Integrand size = 17, antiderivative size = 95 \[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\frac {4 x}{3}-\frac {x^3}{9}+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+2 i \arccos (x) \arctan \left (e^{i \arccos (x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos (x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos (x)}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {4788, 4784, 4804, 4266, 2317, 2438, 8} \[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=2 i \arccos (x) \arctan \left (e^{i \arccos (x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos (x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos (x)}\right )+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+\sqrt {1-x^2} \arccos (x)-\frac {x^3}{9}+\frac {4 x}{3} \]
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Rule 8
Rule 2317
Rule 2438
Rule 4266
Rule 4784
Rule 4788
Rule 4804
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+\frac {1}{3} \int \left (1-x^2\right ) \, dx+\int \frac {\sqrt {1-x^2} \arccos (x)}{x} \, dx \\ & = \frac {x}{3}-\frac {x^3}{9}+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+\int 1 \, dx+\int \frac {\arccos (x)}{x \sqrt {1-x^2}} \, dx \\ & = \frac {4 x}{3}-\frac {x^3}{9}+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)-\text {Subst}(\int x \sec (x) \, dx,x,\arccos (x)) \\ & = \frac {4 x}{3}-\frac {x^3}{9}+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+2 i \arccos (x) \arctan \left (e^{i \arccos (x)}\right )+\text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arccos (x)\right )-\text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arccos (x)\right ) \\ & = \frac {4 x}{3}-\frac {x^3}{9}+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+2 i \arccos (x) \arctan \left (e^{i \arccos (x)}\right )-i \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arccos (x)}\right )+i \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arccos (x)}\right ) \\ & = \frac {4 x}{3}-\frac {x^3}{9}+\sqrt {1-x^2} \arccos (x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \arccos (x)+2 i \arccos (x) \arctan \left (e^{i \arccos (x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos (x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos (x)}\right ) \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.25 \[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=x+\sqrt {1-x^2} \arccos (x)+\frac {1}{36} \left (9 x+12 \left (1-x^2\right )^{3/2} \arccos (x)-\cos (3 \arccos (x))\right )-\arccos (x) \log \left (1-i e^{i \arccos (x)}\right )+\arccos (x) \log \left (1+i e^{i \arccos (x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{i \arccos (x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \arccos (x)}\right ) \]
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Time = 0.59 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.63
method | result | size |
default | \(-\frac {5 \left (-\sqrt {-x^{2}+1}+i x \right ) \left (\arccos \left (x \right )+i\right )}{8}+\frac {5 \left (i x +\sqrt {-x^{2}+1}\right ) \left (\arccos \left (x \right )-i\right )}{8}+\arccos \left (x \right ) \ln \left (1+i \left (i \sqrt {-x^{2}+1}+x \right )\right )-\arccos \left (x \right ) \ln \left (1-i \left (i \sqrt {-x^{2}+1}+x \right )\right )-i \operatorname {dilog}\left (1+i \left (i \sqrt {-x^{2}+1}+x \right )\right )+i \operatorname {dilog}\left (1-i \left (i \sqrt {-x^{2}+1}+x \right )\right )-\frac {\cos \left (3 \arccos \left (x \right )\right )}{36}-\frac {\arccos \left (x \right ) \sin \left (3 \arccos \left (x \right )\right )}{12}\) | \(155\) |
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\[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\int { \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\int { \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )}{x} \,d x } \]
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\[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\int { \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \left (x\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (1-x^2\right )^{3/2} \arccos (x)}{x} \, dx=\int \frac {\mathrm {acos}\left (x\right )\,{\left (1-x^2\right )}^{3/2}}{x} \,d x \]
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