Optimal. Leaf size=82 \[ -\frac{5 \text{CosIntegral}\left (\cos ^{-1}(a x)\right )}{64 a^7}-\frac{27 \text{CosIntegral}\left (3 \cos ^{-1}(a x)\right )}{64 a^7}-\frac{25 \text{CosIntegral}\left (5 \cos ^{-1}(a x)\right )}{64 a^7}-\frac{7 \text{CosIntegral}\left (7 \cos ^{-1}(a x)\right )}{64 a^7}+\frac{x^6 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)} \]
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Rubi [A] time = 0.0789818, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4632, 3302} \[ -\frac{5 \text{CosIntegral}\left (\cos ^{-1}(a x)\right )}{64 a^7}-\frac{27 \text{CosIntegral}\left (3 \cos ^{-1}(a x)\right )}{64 a^7}-\frac{25 \text{CosIntegral}\left (5 \cos ^{-1}(a x)\right )}{64 a^7}-\frac{7 \text{CosIntegral}\left (7 \cos ^{-1}(a x)\right )}{64 a^7}+\frac{x^6 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4632
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^6}{\cos ^{-1}(a x)^2} \, dx &=\frac{x^6 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \left (-\frac{5 \cos (x)}{64 x}-\frac{27 \cos (3 x)}{64 x}-\frac{25 \cos (5 x)}{64 x}-\frac{7 \cos (7 x)}{64 x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^7}\\ &=\frac{x^6 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{5 \operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}-\frac{7 \operatorname{Subst}\left (\int \frac{\cos (7 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}-\frac{25 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}-\frac{27 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\cos ^{-1}(a x)\right )}{64 a^7}\\ &=\frac{x^6 \sqrt{1-a^2 x^2}}{a \cos ^{-1}(a x)}-\frac{5 \text{Ci}\left (\cos ^{-1}(a x)\right )}{64 a^7}-\frac{27 \text{Ci}\left (3 \cos ^{-1}(a x)\right )}{64 a^7}-\frac{25 \text{Ci}\left (5 \cos ^{-1}(a x)\right )}{64 a^7}-\frac{7 \text{Ci}\left (7 \cos ^{-1}(a x)\right )}{64 a^7}\\ \end{align*}
Mathematica [A] time = 0.157698, size = 86, normalized size = 1.05 \[ -\frac{-64 a^6 x^6 \sqrt{1-a^2 x^2}+5 \cos ^{-1}(a x) \text{CosIntegral}\left (\cos ^{-1}(a x)\right )+27 \cos ^{-1}(a x) \text{CosIntegral}\left (3 \cos ^{-1}(a x)\right )+25 \cos ^{-1}(a x) \text{CosIntegral}\left (5 \cos ^{-1}(a x)\right )+7 \cos ^{-1}(a x) \text{CosIntegral}\left (7 \cos ^{-1}(a x)\right )}{64 a^7 \cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 105, normalized size = 1.3 \begin{align*}{\frac{1}{{a}^{7}} \left ({\frac{9\,\sin \left ( 3\,\arccos \left ( ax \right ) \right ) }{64\,\arccos \left ( ax \right ) }}-{\frac{27\,{\it Ci} \left ( 3\,\arccos \left ( ax \right ) \right ) }{64}}+{\frac{5\,\sin \left ( 5\,\arccos \left ( ax \right ) \right ) }{64\,\arccos \left ( ax \right ) }}-{\frac{25\,{\it Ci} \left ( 5\,\arccos \left ( ax \right ) \right ) }{64}}+{\frac{\sin \left ( 7\,\arccos \left ( ax \right ) \right ) }{64\,\arccos \left ( ax \right ) }}-{\frac{7\,{\it Ci} \left ( 7\,\arccos \left ( ax \right ) \right ) }{64}}+{\frac{5}{64\,\arccos \left ( ax \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{5\,{\it Ci} \left ( \arccos \left ( ax \right ) \right ) }{64}} \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^{6}}{\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^{6}}{\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 = 1.18624, size = 97, normalized size = 1.18 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} x^{6}}{a \arccos \left (a x\right )} - \frac{7 \, \operatorname{Ci}\left (7 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac{25 \, \operatorname{Ci}\left (5 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac{27 \, \operatorname{Ci}\left (3 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac{5 \, \operatorname{Ci}\left (\arccos \left (a x\right )\right )}{64 \, a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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