3.99 \(\int \cot ^3(x) \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{2} \cot ^2(x)-\log (\sin (x)) \]

[Out]

-Cot[x]^2/2 - Log[Sin[x]]

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Rubi [A]  time = 0.0138111, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{1}{2} \cot ^2(x)-\log (\sin (x)) \]

Antiderivative was successfully verified.

[In]  Int[Cot[x]^3,x]

[Out]

-Cot[x]^2/2 - Log[Sin[x]]

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Rubi in Sympy [A]  time = 0.497006, size = 14, normalized size = 1. \[ - \log{\left (\sin{\left (x \right )} \right )} - \frac{1}{2 \tan ^{2}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(x)**3,x)

[Out]

-log(sin(x)) - 1/(2*tan(x)**2)

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Mathematica [A]  time = 0.00494694, size = 14, normalized size = 1. \[ -\frac{1}{2} \csc ^2(x)-\log (\sin (x)) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x]^3,x]

[Out]

-Csc[x]^2/2 - Log[Sin[x]]

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Maple [A]  time = 0.005, size = 17, normalized size = 1.2 \[ -{\frac{ \left ( \cot \left ( x \right ) \right ) ^{2}}{2}}+{\frac{\ln \left ( \left ( \cot \left ( x \right ) \right ) ^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(x)^3,x)

[Out]

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

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Maxima [A]  time = 1.34688, size = 19, normalized size = 1.36 \[ -\frac{1}{2 \, \sin \left (x\right )^{2}} - \frac{1}{2} \, \log \left (\sin \left (x\right )^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^3,x, algorithm="maxima")

[Out]

-1/2/sin(x)^2 - 1/2*log(sin(x)^2)

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Fricas [A]  time = 0.24413, size = 38, normalized size = 2.71 \[ -\frac{{\left (\cos \left (2 \, x\right ) - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (2 \, x\right ) + \frac{1}{2}\right ) - 2}{2 \,{\left (\cos \left (2 \, x\right ) - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.091527, size = 14, normalized size = 1. \[ - \log{\left (\sin{\left (x \right )} \right )} - \frac{1}{2 \sin ^{2}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)**3,x)

[Out]

-log(sin(x)) - 1/(2*sin(x)**2)

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GIAC/XCAS [A]  time = 0.20577, size = 30, normalized size = 2.14 \[ \frac{1}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} - \frac{1}{2} \,{\rm ln}\left (-\cos \left (x\right )^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^3,x, algorithm="giac")

[Out]

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