3.666 \(\int \frac{\cos ^{-1}(x)}{x^4 \sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=54 \[ \frac{1}{6 x^2}-\frac{2 \sqrt{1-x^2} \cos ^{-1}(x)}{3 x}-\frac{\sqrt{1-x^2} \cos ^{-1}(x)}{3 x^3}-\frac{2 \log (x)}{3} \]

[Out]

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

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Rubi [A]  time = 0.154835, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{1}{6 x^2}-\frac{2 \sqrt{1-x^2} \cos ^{-1}(x)}{3 x}-\frac{\sqrt{1-x^2} \cos ^{-1}(x)}{3 x^3}-\frac{2 \log (x)}{3} \]

Antiderivative was successfully verified.

[In]  Int[ArcCos[x]/(x^4*Sqrt[1 - x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.68549, size = 46, normalized size = 0.85 \[ - \frac{2 \log{\left (x \right )}}{3} - \frac{2 \sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}}{3 x} + \frac{1}{6 x^{2}} - \frac{\sqrt{- x^{2} + 1} \operatorname{acos}{\left (x \right )}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(acos(x)/x**4/(-x**2+1)**(1/2),x)

[Out]

-2*log(x)/3 - 2*sqrt(-x**2 + 1)*acos(x)/(3*x) + 1/(6*x**2) - sqrt(-x**2 + 1)*aco
s(x)/(3*x**3)

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Mathematica [A]  time = 0.0371372, size = 38, normalized size = 0.7 \[ \frac{-4 x^3 \log (x)-2 \sqrt{1-x^2} \left (2 x^2+1\right ) \cos ^{-1}(x)+x}{6 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[ArcCos[x]/(x^4*Sqrt[1 - x^2]),x]

[Out]

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

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Maple [A]  time = 0.075, size = 43, normalized size = 0.8 \[{\frac{1}{6\,{x}^{2}}}-{\frac{2\,\ln \left ( x \right ) }{3}}-{\frac{\arccos \left ( x \right ) }{3\,{x}^{3}}\sqrt{-{x}^{2}+1}}-{\frac{2\,\arccos \left ( x \right ) }{3\,x}\sqrt{-{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arccos(x)/x^4/(-x^2+1)^(1/2),x)

[Out]

1/6/x^2-2/3*ln(x)-1/3*arccos(x)*(-x^2+1)^(1/2)/x^3-2/3*arccos(x)*(-x^2+1)^(1/2)/
x

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Maxima [A]  time = 1.49825, size = 57, normalized size = 1.06 \[ -\frac{1}{3} \,{\left (\frac{2 \, \sqrt{-x^{2} + 1}}{x} + \frac{\sqrt{-x^{2} + 1}}{x^{3}}\right )} \arccos \left (x\right ) + \frac{1}{6 \, x^{2}} - \frac{2}{3} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arccos(x)/(sqrt(-x^2 + 1)*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.227472, size = 49, normalized size = 0.91 \[ -\frac{4 \, x^{3} \log \left (x\right ) + 2 \,{\left (2 \, x^{2} + 1\right )} \sqrt{-x^{2} + 1} \arccos \left (x\right ) - x}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arccos(x)/(sqrt(-x^2 + 1)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(acos(x)/x**4/(-x**2+1)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217484, size = 128, normalized size = 2.37 \[ \frac{1}{24} \,{\left (\frac{x^{3}{\left (\frac{9 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}} - \frac{9 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}}{x^{3}}\right )} \arccos \left (x\right ) + \frac{2 \, x^{2} + 1}{6 \, x^{2}} - \frac{1}{3} \,{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arccos(x)/(sqrt(-x^2 + 1)*x^4),x, algorithm="giac")

[Out]

1/24*(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) + 1/6*(2*x^2 + 1)/x^2 - 1
/3*ln(x^2)