Integrand size = 8, antiderivative size = 40 \[ \int \frac {1}{\arccos (a+b x)^2} \, dx=\frac {\sqrt {1-(a+b x)^2}}{b \arccos (a+b x)}-\frac {\operatorname {CosIntegral}(\arccos (a+b x))}{b} \]
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Time = 0.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4888, 4718, 4810, 3383} \[ \int \frac {1}{\arccos (a+b x)^2} \, dx=\frac {\sqrt {1-(a+b x)^2}}{b \arccos (a+b x)}-\frac {\operatorname {CosIntegral}(\arccos (a+b x))}{b} \]
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Rule 3383
Rule 4718
Rule 4810
Rule 4888
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\arccos (x)^2} \, dx,x,a+b x\right )}{b} \\ & = \frac {\sqrt {1-(a+b x)^2}}{b \arccos (a+b x)}+\frac {\text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \arccos (x)} \, dx,x,a+b x\right )}{b} \\ & = \frac {\sqrt {1-(a+b x)^2}}{b \arccos (a+b x)}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arccos (a+b x)\right )}{b} \\ & = \frac {\sqrt {1-(a+b x)^2}}{b \arccos (a+b x)}-\frac {\operatorname {CosIntegral}(\arccos (a+b x))}{b} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\arccos (a+b x)^2} \, dx=\frac {\sqrt {1-(a+b x)^2}}{b \arccos (a+b x)}-\frac {\operatorname {CosIntegral}(\arccos (a+b x))}{b} \]
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Time = 0.71 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {1-\left (b x +a \right )^{2}}}{\arccos \left (b x +a \right )}-\operatorname {Ci}\left (\arccos \left (b x +a \right )\right )}{b}\) | \(37\) |
default | \(\frac {\frac {\sqrt {1-\left (b x +a \right )^{2}}}{\arccos \left (b x +a \right )}-\operatorname {Ci}\left (\arccos \left (b x +a \right )\right )}{b}\) | \(37\) |
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\[ \int \frac {1}{\arccos (a+b x)^2} \, dx=\int { \frac {1}{\arccos \left (b x + a\right )^{2}} \,d x } \]
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\[ \int \frac {1}{\arccos (a+b x)^2} \, dx=\int \frac {1}{\operatorname {acos}^{2}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {1}{\arccos (a+b x)^2} \, dx=\int { \frac {1}{\arccos \left (b x + a\right )^{2}} \,d x } \]
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none
Time = 0.29 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\arccos (a+b x)^2} \, dx=-\frac {\operatorname {Ci}\left (\arccos \left (b x + a\right )\right )}{b} + \frac {\sqrt {-{\left (b x + a\right )}^{2} + 1}}{b \arccos \left (b x + a\right )} \]
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Timed out. \[ \int \frac {1}{\arccos (a+b x)^2} \, dx=\int \frac {1}{{\mathrm {acos}\left (a+b\,x\right )}^2} \,d x \]
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