Integrand size = 10, antiderivative size = 54 \[ \int \frac {1}{a+b \arccos (c x)} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b c}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b c} \]
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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 = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4720, 3384, 3380, 3383} \[ \int \frac {1}{a+b \arccos (c x)} \, dx=\frac {\sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b c}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b c} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 4720
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{b c} \\ & = -\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{b c}+\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{b c} \\ & = \frac {\operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b c}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b c} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int \frac {1}{a+b \arccos (c x)} \, dx=-\frac {-\operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right ) \sin \left (\frac {a}{b}\right )+\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )}{b c} \]
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Time = 0.49 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )}{b}+\frac {\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )}{b}}{c}\) | \(49\) |
default | \(\frac {-\frac {\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )}{b}+\frac {\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )}{b}}{c}\) | \(49\) |
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\[ \int \frac {1}{a+b \arccos (c x)} \, dx=\int { \frac {1}{b \arccos \left (c x\right ) + a} \,d x } \]
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\[ \int \frac {1}{a+b \arccos (c x)} \, dx=\int \frac {1}{a + b \operatorname {acos}{\left (c x \right )}}\, dx \]
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\[ \int \frac {1}{a+b \arccos (c x)} \, dx=\int { \frac {1}{b \arccos \left (c x\right ) + a} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93 \[ \int \frac {1}{a+b \arccos (c x)} \, dx=\frac {\operatorname {Ci}\left (\frac {a}{b} + \arccos \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c} - \frac {\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{b c} \]
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Timed out. \[ \int \frac {1}{a+b \arccos (c x)} \, dx=\int \frac {1}{a+b\,\mathrm {acos}\left (c\,x\right )} \,d x \]
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