Integrand size = 10, antiderivative size = 74 \[ \int \frac {\arccos (a x)^2}{x^2} \, dx=-\frac {\arccos (a x)^2}{x}-4 i a \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+2 i a \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i a \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4724, 4804, 4266, 2317, 2438} \[ \int \frac {\arccos (a x)^2}{x^2} \, dx=-4 i a \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+2 i a \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i a \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right )-\frac {\arccos (a x)^2}{x} \]
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Rule 2317
Rule 2438
Rule 4266
Rule 4724
Rule 4804
Rubi steps \begin{align*} \text {integral}& = -\frac {\arccos (a x)^2}{x}-(2 a) \int \frac {\arccos (a x)}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {\arccos (a x)^2}{x}+(2 a) \text {Subst}(\int x \sec (x) \, dx,x,\arccos (a x)) \\ & = -\frac {\arccos (a x)^2}{x}-4 i a \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )-(2 a) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arccos (a x)\right )+(2 a) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arccos (a x)\right ) \\ & = -\frac {\arccos (a x)^2}{x}-4 i a \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+(2 i a) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arccos (a x)}\right )-(2 i a) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arccos (a x)}\right ) \\ & = -\frac {\arccos (a x)^2}{x}-4 i a \arccos (a x) \arctan \left (e^{i \arccos (a x)}\right )+2 i a \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i a \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right ) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.32 \[ \int \frac {\arccos (a x)^2}{x^2} \, dx=-\frac {\arccos (a x) \left (\arccos (a x)+2 a x \left (-\log \left (1-i e^{i \arccos (a x)}\right )+\log \left (1+i e^{i \arccos (a x)}\right )\right )\right )}{x}+2 i a \operatorname {PolyLog}\left (2,-i e^{i \arccos (a x)}\right )-2 i a \operatorname {PolyLog}\left (2,i e^{i \arccos (a x)}\right ) \]
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Time = 0.43 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.84
method | result | size |
derivativedivides | \(a \left (-\frac {\arccos \left (a x \right )^{2}}{a x}-2 \arccos \left (a x \right ) \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+2 \arccos \left (a x \right ) \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+2 i \operatorname {dilog}\left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-2 i \operatorname {dilog}\left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )\right )\) | \(136\) |
default | \(a \left (-\frac {\arccos \left (a x \right )^{2}}{a x}-2 \arccos \left (a x \right ) \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+2 \arccos \left (a x \right ) \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+2 i \operatorname {dilog}\left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-2 i \operatorname {dilog}\left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )\right )\) | \(136\) |
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\[ \int \frac {\arccos (a x)^2}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\arccos (a x)^2}{x^2} \, dx=\int \frac {\operatorname {acos}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\arccos (a x)^2}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\arccos (a x)^2}{x^2} \, dx=\int { \frac {\arccos \left (a x\right )^{2}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\arccos (a x)^2}{x^2} \, dx=\int \frac {{\mathrm {acos}\left (a\,x\right )}^2}{x^2} \,d x \]
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