3.79 \(\int \cos ^2(x) \cot ^3(x) \, dx\)

Optimal. Leaf size=22 \[ \frac{\sin ^2(x)}{2}-\frac{1}{2} \csc ^2(x)-2 \log (\sin (x)) \]

[Out]

-Csc[x]^2/2 - 2*Log[Sin[x]] + Sin[x]^2/2

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Rubi [A]  time = 0.05361, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sin ^2(x)}{2}-\frac{1}{2} \csc ^2(x)-2 \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^2*Cot[x]^3,x]

[Out]

-Csc[x]^2/2 - 2*Log[Sin[x]] + Sin[x]^2/2

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Rubi in Sympy [A]  time = 3.6298, size = 26, normalized size = 1.18 \[ - \log{\left (- \cos ^{2}{\left (x \right )} + 1 \right )} - \frac{\cos ^{2}{\left (x \right )}}{2} - \frac{1}{2 \left (- \cos ^{2}{\left (x \right )} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(x)**5*sin(x)**2,x)

[Out]

-log(-cos(x)**2 + 1) - cos(x)**2/2 - 1/(2*(-cos(x)**2 + 1))

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Mathematica [A]  time = 0.00890417, size = 22, normalized size = 1. \[ -\frac{1}{4} \cos (2 x)-\frac{1}{2} \csc ^2(x)-2 \log (\sin (x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^2*Cot[x]^3,x]

[Out]

-Cos[2*x]/4 - Csc[x]^2/2 - 2*Log[Sin[x]]

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Maple [A]  time = 0.02, size = 29, normalized size = 1.3 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{6}}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{2}}- \left ( \cos \left ( x \right ) \right ) ^{2}-2\,\ln \left ( \sin \left ( x \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(x)^5*sin(x)^2,x)

[Out]

-1/2/sin(x)^2*cos(x)^6-1/2*cos(x)^4-cos(x)^2-2*ln(sin(x))

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Maxima [A]  time = 1.33848, size = 27, normalized size = 1.23 \[ \frac{1}{2} \, \sin \left (x\right )^{2} - \frac{1}{2 \, \sin \left (x\right )^{2}} - \log \left (\sin \left (x\right )^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^5*sin(x)^2,x, algorithm="maxima")

[Out]

1/2*sin(x)^2 - 1/2/sin(x)^2 - log(sin(x)^2)

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Fricas [A]  time = 0.232614, size = 50, normalized size = 2.27 \[ -\frac{2 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} + 8 \,{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (x\right )\right ) - 1}{4 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^5*sin(x)^2,x, algorithm="fricas")

[Out]

-1/4*(2*cos(x)^4 - 3*cos(x)^2 + 8*(cos(x)^2 - 1)*log(1/2*sin(x)) - 1)/(cos(x)^2
- 1)

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Sympy [A]  time = 0.098456, size = 20, normalized size = 0.91 \[ - 2 \log{\left (\sin{\left (x \right )} \right )} + \frac{\sin ^{2}{\left (x \right )}}{2} - \frac{1}{2 \sin ^{2}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)**5*sin(x)**2,x)

[Out]

-2*log(sin(x)) + sin(x)**2/2 - 1/(2*sin(x)**2)

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GIAC/XCAS [A]  time = 0.209236, size = 49, normalized size = 2.23 \[ -\frac{1}{2} \, \cos \left (x\right )^{2} + \frac{2 \, \cos \left (x\right )^{2} - 1}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} -{\rm ln}\left (-\cos \left (x\right )^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^5*sin(x)^2,x, algorithm="giac")

[Out]

-1/2*cos(x)^2 + 1/2*(2*cos(x)^2 - 1)/(cos(x)^2 - 1) - ln(-cos(x)^2 + 1)