Integrand size = 10, antiderivative size = 54 \[ \int \frac {x^2}{\arccos (a x)^2} \, dx=\frac {x^2 \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(\arccos (a x))}{4 a^3}-\frac {3 \operatorname {CosIntegral}(3 \arccos (a x))}{4 a^3} \]
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Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4728, 3383} \[ \int \frac {x^2}{\arccos (a x)^2} \, dx=-\frac {\operatorname {CosIntegral}(\arccos (a x))}{4 a^3}-\frac {3 \operatorname {CosIntegral}(3 \arccos (a x))}{4 a^3}+\frac {x^2 \sqrt {1-a^2 x^2}}{a \arccos (a x)} \]
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Rule 3383
Rule 4728
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \sqrt {1-a^2 x^2}}{a \arccos (a x)}+\frac {\text {Subst}\left (\int \left (-\frac {\cos (x)}{4 x}-\frac {3 \cos (3 x)}{4 x}\right ) \, dx,x,\arccos (a x)\right )}{a^3} \\ & = \frac {x^2 \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arccos (a x)\right )}{4 a^3}-\frac {3 \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arccos (a x)\right )}{4 a^3} \\ & = \frac {x^2 \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(\arccos (a x))}{4 a^3}-\frac {3 \operatorname {CosIntegral}(3 \arccos (a x))}{4 a^3} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93 \[ \int \frac {x^2}{\arccos (a x)^2} \, dx=-\frac {-\frac {4 a^2 x^2 \sqrt {1-a^2 x^2}}{\arccos (a x)}+\operatorname {CosIntegral}(\arccos (a x))+3 \operatorname {CosIntegral}(3 \arccos (a x))}{4 a^3} \]
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Time = 0.68 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (3 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}-\frac {3 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{4}+\frac {\sqrt {-a^{2} x^{2}+1}}{4 \arccos \left (a x \right )}-\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{4}}{a^{3}}\) | \(57\) |
default | \(\frac {\frac {\sin \left (3 \arccos \left (a x \right )\right )}{4 \arccos \left (a x \right )}-\frac {3 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{4}+\frac {\sqrt {-a^{2} x^{2}+1}}{4 \arccos \left (a x \right )}-\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{4}}{a^{3}}\) | \(57\) |
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\[ \int \frac {x^2}{\arccos (a x)^2} \, dx=\int { \frac {x^{2}}{\arccos \left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {x^2}{\arccos (a x)^2} \, dx=\int \frac {x^{2}}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x^2}{\arccos (a x)^2} \, dx=\int { \frac {x^{2}}{\arccos \left (a x\right )^{2}} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\arccos (a x)^2} \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a \arccos \left (a x\right )} - \frac {3 \, \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{4 \, a^{3}} - \frac {\operatorname {Ci}\left (\arccos \left (a x\right )\right )}{4 \, a^{3}} \]
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Timed out. \[ \int \frac {x^2}{\arccos (a x)^2} \, dx=\int \frac {x^2}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \]
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