Optimal. Leaf size=95 \[ -i \operatorname {PolyLog}\left (2,-i e^{i \cos ^{-1}(x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \cos ^{-1}(x)}\right )-\frac {x^3}{9}+\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)+\sqrt {1-x^2} \cos ^{-1}(x)+\frac {4 x}{3}+2 i \cos ^{-1}(x) \tan ^{-1}\left (e^{i \cos ^{-1}(x)}\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {4700, 4698, 4710, 4181, 2279, 2391, 8} \[ -i \text {PolyLog}\left (2,-i e^{i \cos ^{-1}(x)}\right )+i \text {PolyLog}\left (2,i e^{i \cos ^{-1}(x)}\right )-\frac {x^3}{9}+\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)+\sqrt {1-x^2} \cos ^{-1}(x)+\frac {4 x}{3}+2 i \cos ^{-1}(x) \tan ^{-1}\left (e^{i \cos ^{-1}(x)}\right ) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2279
Rule 2391
Rule 4181
Rule 4698
Rule 4700
Rule 4710
Rubi steps
\begin {align*} \int \frac {\left (1-x^2\right )^{3/2} \cos ^{-1}(x)}{x} \, dx &=\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)+\frac {1}{3} \int \left (1-x^2\right ) \, dx+\int \frac {\sqrt {1-x^2} \cos ^{-1}(x)}{x} \, dx\\ &=\frac {x}{3}-\frac {x^3}{9}+\sqrt {1-x^2} \cos ^{-1}(x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)+\int 1 \, dx+\int \frac {\cos ^{-1}(x)}{x \sqrt {1-x^2}} \, dx\\ &=\frac {4 x}{3}-\frac {x^3}{9}+\sqrt {1-x^2} \cos ^{-1}(x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)-\operatorname {Subst}\left (\int x \sec (x) \, dx,x,\cos ^{-1}(x)\right )\\ &=\frac {4 x}{3}-\frac {x^3}{9}+\sqrt {1-x^2} \cos ^{-1}(x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)+2 i \cos ^{-1}(x) \tan ^{-1}\left (e^{i \cos ^{-1}(x)}\right )+\operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(x)\right )-\operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(x)\right )\\ &=\frac {4 x}{3}-\frac {x^3}{9}+\sqrt {1-x^2} \cos ^{-1}(x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)+2 i \cos ^{-1}(x) \tan ^{-1}\left (e^{i \cos ^{-1}(x)}\right )-i \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \cos ^{-1}(x)}\right )+i \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \cos ^{-1}(x)}\right )\\ &=\frac {4 x}{3}-\frac {x^3}{9}+\sqrt {1-x^2} \cos ^{-1}(x)+\frac {1}{3} \left (1-x^2\right )^{3/2} \cos ^{-1}(x)+2 i \cos ^{-1}(x) \tan ^{-1}\left (e^{i \cos ^{-1}(x)}\right )-i \text {Li}_2\left (-i e^{i \cos ^{-1}(x)}\right )+i \text {Li}_2\left (i e^{i \cos ^{-1}(x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.28, size = 119, normalized size = 1.25 \[ -i \operatorname {PolyLog}\left (2,-i e^{i \cos ^{-1}(x)}\right )+i \operatorname {PolyLog}\left (2,i e^{i \cos ^{-1}(x)}\right )+\sqrt {1-x^2} \cos ^{-1}(x)+\frac {1}{36} \left (12 \left (1-x^2\right )^{3/2} \cos ^{-1}(x)+9 x-\cos \left (3 \cos ^{-1}(x)\right )\right )+x-\cos ^{-1}(x) \log \left (1-i e^{i \cos ^{-1}(x)}\right )+\cos ^{-1}(x) \log \left (1+i e^{i \cos ^{-1}(x)}\right ) \]
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (1-x^2\right )^{3/2} \cos ^{-1}(x)}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (x^{2} - 1\right )} \sqrt {-x^{2} + 1} \arccos \relax (x)}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \relax (x)}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 230, normalized size = 2.42
method | result | size |
default | \(\frac {\left (i+3 \arccos \relax (x )\right ) \left (4 i x^{3}-4 \sqrt {-x^{2}+1}\, x^{2}-3 i x +\sqrt {-x^{2}+1}\right )}{72}-\frac {5 \left (\arccos \relax (x )+i\right ) \left (i x -\sqrt {-x^{2}+1}\right )}{8}+\frac {5 \left (\arccos \relax (x )-i\right ) \left (i x +\sqrt {-x^{2}+1}\right )}{8}-\frac {\left (-i+3 \arccos \relax (x )\right ) \left (4 i x^{3}+4 \sqrt {-x^{2}+1}\, x^{2}-3 i x -\sqrt {-x^{2}+1}\right )}{72}-i \left (i \arccos \relax (x ) \ln \left (1+i \left (x +i \sqrt {-x^{2}+1}\right )\right )-i \arccos \relax (x ) \ln \left (1-i \left (x +i \sqrt {-x^{2}+1}\right )\right )+\dilog \left (1+i \left (x +i \sqrt {-x^{2}+1}\right )\right )-\dilog \left (1-i \left (x +i \sqrt {-x^{2}+1}\right )\right )\right )\) | \(230\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{2} + 1\right )}^{\frac {3}{2}} \arccos \relax (x)}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acos}\relax (x)\,{\left (1-x^2\right )}^{3/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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