3.645 \(\int \frac{\cos ^{-1}(x)^2}{x^5} \, dx\)

Optimal. Leaf size=65 \[ -\frac{\cos ^{-1}(x)^2}{4 x^4}-\frac{1}{12 x^2}+\frac{\sqrt{1-x^2} \cos ^{-1}(x)}{3 x}+\frac{\sqrt{1-x^2} \cos ^{-1}(x)}{6 x^3}+\frac{\log (x)}{3} \]

[Out]

-1/(12*x^2) + (Sqrt[1 - x^2]*ArcCos[x])/(6*x^3) + (Sqrt[1 - x^2]*ArcCos[x])/(3*x
) - ArcCos[x]^2/(4*x^4) + Log[x]/3

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Rubi [A]  time = 0.173195, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625 \[ -\frac{\cos ^{-1}(x)^2}{4 x^4}-\frac{1}{12 x^2}+\frac{\sqrt{1-x^2} \cos ^{-1}(x)}{3 x}+\frac{\sqrt{1-x^2} \cos ^{-1}(x)}{6 x^3}+\frac{\log (x)}{3} \]

Antiderivative was successfully verified.

[In]  Int[ArcCos[x]^2/x^5,x]

[Out]

-1/(12*x^2) + (Sqrt[1 - x^2]*ArcCos[x])/(6*x^3) + (Sqrt[1 - x^2]*ArcCos[x])/(3*x
) - ArcCos[x]^2/(4*x^4) + Log[x]/3

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Rubi in Sympy [A]  time = 7.98202, size = 53, normalized size = 0.82 \[ \frac{\log{\left (x \right )}}{3} + \frac{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}}{3 x} - \frac{1}{12 x^{2}} + \frac{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}}{6 x^{3}} - \frac{\operatorname{acos}^{2}{\left (x \right )}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(acos(x)**2/x**5,x)

[Out]

log(x)/3 + sqrt(-x**2 + 1)*acos(x)/(3*x) - 1/(12*x**2) + sqrt(-x**2 + 1)*acos(x)
/(6*x**3) - acos(x)**2/(4*x**4)

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Mathematica [A]  time = 0.0347489, size = 52, normalized size = 0.8 \[ -\frac{\cos ^{-1}(x)^2}{4 x^4}-\frac{1}{12 x^2}+\frac{\sqrt{1-x^2} \left (2 x^2+1\right ) \cos ^{-1}(x)}{6 x^3}+\frac{\log (x)}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[ArcCos[x]^2/x^5,x]

[Out]

-1/(12*x^2) + (Sqrt[1 - x^2]*(1 + 2*x^2)*ArcCos[x])/(6*x^3) - ArcCos[x]^2/(4*x^4
) + Log[x]/3

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Maple [A]  time = 0.066, size = 52, normalized size = 0.8 \[ -{\frac{1}{12\,{x}^{2}}}-{\frac{ \left ( \arccos \left ( x \right ) \right ) ^{2}}{4\,{x}^{4}}}+{\frac{\ln \left ( x \right ) }{3}}+{\frac{\arccos \left ( x \right ) }{6\,{x}^{3}}\sqrt{-{x}^{2}+1}}+{\frac{\arccos \left ( x \right ) }{3\,x}\sqrt{-{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arccos(x)^2/x^5,x)

[Out]

-1/12/x^2-1/4*arccos(x)^2/x^4+1/3*ln(x)+1/6*arccos(x)*(-x^2+1)^(1/2)/x^3+1/3*arc
cos(x)*(-x^2+1)^(1/2)/x

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Maxima [A]  time = 1.84669, size = 69, normalized size = 1.06 \[ \frac{1}{6} \,{\left (\frac{2 \, \sqrt{-x^{2} + 1}}{x} + \frac{\sqrt{-x^{2} + 1}}{x^{3}}\right )} \arccos \left (x\right ) - \frac{1}{12 \, x^{2}} - \frac{\arccos \left (x\right )^{2}}{4 \, x^{4}} + \frac{1}{3} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arccos(x)^2/x^5,x, algorithm="maxima")

[Out]

1/6*(2*sqrt(-x^2 + 1)/x + sqrt(-x^2 + 1)/x^3)*arccos(x) - 1/12/x^2 - 1/4*arccos(
x)^2/x^4 + 1/3*log(x)

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Fricas [A]  time = 0.24294, size = 59, normalized size = 0.91 \[ \frac{4 \, x^{4} \log \left (x\right ) + 2 \,{\left (2 \, x^{3} + x\right )} \sqrt{-x^{2} + 1} \arccos \left (x\right ) - x^{2} - 3 \, \arccos \left (x\right )^{2}}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arccos(x)^2/x^5,x, algorithm="fricas")

[Out]

1/12*(4*x^4*log(x) + 2*(2*x^3 + x)*sqrt(-x^2 + 1)*arccos(x) - x^2 - 3*arccos(x)^
2)/x^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\operatorname{acos}^{2}{\left (x \right )}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(acos(x)**2/x**5,x)

[Out]

Integral(acos(x)**2/x**5, x)

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GIAC/XCAS [A]  time = 0.229496, size = 140, normalized size = 2.15 \[ -\frac{1}{48} \,{\left (\frac{x^{3}{\left (\frac{9 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}} - \frac{9 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}}{x^{3}}\right )} \arccos \left (x\right ) - \frac{2 \, x^{2} + 1}{12 \, x^{2}} - \frac{\arccos \left (x\right )^{2}}{4 \, x^{4}} + \frac{1}{6} \,{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arccos(x)^2/x^5,x, algorithm="giac")

[Out]

-1/48*(x^3*(9*(sqrt(-x^2 + 1) - 1)^2/x^2 + 1)/(sqrt(-x^2 + 1) - 1)^3 - 9*(sqrt(-
x^2 + 1) - 1)/x - (sqrt(-x^2 + 1) - 1)^3/x^3)*arccos(x) - 1/12*(2*x^2 + 1)/x^2 -
1/4*arccos(x)^2/x^4 + 1/6*ln(x^2)