Integrand size = 10, antiderivative size = 73 \[ \int \frac {\arccos (a x)^2}{x} \, dx=-\frac {1}{3} i \arccos (a x)^3+\arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4722, 3800, 2221, 2611, 2320, 6724} \[ \int \frac {\arccos (a x)^2}{x} \, dx=-i \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )-\frac {1}{3} i \arccos (a x)^3+\arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right ) \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 4722
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x^2 \tan (x) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {1}{3} i \arccos (a x)^3+2 i \text {Subst}\left (\int \frac {e^{2 i x} x^2}{1+e^{2 i x}} \, dx,x,\arccos (a x)\right ) \\ & = -\frac {1}{3} i \arccos (a x)^3+\arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-2 \text {Subst}\left (\int x \log \left (1+e^{2 i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {1}{3} i \arccos (a x)^3+\arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+i \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {1}{3} i \arccos (a x)^3+\arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i \arccos (a x)}\right ) \\ & = -\frac {1}{3} i \arccos (a x)^3+\arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00 \[ \int \frac {\arccos (a x)^2}{x} \, dx=-\frac {1}{3} i \arccos (a x)^3+\arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-i \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right ) \]
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Time = 0.64 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.38
method | result | size |
derivativedivides | \(-\frac {i \arccos \left (a x \right )^{3}}{3}+\arccos \left (a x \right )^{2} \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}\) | \(101\) |
default | \(-\frac {i \arccos \left (a x \right )^{3}}{3}+\arccos \left (a x \right )^{2} \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )+\frac {\operatorname {polylog}\left (3, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}\) | \(101\) |
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\[ \int \frac {\arccos (a x)^2}{x} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{x} \,d x } \]
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\[ \int \frac {\arccos (a x)^2}{x} \, dx=\int \frac {\operatorname {acos}^{2}{\left (a x \right )}}{x}\, dx \]
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\[ \int \frac {\arccos (a x)^2}{x} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{x} \,d x } \]
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\[ \int \frac {\arccos (a x)^2}{x} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{x} \,d x } \]
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Timed out. \[ \int \frac {\arccos (a x)^2}{x} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^2}{x} \,d x \]
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