Integrand size = 8, antiderivative size = 51 \[ \int \frac {\arccos (a x)}{x} \, dx=-\frac {1}{2} i \arccos (a x)^2+\arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4722, 3800, 2221, 2317, 2438} \[ \int \frac {\arccos (a x)}{x} \, dx=-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )-\frac {1}{2} i \arccos (a x)^2+\arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right ) \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4722
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}(\int x \tan (x) \, dx,x,\arccos (a x)) \\ & = -\frac {1}{2} i \arccos (a x)^2+2 i \text {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\arccos (a x)\right ) \\ & = -\frac {1}{2} i \arccos (a x)^2+\arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-\text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {1}{2} i \arccos (a x)^2+\arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )+\frac {1}{2} i \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \arccos (a x)}\right ) \\ & = -\frac {1}{2} i \arccos (a x)^2+\arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \frac {\arccos (a x)}{x} \, dx=-\frac {1}{2} i \arccos (a x)^2+\arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {1}{2} i \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right ) \]
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Time = 0.71 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(-\frac {i \arccos \left (a x \right )^{2}}{2}+\arccos \left (a x \right ) \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}\) | \(68\) |
default | \(-\frac {i \arccos \left (a x \right )^{2}}{2}+\arccos \left (a x \right ) \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}\) | \(68\) |
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\[ \int \frac {\arccos (a x)}{x} \, dx=\int { \frac {\arccos \left (a x\right )}{x} \,d x } \]
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\[ \int \frac {\arccos (a x)}{x} \, dx=\int \frac {\operatorname {acos}{\left (a x \right )}}{x}\, dx \]
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\[ \int \frac {\arccos (a x)}{x} \, dx=\int { \frac {\arccos \left (a x\right )}{x} \,d x } \]
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\[ \int \frac {\arccos (a x)}{x} \, dx=\int { \frac {\arccos \left (a x\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\arccos (a x)}{x} \, dx=\int \frac {\mathrm {acos}\left (a\,x\right )}{x} \,d x \]
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