Integrand size = 10, antiderivative size = 86 \[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=\frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c} \]
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Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4718, 4810, 3384, 3380, 3383} \[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c}+\frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 4718
Rule 4810
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}+\frac {c \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arccos (c x))} \, dx}{b} \\ & = \frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{b^2 c} \\ & = \frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\cos \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^2 c}-\frac {\sin \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^2 c} \\ & = \frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=\frac {\frac {b \sqrt {1-c^2 x^2}}{a+b \arccos (c x)}-\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right )-\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )}{b^2 c} \]
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Time = 0.65 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {-c^{2} x^{2}+1}}{\left (a +b \arccos \left (c x \right )\right ) b}-\frac {\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )}{b^{2}}}{c}\) | \(74\) |
default | \(\frac {\frac {\sqrt {-c^{2} x^{2}+1}}{\left (a +b \arccos \left (c x \right )\right ) b}-\frac {\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )}{b^{2}}}{c}\) | \(74\) |
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\[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=\int { \frac {1}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=\int { \frac {1}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (84) = 168\).
Time = 0.29 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.24 \[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=-\frac {b \arccos \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{b^{3} c \arccos \left (c x\right ) + a b^{2} c} - \frac {b \arccos \left (c x\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{b^{3} c \arccos \left (c x\right ) + a b^{2} c} - \frac {a \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{b^{3} c \arccos \left (c x\right ) + a b^{2} c} - \frac {a \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{b^{3} c \arccos \left (c x\right ) + a b^{2} c} + \frac {\sqrt {-c^{2} x^{2} + 1} b}{b^{3} c \arccos \left (c x\right ) + a b^{2} c} \]
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Timed out. \[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2} \,d x \]
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