Integrand size = 10, antiderivative size = 102 \[ \int \frac {\arccos (a x)^3}{x^3} \, dx=-\frac {3}{2} i a^2 \arccos (a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x}-\frac {\arccos (a x)^3}{2 x^2}+3 a^2 \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 102, 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 = {4724, 4772, 4722, 3800, 2221, 2317, 2438} \[ \int \frac {\arccos (a x)^3}{x^3} \, dx=-\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\frac {3 a \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x}-\frac {3}{2} i a^2 \arccos (a x)^2+3 a^2 \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {\arccos (a x)^3}{2 x^2} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4722
Rule 4724
Rule 4772
Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (a x)^3}{2 x^2}-\frac {1}{2} (3 a) \int \frac {\arccos (a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx \\ & = \frac {3 a \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x}-\frac {\arccos (a x)^3}{2 x^2}+\left (3 a^2\right ) \int \frac {\arccos (a x)}{x} \, dx \\ & = \frac {3 a \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x}-\frac {\arccos (a x)^3}{2 x^2}-\left (3 a^2\right ) \text {Subst}(\int x \tan (x) \, dx,x,\arccos (a x)) \\ & = -\frac {3}{2} i a^2 \arccos (a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x}-\frac {\arccos (a x)^3}{2 x^2}+\left (6 i a^2\right ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\arccos (a x)\right ) \\ & = -\frac {3}{2} i a^2 \arccos (a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x}-\frac {\arccos (a x)^3}{2 x^2}+3 a^2 \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-\left (3 a^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {3}{2} i a^2 \arccos (a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x}-\frac {\arccos (a x)^3}{2 x^2}+3 a^2 \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )+\frac {1}{2} \left (3 i a^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \arccos (a x)}\right ) \\ & = -\frac {3}{2} i a^2 \arccos (a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \arccos (a x)^2}{2 x}-\frac {\arccos (a x)^3}{2 x^2}+3 a^2 \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {3}{2} i a^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right ) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.90 \[ \int \frac {\arccos (a x)^3}{x^3} \, dx=\frac {1}{2} \left (\frac {3 a \left (-i a x+\sqrt {1-a^2 x^2}\right ) \arccos (a x)^2}{x}-\frac {\arccos (a x)^3}{x^2}+6 a^2 \arccos (a x) \log \left (1+e^{2 i \arccos (a x)}\right )-3 i a^2 \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )\right ) \]
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Time = 0.76 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.15
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\arccos \left (a x \right )^{2} \left (-3 i a^{2} x^{2}-3 a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2 a^{2} x^{2}}-3 i \arccos \left (a x \right )^{2}+3 \arccos \left (a x \right ) \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-\frac {3 i \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}\right )\) | \(117\) |
default | \(a^{2} \left (-\frac {\arccos \left (a x \right )^{2} \left (-3 i a^{2} x^{2}-3 a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2 a^{2} x^{2}}-3 i \arccos \left (a x \right )^{2}+3 \arccos \left (a x \right ) \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-\frac {3 i \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}\right )\) | \(117\) |
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\[ \int \frac {\arccos (a x)^3}{x^3} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{3}} \,d x } \]
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\[ \int \frac {\arccos (a x)^3}{x^3} \, dx=\int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\arccos (a x)^3}{x^3} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{3}} \,d x } \]
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\[ \int \frac {\arccos (a x)^3}{x^3} \, dx=\int { \frac {\arccos \left (a x\right )^{3}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\arccos (a x)^3}{x^3} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{x^3} \,d x \]
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