Integrand size = 10, antiderivative size = 176 \[ \int \frac {\arccos (a x)^4}{x^2} \, dx=-\frac {\arccos (a x)^4}{x}-8 i a \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-24 a \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+24 a \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )-24 i a \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )+24 i a \operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4724, 4804, 4266, 2611, 6744, 2320, 6724} \[ \int \frac {\arccos (a x)^4}{x^2} \, dx=-8 i a \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-24 a \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+24 a \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )-24 i a \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )+24 i a \operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right )-\frac {\arccos (a x)^4}{x} \]
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Rule 2320
Rule 2611
Rule 4266
Rule 4724
Rule 4804
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (a x)^4}{x}-(4 a) \int \frac {\arccos (a x)^3}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {\arccos (a x)^4}{x}+(4 a) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {\arccos (a x)^4}{x}-8 i a \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )-(12 a) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\arccos (a x)\right )+(12 a) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {\arccos (a x)^4}{x}-8 i a \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-(24 i a) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arccos (a x)\right )+(24 i a) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {\arccos (a x)^4}{x}-8 i a \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-24 a \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+24 a \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )+(24 a) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^{i x}\right ) \, dx,x,\arccos (a x)\right )-(24 a) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^{i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {\arccos (a x)^4}{x}-8 i a \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-24 a \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+24 a \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )-(24 i a) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i \arccos (a x)}\right )+(24 i a) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i \arccos (a x)}\right ) \\ & = -\frac {\arccos (a x)^4}{x}-8 i a \arccos (a x)^3 \arctan \left (e^{i \arccos (a x)}\right )+12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-12 i a \arccos (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-24 a \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )+24 a \arccos (a x) \operatorname {PolyLog}\left (3,i e^{i \arccos (a x)}\right )-24 i a \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )+24 i a \operatorname {PolyLog}\left (4,i e^{i \arccos (a x)}\right ) \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(549\) vs. \(2(176)=352\).
Time = 0.74 (sec) , antiderivative size = 549, normalized size of antiderivative = 3.12 \[ \int \frac {\arccos (a x)^4}{x^2} \, dx=a \left (-\frac {7 i \pi ^4}{16}-\frac {1}{2} i \pi ^3 \arccos (a x)+\frac {3}{2} i \pi ^2 \arccos (a x)^2-2 i \pi \arccos (a x)^3+i \arccos (a x)^4-\frac {\arccos (a x)^4}{a x}+3 \pi ^2 \arccos (a x) \log \left (1-i e^{-i \arccos (a x)}\right )-6 \pi \arccos (a x)^2 \log \left (1-i e^{-i \arccos (a x)}\right )-\frac {1}{2} \pi ^3 \log \left (1+i e^{-i \arccos (a x)}\right )+4 \arccos (a x)^3 \log \left (1+i e^{-i \arccos (a x)}\right )+\frac {1}{2} \pi ^3 \log \left (1+i e^{i \arccos (a x)}\right )-3 \pi ^2 \arccos (a x) \log \left (1+i e^{i \arccos (a x)}\right )+6 \pi \arccos (a x)^2 \log \left (1+i e^{i \arccos (a x)}\right )-4 \arccos (a x)^3 \log \left (1+i e^{i \arccos (a x)}\right )+\frac {1}{2} \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 \arccos (a x))\right )\right )+12 i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{-i \arccos (a x)}\right )+3 i \pi (\pi -4 \arccos (a x)) \operatorname {PolyLog}\left (2,i e^{-i \arccos (a x)}\right )+3 i \pi ^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-12 i \pi \arccos (a x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+12 i \arccos (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )+24 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{-i \arccos (a x)}\right )-12 \pi \operatorname {PolyLog}\left (3,i e^{-i \arccos (a x)}\right )+12 \pi \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )-24 \arccos (a x) \operatorname {PolyLog}\left (3,-i e^{i \arccos (a x)}\right )-24 i \operatorname {PolyLog}\left (4,-i e^{-i \arccos (a x)}\right )-24 i \operatorname {PolyLog}\left (4,-i e^{i \arccos (a x)}\right )\right ) \]
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\[\int \frac {\arccos \left (a x \right )^{4}}{x^{2}}d x\]
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\[ \int \frac {\arccos (a x)^4}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{2}} \,d x } \]
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\[ \int \frac {\arccos (a x)^4}{x^2} \, dx=\int \frac {\operatorname {acos}^{4}{\left (a x \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\arccos (a x)^4}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{2}} \,d x } \]
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\[ \int \frac {\arccos (a x)^4}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{4}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\arccos (a x)^4}{x^2} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^4}{x^2} \,d x \]
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