Integrand size = 8, antiderivative size = 38 \[ \int \frac {x}{\arccos (a x)^2} \, dx=\frac {x \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(2 \arccos (a x))}{a^2} \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4728, 3383} \[ \int \frac {x}{\arccos (a x)^2} \, dx=\frac {x \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(2 \arccos (a x))}{a^2} \]
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Rule 3383
Rule 4728
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arccos (a x)\right )}{a^2} \\ & = \frac {x \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(2 \arccos (a x))}{a^2} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {x}{\arccos (a x)^2} \, dx=\frac {\frac {a x \sqrt {1-a^2 x^2}}{\arccos (a x)}-\operatorname {CosIntegral}(2 \arccos (a x))}{a^2} \]
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Time = 0.61 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{2 \arccos \left (a x \right )}-\operatorname {Ci}\left (2 \arccos \left (a x \right )\right )}{a^{2}}\) | \(30\) |
default | \(\frac {\frac {\sin \left (2 \arccos \left (a x \right )\right )}{2 \arccos \left (a x \right )}-\operatorname {Ci}\left (2 \arccos \left (a x \right )\right )}{a^{2}}\) | \(30\) |
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\[ \int \frac {x}{\arccos (a x)^2} \, dx=\int { \frac {x}{\arccos \left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {x}{\arccos (a x)^2} \, dx=\int \frac {x}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x}{\arccos (a x)^2} \, dx=\int { \frac {x}{\arccos \left (a x\right )^{2}} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \frac {x}{\arccos (a x)^2} \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} x}{a \arccos \left (a x\right )} - \frac {\operatorname {Ci}\left (2 \, \arccos \left (a x\right )\right )}{a^{2}} \]
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Timed out. \[ \int \frac {x}{\arccos (a x)^2} \, dx=\int \frac {x}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \]
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