Integrand size = 10, antiderivative size = 121 \[ \int \frac {\arccos (a x)^4}{x^3} \, dx=-2 i a^2 \arccos (a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{x}-\frac {\arccos (a x)^4}{2 x^2}+6 a^2 \arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-6 i a^2 \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+3 a^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right ) \]
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Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {4724, 4772, 4722, 3800, 2221, 2611, 2320, 6724} \[ \int \frac {\arccos (a x)^4}{x^3} \, dx=-6 i a^2 \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+3 a^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{x}-2 i a^2 \arccos (a x)^3+6 a^2 \arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-\frac {\arccos (a x)^4}{2 x^2} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 4722
Rule 4724
Rule 4772
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (a x)^4}{2 x^2}-(2 a) \int \frac {\arccos (a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx \\ & = \frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{x}-\frac {\arccos (a x)^4}{2 x^2}+\left (6 a^2\right ) \int \frac {\arccos (a x)^2}{x} \, dx \\ & = \frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{x}-\frac {\arccos (a x)^4}{2 x^2}-\left (6 a^2\right ) \text {Subst}\left (\int x^2 \tan (x) \, dx,x,\arccos (a x)\right ) \\ & = -2 i a^2 \arccos (a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{x}-\frac {\arccos (a x)^4}{2 x^2}+\left (12 i a^2\right ) \text {Subst}\left (\int \frac {e^{2 i x} x^2}{1+e^{2 i x}} \, dx,x,\arccos (a x)\right ) \\ & = -2 i a^2 \arccos (a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{x}-\frac {\arccos (a x)^4}{2 x^2}+6 a^2 \arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-\left (12 a^2\right ) \text {Subst}\left (\int x \log \left (1+e^{2 i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -2 i a^2 \arccos (a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{x}-\frac {\arccos (a x)^4}{2 x^2}+6 a^2 \arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-6 i a^2 \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\left (6 i a^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{2 i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -2 i a^2 \arccos (a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{x}-\frac {\arccos (a x)^4}{2 x^2}+6 a^2 \arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-6 i a^2 \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+\left (3 a^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i \arccos (a x)}\right ) \\ & = -2 i a^2 \arccos (a x)^3+\frac {2 a \sqrt {1-a^2 x^2} \arccos (a x)^3}{x}-\frac {\arccos (a x)^4}{2 x^2}+6 a^2 \arccos (a x)^2 \log \left (1+e^{2 i \arccos (a x)}\right )-6 i a^2 \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )+3 a^2 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right ) \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.95 \[ \int \frac {\arccos (a x)^4}{x^3} \, dx=-\frac {\arccos (a x)^4}{2 x^2}-a^2 \left (-2 \arccos (a x)^2 \left (-i \arccos (a x)+\frac {\sqrt {1-a^2 x^2} \arccos (a x)}{a x}+3 \log \left (1+e^{2 i \arccos (a x)}\right )\right )+6 i \arccos (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arccos (a x)}\right )-3 \operatorname {PolyLog}\left (3,-e^{2 i \arccos (a x)}\right )\right ) \]
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Time = 0.86 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.24
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\arccos \left (a x \right )^{3} \left (-4 i a^{2} x^{2}-4 a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2 a^{2} x^{2}}-4 i \arccos \left (a x \right )^{3}+6 \arccos \left (a x \right )^{2} \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-6 i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )+3 \operatorname {polylog}\left (3, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )\right )\) | \(150\) |
default | \(a^{2} \left (-\frac {\arccos \left (a x \right )^{3} \left (-4 i a^{2} x^{2}-4 a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{2 a^{2} x^{2}}-4 i \arccos \left (a x \right )^{3}+6 \arccos \left (a x \right )^{2} \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-6 i \arccos \left (a x \right ) \operatorname {polylog}\left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )+3 \operatorname {polylog}\left (3, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )\right )\) | \(150\) |
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\[ \int \frac {\arccos (a x)^4}{x^3} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{3}} \,d x } \]
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\[ \int \frac {\arccos (a x)^4}{x^3} \, dx=\int \frac {\operatorname {acos}^{4}{\left (a x \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\arccos (a x)^4}{x^3} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{3}} \,d x } \]
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\[ \int \frac {\arccos (a x)^4}{x^3} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\arccos (a x)^4}{x^3} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^4}{x^3} \,d x \]
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