Optimal. Leaf size=82 \[ -\frac {5 \text {Ci}\left (\cos ^{-1}(a x)\right )}{64 a^7}-\frac {27 \text {Ci}\left (3 \cos ^{-1}(a x)\right )}{64 a^7}-\frac {25 \text {Ci}\left (5 \cos ^{-1}(a x)\right )}{64 a^7}-\frac {7 \text {Ci}\left (7 \cos ^{-1}(a x)\right )}{64 a^7}+\frac {x^6 \sqrt {1-a^2 x^2}}{a \cos ^{-1}(a x)} \]
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Rubi [A] time = 0.08, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4632, 3302} \[ -\frac {5 \text {CosIntegral}\left (\cos ^{-1}(a x)\right )}{64 a^7}-\frac {27 \text {CosIntegral}\left (3 \cos ^{-1}(a x)\right )}{64 a^7}-\frac {25 \text {CosIntegral}\left (5 \cos ^{-1}(a x)\right )}{64 a^7}-\frac {7 \text {CosIntegral}\left (7 \cos ^{-1}(a x)\right )}{64 a^7}+\frac {x^6 \sqrt {1-a^2 x^2}}{a \cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 4632
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*}
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Mathematica [A] time = 0.17, size = 86, normalized size = 1.05 \[ -\frac {-64 a^6 x^6 \sqrt {1-a^2 x^2}+5 \cos ^{-1}(a x) \text {Ci}\left (\cos ^{-1}(a x)\right )+27 \cos ^{-1}(a x) \text {Ci}\left (3 \cos ^{-1}(a x)\right )+25 \cos ^{-1}(a x) \text {Ci}\left (5 \cos ^{-1}(a x)\right )+7 \cos ^{-1}(a x) \text {Ci}\left (7 \cos ^{-1}(a x)\right )}{64 a^7 \cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{6}}{\arccos \left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 72, normalized size = 0.88 \[ \frac {\sqrt {-a^{2} x^{2} + 1} x^{6}}{a \arccos \left (a x\right )} - \frac {7 \, \operatorname {Ci}\left (7 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {25 \, \operatorname {Ci}\left (5 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {27 \, \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {5 \, \operatorname {Ci}\left (\arccos \left (a x\right )\right )}{64 \, a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 105, normalized size = 1.28 \[ \frac {\frac {9 \sin \left (3 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {27 \Ci \left (3 \arccos \left (a x \right )\right )}{64}+\frac {5 \sin \left (5 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {25 \Ci \left (5 \arccos \left (a x \right )\right )}{64}+\frac {\sin \left (7 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {7 \Ci \left (7 \arccos \left (a x \right )\right )}{64}+\frac {5 \sqrt {-a^{2} x^{2}+1}}{64 \arccos \left (a x \right )}-\frac {5 \Ci \left (\arccos \left (a x \right )\right )}{64}}{a^{7}} \]
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 \[ \int \frac {x^6}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \]
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 \[ \int \frac {x^{6}}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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