Integrand size = 8, antiderivative size = 14 \[ \int \frac {x}{\arccos (a x)} \, dx=-\frac {\text {Si}(2 \arccos (a x))}{2 a^2} \]
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Time = 0.03 (sec) , antiderivative size = 14, 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 = {4732, 4491, 12, 3380} \[ \int \frac {x}{\arccos (a x)} \, dx=-\frac {\text {Si}(2 \arccos (a x))}{2 a^2} \]
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Rule 12
Rule 3380
Rule 4491
Rule 4732
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\cos (x) \sin (x)}{x} \, dx,x,\arccos (a x)\right )}{a^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{2 x} \, dx,x,\arccos (a x)\right )}{a^2} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (2 x)}{x} \, dx,x,\arccos (a x)\right )}{2 a^2} \\ & = -\frac {\text {Si}(2 \arccos (a x))}{2 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\arccos (a x)} \, dx=-\frac {\text {Si}(2 \arccos (a x))}{2 a^2} \]
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Time = 0.69 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(-\frac {\operatorname {Si}\left (2 \arccos \left (a x \right )\right )}{2 a^{2}}\) | \(13\) |
default | \(-\frac {\operatorname {Si}\left (2 \arccos \left (a x \right )\right )}{2 a^{2}}\) | \(13\) |
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\[ \int \frac {x}{\arccos (a x)} \, dx=\int { \frac {x}{\arccos \left (a x\right )} \,d x } \]
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\[ \int \frac {x}{\arccos (a x)} \, dx=\int \frac {x}{\operatorname {acos}{\left (a x \right )}}\, dx \]
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\[ \int \frac {x}{\arccos (a x)} \, dx=\int { \frac {x}{\arccos \left (a x\right )} \,d x } \]
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none
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {x}{\arccos (a x)} \, dx=-\frac {\operatorname {Si}\left (2 \, \arccos \left (a x\right )\right )}{2 \, a^{2}} \]
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Timed out. \[ \int \frac {x}{\arccos (a x)} \, dx=\int \frac {x}{\mathrm {acos}\left (a\,x\right )} \,d x \]
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