Integrand size = 10, antiderivative size = 82 \[ \int \frac {x^6}{\arccos (a x)^2} \, dx=\frac {x^6 \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {5 \operatorname {CosIntegral}(\arccos (a x))}{64 a^7}-\frac {27 \operatorname {CosIntegral}(3 \arccos (a x))}{64 a^7}-\frac {25 \operatorname {CosIntegral}(5 \arccos (a x))}{64 a^7}-\frac {7 \operatorname {CosIntegral}(7 \arccos (a x))}{64 a^7} \]
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Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4728, 3383} \[ \int \frac {x^6}{\arccos (a x)^2} \, dx=-\frac {5 \operatorname {CosIntegral}(\arccos (a x))}{64 a^7}-\frac {27 \operatorname {CosIntegral}(3 \arccos (a x))}{64 a^7}-\frac {25 \operatorname {CosIntegral}(5 \arccos (a x))}{64 a^7}-\frac {7 \operatorname {CosIntegral}(7 \arccos (a x))}{64 a^7}+\frac {x^6 \sqrt {1-a^2 x^2}}{a \arccos (a x)} \]
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Rule 3383
Rule 4728
Rubi steps \begin{align*} \text {integral}& = \frac {x^6 \sqrt {1-a^2 x^2}}{a \arccos (a x)}+\frac {\text {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,\arccos (a x)\right )}{a^7} \\ & = \frac {x^6 \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {5 \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arccos (a x)\right )}{64 a^7}-\frac {7 \text {Subst}\left (\int \frac {\cos (7 x)}{x} \, dx,x,\arccos (a x)\right )}{64 a^7}-\frac {25 \text {Subst}\left (\int \frac {\cos (5 x)}{x} \, dx,x,\arccos (a x)\right )}{64 a^7}-\frac {27 \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arccos (a x)\right )}{64 a^7} \\ & = \frac {x^6 \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {5 \operatorname {CosIntegral}(\arccos (a x))}{64 a^7}-\frac {27 \operatorname {CosIntegral}(3 \arccos (a x))}{64 a^7}-\frac {25 \operatorname {CosIntegral}(5 \arccos (a x))}{64 a^7}-\frac {7 \operatorname {CosIntegral}(7 \arccos (a x))}{64 a^7} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05 \[ \int \frac {x^6}{\arccos (a x)^2} \, dx=-\frac {-64 a^6 x^6 \sqrt {1-a^2 x^2}+5 \arccos (a x) \operatorname {CosIntegral}(\arccos (a x))+27 \arccos (a x) \operatorname {CosIntegral}(3 \arccos (a x))+25 \arccos (a x) \operatorname {CosIntegral}(5 \arccos (a x))+7 \arccos (a x) \operatorname {CosIntegral}(7 \arccos (a x))}{64 a^7 \arccos (a x)} \]
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Time = 0.88 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(\frac {\frac {9 \sin \left (3 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {27 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{64}+\frac {5 \sin \left (5 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {25 \,\operatorname {Ci}\left (5 \arccos \left (a x \right )\right )}{64}+\frac {\sin \left (7 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {7 \,\operatorname {Ci}\left (7 \arccos \left (a x \right )\right )}{64}+\frac {5 \sqrt {-a^{2} x^{2}+1}}{64 \arccos \left (a x \right )}-\frac {5 \,\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{64}}{a^{7}}\) | \(105\) |
default | \(\frac {\frac {9 \sin \left (3 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {27 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{64}+\frac {5 \sin \left (5 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {25 \,\operatorname {Ci}\left (5 \arccos \left (a x \right )\right )}{64}+\frac {\sin \left (7 \arccos \left (a x \right )\right )}{64 \arccos \left (a x \right )}-\frac {7 \,\operatorname {Ci}\left (7 \arccos \left (a x \right )\right )}{64}+\frac {5 \sqrt {-a^{2} x^{2}+1}}{64 \arccos \left (a x \right )}-\frac {5 \,\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{64}}{a^{7}}\) | \(105\) |
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\[ \int \frac {x^6}{\arccos (a x)^2} \, dx=\int { \frac {x^{6}}{\arccos \left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {x^6}{\arccos (a x)^2} \, dx=\int \frac {x^{6}}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^6}{\arccos (a x)^2} \, dx=\int { \frac {x^{6}}{\arccos \left (a x\right )^{2}} \,d x } \]
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none
Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.88 \[ \int \frac {x^6}{\arccos (a x)^2} \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} x^{6}}{a \arccos \left (a x\right )} - \frac {7 \, \operatorname {Ci}\left (7 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {25 \, \operatorname {Ci}\left (5 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {27 \, \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{64 \, a^{7}} - \frac {5 \, \operatorname {Ci}\left (\arccos \left (a x\right )\right )}{64 \, a^{7}} \]
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Timed out. \[ \int \frac {x^6}{\arccos (a x)^2} \, dx=\int \frac {x^6}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \]
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