632, 3302} \[ -\frac{5 \text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )}{16 a^6}-\frac{\text{CosIntegral}\left (4 \cos ^{-1}(a x)\right )}{2 a^6}-\frac{3 \text{CosIntegral}\left (6 \cos ^{-1}(a x)\right )}{16 a^6}+\frac{x^5 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4632
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^5}{\cos ^{-1}(a x)^2} \, dx &=\frac{x^5 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \left (-\frac{5 \cos (2 x)}{16 x}-\frac{\cos (4 x)}{2 x}-\frac{3 \cos (6 x)}{16 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^6}\\ &=\frac{x^5 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (6 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^6}-\frac{5 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{16 a^6}-\frac{\operatorname{Subst}\left (\int \frac{\cos (4 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^6}\\ &=\frac{x^5 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{5 \text{Ci}\left (2 \cos ^{-1}(a x)\right )}{16 a^6}-\frac{\text{Ci}\left (4 \cos ^{-1}(a x)\right )}{2 a^6}-\frac{3 \text{Ci}\left (6 \cos ^{-1}(a x)\right )}{16 a^6}\\ \end{align*}
Mathematica [A] time = 0.162543, size = 63, normalized size = 0.9 \[ -\frac{-\frac{16 a^5 x^5 \sqrt{1-a^2 x^2}}{\cos ^{-1}(a x)}+5 \text{CosIntegral}\left (2 \cos ^{-1}(a x)\right )+8 \text{CosIntegral}\left (4 \cos ^{-1}(a x)\right )+3 \text{CosIntegral}\left (6 \cos ^{-1}(a x)\right )}{16 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 78, normalized size = 1.1 \begin{align*}{\frac{1}{{a}^{6}} \left ({\frac{5\,\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{32\,\arccos \left ( ax \right ) }}-{\frac{5\,{\it Ci} \left ( 2\,\arccos \left ( ax \right ) \right ) }{16}}+{\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}}+{\frac{\sin \left ( 6\,\arccos \left ( ax \right ) \right ) }{32\,\arccos \left ( ax \right ) }}-{\frac{3\,{\it Ci} \left ( 6\,\arccos \left ( ax \right ) \right ) }{16}} \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^{5}}{\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^{5}}{\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 = 1.16818, size = 84, normalized size = 1.2 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} x^{5}}{a \arccos \left (a x\right )} - \frac{3 \, \operatorname{Ci}\left (6 \, \arccos \left (a x\right )\right )}{16 \, a^{6}} - \frac{\operatorname{Ci}\left (4 \, \arccos \left (a x\right )\right )}{2 \, a^{6}} - \frac{5 \, \operatorname{Ci}\left (2 \, \arccos \left (a x\right )\right )}{16 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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