Integrand size = 10, antiderivative size = 124 \[ \int \frac {\arccos (a x)^2}{x^4} \, dx=-\frac {a^2}{3 x}+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{3 x^2}-\frac {\arccos (a x)^2}{3 x^3}-\frac {2}{3} i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right ) \]
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Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4724, 4790, 4804, 4266, 2317, 2438, 30} \[ \int \frac {\arccos (a x)^2}{x^4} \, dx=-\frac {2}{3} i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{3 x^2}-\frac {a^2}{3 x}-\frac {\arccos (a x)^2}{3 x^3} \]
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Rule 30
Rule 2317
Rule 2438
Rule 4266
Rule 4724
Rule 4790
Rule 4804
Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (a x)^2}{3 x^3}-\frac {1}{3} (2 a) \int \frac {\arccos (a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx \\ & = \frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{3 x^2}-\frac {\arccos (a x)^2}{3 x^3}+\frac {1}{3} a^2 \int \frac {1}{x^2} \, dx-\frac {1}{3} a^3 \int \frac {\arccos (a x)}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {a^2}{3 x}+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{3 x^2}-\frac {\arccos (a x)^2}{3 x^3}+\frac {1}{3} a^3 \text {Subst}(\int x \sec (x) \, dx,x,\arccos (a x)) \\ & = -\frac {a^2}{3 x}+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{3 x^2}-\frac {\arccos (a x)^2}{3 x^3}-\frac {2}{3} i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )-\frac {1}{3} a^3 \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arccos (a x)\right )+\frac {1}{3} a^3 \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {a^2}{3 x}+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{3 x^2}-\frac {\arccos (a x)^2}{3 x^3}-\frac {2}{3} i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+\frac {1}{3} \left (i a^3\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arccos (a x)}\right )-\frac {1}{3} \left (i a^3\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arccos (a x)}\right ) \\ & = -\frac {a^2}{3 x}+\frac {a \sqrt {1-a^2 x^2} \arccos (a x)}{3 x^2}-\frac {\arccos (a x)^2}{3 x^3}-\frac {2}{3} i a^3 \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right ) \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.23 \[ \int \frac {\arccos (a x)^2}{x^4} \, dx=-\frac {a^2 x^2-a x \sqrt {1-a^2 x^2} \arccos (a x)+\arccos (a x)^2-a^3 x^3 \arccos (a x) \log \left (1-i e^{i \arccos (a x)}\right )+a^3 x^3 \arccos (a x) \log \left (1+i e^{i \arccos (a x)}\right )-i a^3 x^3 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+i a^3 x^3 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )}{3 x^3} \]
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Time = 1.22 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.34
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {-\sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a x +\arccos \left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}-\frac {\arccos \left (a x \right ) \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}+\frac {\arccos \left (a x \right ) \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}+\frac {i \operatorname {dilog}\left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}-\frac {i \operatorname {dilog}\left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}\right )\) | \(166\) |
default | \(a^{3} \left (-\frac {-\sqrt {-a^{2} x^{2}+1}\, \arccos \left (a x \right ) a x +\arccos \left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}-\frac {\arccos \left (a x \right ) \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}+\frac {\arccos \left (a x \right ) \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}+\frac {i \operatorname {dilog}\left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}-\frac {i \operatorname {dilog}\left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{3}\right )\) | \(166\) |
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\[ \int \frac {\arccos (a x)^2}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{x^{4}} \,d x } \]
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\[ \int \frac {\arccos (a x)^2}{x^4} \, dx=\int \frac {\operatorname {acos}^{2}{\left (a x \right )}}{x^{4}}\, dx \]
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\[ \int \frac {\arccos (a x)^2}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{x^{4}} \,d x } \]
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\[ \int \frac {\arccos (a x)^2}{x^4} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\arccos (a x)^2}{x^4} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^2}{x^4} \,d x \]
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