Integrand size = 8, antiderivative size = 12 \[ \int \frac {1}{\arccos (a+b x)} \, dx=-\frac {\text {Si}(\arccos (a+b x))}{b} \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4888, 4720, 3380} \[ \int \frac {1}{\arccos (a+b x)} \, dx=-\frac {\text {Si}(\arccos (a+b x))}{b} \]
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Rule 3380
Rule 4720
Rule 4888
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\arccos (x)} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arccos (a+b x)\right )}{b} \\ & = -\frac {\text {Si}(\arccos (a+b x))}{b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\arccos (a+b x)} \, dx=-\frac {\text {Si}(\arccos (a+b x))}{b} \]
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Time = 0.58 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(-\frac {\operatorname {Si}\left (\arccos \left (b x +a \right )\right )}{b}\) | \(13\) |
default | \(-\frac {\operatorname {Si}\left (\arccos \left (b x +a \right )\right )}{b}\) | \(13\) |
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\[ \int \frac {1}{\arccos (a+b x)} \, dx=\int { \frac {1}{\arccos \left (b x + a\right )} \,d x } \]
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\[ \int \frac {1}{\arccos (a+b x)} \, dx=\int \frac {1}{\operatorname {acos}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {1}{\arccos (a+b x)} \, dx=\int { \frac {1}{\arccos \left (b x + a\right )} \,d x } \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\arccos (a+b x)} \, dx=-\frac {\operatorname {Si}\left (\arccos \left (b x + a\right )\right )}{b} \]
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Timed out. \[ \int \frac {1}{\arccos (a+b x)} \, dx=\int \frac {1}{\mathrm {acos}\left (a+b\,x\right )} \,d x \]
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