Optimal. Leaf size=70 \[ -\frac {5 \text {Ci}\left (2 \cos ^{-1}(a x)\right )}{16 a^6}-\frac {\text {Ci}\left (4 \cos ^{-1}(a x)\right )}{2 a^6}-\frac {3 \text {Ci}\left (6 \cos ^{-1}(a x)\right )}{16 a^6}+\frac {x^5 \sqrt {1-a^2 x^2}}{a \cos ^{-1}(a x)} \]
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Rubi [A] time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 3302
Rule 4632
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*}
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Mathematica [A] time = 0.17, size = 63, normalized size = 0.90 \[ -\frac {-\frac {16 a^5 x^5 \sqrt {1-a^2 x^2}}{\cos ^{-1}(a x)}+5 \text {Ci}\left (2 \cos ^{-1}(a x)\right )+8 \text {Ci}\left (4 \cos ^{-1}(a x)\right )+3 \text {Ci}\left (6 \cos ^{-1}(a x)\right )}{16 a^6} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{5}}{\arccos \left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 62, normalized size = 0.89 \[ \frac {\sqrt {-a^{2} x^{2} + 1} x^{5}}{a \arccos \left (a x\right )} - \frac {3 \, \operatorname {Ci}\left (6 \, \arccos \left (a x\right )\right )}{16 \, a^{6}} - \frac {\operatorname {Ci}\left (4 \, \arccos \left (a x\right )\right )}{2 \, a^{6}} - \frac {5 \, \operatorname {Ci}\left (2 \, \arccos \left (a x\right )\right )}{16 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 78, normalized size = 1.11 \[ \frac {\frac {5 \sin \left (2 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )}-\frac {5 \Ci \left (2 \arccos \left (a x \right )\right )}{16}+\frac {\sin \left (4 \arccos \left (a x \right )\right )}{8 \arccos \left (a x \right )}-\frac {\Ci \left (4 \arccos \left (a x \right )\right )}{2}+\frac {\sin \left (6 \arccos \left (a x \right )\right )}{32 \arccos \left (a x \right )}-\frac {3 \Ci \left (6 \arccos \left (a x \right )\right )}{16}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^5}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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