Integrand size = 6, antiderivative size = 35 \[ \int \frac {1}{\arccos (a x)^2} \, dx=\frac {\sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(\arccos (a x))}{a} \]
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Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4718, 4810, 3383} \[ \int \frac {1}{\arccos (a x)^2} \, dx=\frac {\sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(\arccos (a x))}{a} \]
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Rule 3383
Rule 4718
Rule 4810
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-a^2 x^2}}{a \arccos (a x)}+a \int \frac {x}{\sqrt {1-a^2 x^2} \arccos (a x)} \, dx \\ & = \frac {\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 )}{a} \\ & = \frac {\sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(\arccos (a x))}{a} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\arccos (a x)^2} \, dx=\frac {\sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(\arccos (a x))}{a} \]
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Time = 0.53 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{\arccos \left (a x \right )}-\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{a}\) | \(32\) |
default | \(\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{\arccos \left (a x \right )}-\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{a}\) | \(32\) |
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\[ \int \frac {1}{\arccos (a x)^2} \, dx=\int { \frac {1}{\arccos \left (a x\right )^{2}} \,d x } \]
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\[ \int \frac {1}{\arccos (a x)^2} \, dx=\int \frac {1}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \]
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\[ \int \frac {1}{\arccos (a x)^2} \, dx=\int { \frac {1}{\arccos \left (a x\right )^{2}} \,d x } \]
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none
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\arccos (a x)^2} \, dx=-\frac {\operatorname {Ci}\left (\arccos \left (a x\right )\right )}{a} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a \arccos \left (a x\right )} \]
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Timed out. \[ \int \frac {1}{\arccos (a x)^2} \, dx=\int \frac {1}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \]
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