3.342 \(\int \cot ^5(x) \, dx\)

Optimal. Leaf size=20 \[ -\frac{1}{4} \cot ^4(x)+\frac{\cot ^2(x)}{2}+\log (\sin (x)) \]

[Out]

Cot[x]^2/2 - Cot[x]^4/4 + Log[Sin[x]]

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Rubi [A]  time = 0.024199, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{1}{4} \cot ^4(x)+\frac{\cot ^2(x)}{2}+\log (\sin (x)) \]

Antiderivative was successfully verified.

[In]  Int[Cot[x]^5,x]

[Out]

Cot[x]^2/2 - Cot[x]^4/4 + Log[Sin[x]]

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Rubi in Sympy [A]  time = 0.529923, size = 20, normalized size = 1. \[ \log{\left (\sin{\left (x \right )} \right )} + \frac{1}{2 \tan ^{2}{\left (x \right )}} - \frac{1}{4 \tan ^{4}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/tan(x)**5,x)

[Out]

log(sin(x)) + 1/(2*tan(x)**2) - 1/(4*tan(x)**4)

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Mathematica [A]  time = 0.00550819, size = 16, normalized size = 0.8 \[ -\frac{1}{4} \csc ^4(x)+\csc ^2(x)+\log (\sin (x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x]^5,x]

[Out]

Csc[x]^2 - Csc[x]^4/4 + Log[Sin[x]]

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Maple [A]  time = 0.01, size = 26, normalized size = 1.3 \[ -{\frac{\ln \left ( 1+ \left ( \tan \left ( x \right ) \right ) ^{2} \right ) }{2}}-{\frac{1}{4\, \left ( \tan \left ( x \right ) \right ) ^{4}}}+\ln \left ( \tan \left ( x \right ) \right ) +{\frac{1}{2\, \left ( \tan \left ( x \right ) \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/tan(x)^5,x)

[Out]

-1/2*ln(1+tan(x)^2)-1/4/tan(x)^4+ln(tan(x))+1/2/tan(x)^2

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Maxima [A]  time = 1.44574, size = 30, normalized size = 1.5 \[ \frac{4 \, \sin \left (x\right )^{2} - 1}{4 \, \sin \left (x\right )^{4}} + \frac{1}{2} \, \log \left (\sin \left (x\right )^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)^(-5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.214313, size = 54, normalized size = 2.7 \[ \frac{2 \, \log \left (\frac{\tan \left (x\right )^{2}}{\tan \left (x\right )^{2} + 1}\right ) \tan \left (x\right )^{4} + 3 \, \tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} - 1}{4 \, \tan \left (x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)^(-5),x, algorithm="fricas")

[Out]

1/4*(2*log(tan(x)^2/(tan(x)^2 + 1))*tan(x)^4 + 3*tan(x)^4 + 2*tan(x)^2 - 1)/tan(
x)^4

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Sympy [A]  time = 0.114924, size = 19, normalized size = 0.95 \[ \frac{4 \sin ^{2}{\left (x \right )} - 1}{4 \sin ^{4}{\left (x \right )}} + \log{\left (\sin{\left (x \right )} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/tan(x)**5,x)

[Out]

(4*sin(x)**2 - 1)/(4*sin(x)**4) + log(sin(x))

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GIAC/XCAS [A]  time = 0.208124, size = 50, normalized size = 2.5 \[ -\frac{3 \, \tan \left (x\right )^{4} - 2 \, \tan \left (x\right )^{2} + 1}{4 \, \tan \left (x\right )^{4}} - \frac{1}{2} \,{\rm ln}\left (\tan \left (x\right )^{2} + 1\right ) + \frac{1}{2} \,{\rm ln}\left (\tan \left (x\right )^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(tan(x)^(-5),x, algorithm="giac")

[Out]

-1/4*(3*tan(x)^4 - 2*tan(x)^2 + 1)/tan(x)^4 - 1/2*ln(tan(x)^2 + 1) + 1/2*ln(tan(
x)^2)