Integrand size = 10, antiderivative size = 101 \[ \int \frac {\arccos (a x)^3}{x} \, dx=-\frac {1}{4} i \arccos (a x)^4+\arccos (a x)^3 \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {3}{2} i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {3}{2} \arccos (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )+\frac {3}{4} i \operatorname {PolyLog}\left (4,-e^{2 i \arccos (a x)}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4722, 3800, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {\arccos (a x)^3}{x} \, dx=-\frac {3}{2} i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {3}{2} \arccos (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )+\frac {3}{4} i \operatorname {PolyLog}\left (4,-e^{2 i \arccos (a x)}\right )-\frac {1}{4} i \arccos (a x)^4+\arccos (a x)^3 \log \left (1+e^{2 i \arccos (a x)}\right ) \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 4722
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int x^3 \tan (x) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {1}{4} i \arccos (a x)^4+2 i \text {Subst}\left (\int \frac {e^{2 i x} x^3}{1+e^{2 i x}} \, dx,x,\arccos (a x)\right ) \\ & = -\frac {1}{4} i \arccos (a x)^4+\arccos (a x)^3 \log \left (1+e^{2 i \arccos (a x)}\right )-3 \text {Subst}\left (\int x^2 \log \left (1+e^{2 i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {1}{4} i \arccos (a x)^4+\arccos (a x)^3 \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {3}{2} i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+3 i \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {1}{4} i \arccos (a x)^4+\arccos (a x)^3 \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {3}{2} i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {3}{2} \arccos (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )-\frac {3}{2} \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{2 i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {1}{4} i \arccos (a x)^4+\arccos (a x)^3 \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {3}{2} i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {3}{2} \arccos (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )+\frac {3}{4} i \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 i \arccos (a x)}\right ) \\ & = -\frac {1}{4} i \arccos (a x)^4+\arccos (a x)^3 \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {3}{2} i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {3}{2} \arccos (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )+\frac {3}{4} i \operatorname {PolyLog}\left (4,-e^{2 i \arccos (a x)}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {\arccos (a x)^3}{x} \, dx=-\frac {1}{4} i \arccos (a x)^4+\arccos (a x)^3 \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {3}{2} i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {3}{2} \arccos (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )+\frac {3}{4} i \operatorname {PolyLog}\left (4,-e^{2 i \arccos (a x)}\right ) \]
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Time = 0.66 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.34
method | result | size |
derivativedivides | \(-\frac {i \arccos \left (a x \right )^{4}}{4}+\arccos \left (a x \right )^{3} \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-\frac {3 i \arccos \left (a x \right )^{2} \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}+\frac {3 \arccos \left (a x \right ) \operatorname {polylog}\left (3, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}+\frac {3 i \operatorname {polylog}\left (4, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{4}\) | \(135\) |
default | \(-\frac {i \arccos \left (a x \right )^{4}}{4}+\arccos \left (a x \right )^{3} \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-\frac {3 i \arccos \left (a x \right )^{2} \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}+\frac {3 \arccos \left (a x \right ) \operatorname {polylog}\left (3, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}+\frac {3 i \operatorname {polylog}\left (4, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{4}\) | \(135\) |
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\[ \int \frac {\arccos (a x)^3}{x} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\arccos (a x)^3}{x} \, dx=\int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x}\, dx \]
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\[ \int \frac {\arccos (a x)^3}{x} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\arccos (a x)^3}{x} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x} \,d x } \]
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Timed out. \[ \int \frac {\arccos (a x)^3}{x} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{x} \,d x \]
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