3.245 \(\int \cot ^4(x) \, dx\)

Optimal. Leaf size=12 \[ x-\frac{1}{3} \cot ^3(x)+\cot (x) \]

[Out]

x + Cot[x] - Cot[x]^3/3

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Rubi [A]  time = 0.0149954, antiderivative size = 12, 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 \[ x-\frac{1}{3} \cot ^3(x)+\cot (x) \]

Antiderivative was successfully verified.

[In]  Int[Cot[x]^4,x]

[Out]

x + Cot[x] - Cot[x]^3/3

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Rubi in Sympy [A]  time = 0.520162, size = 14, normalized size = 1.17 \[ x + \frac{1}{\tan{\left (x \right )}} - \frac{1}{3 \tan ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(x)**4,x)

[Out]

x + 1/tan(x) - 1/(3*tan(x)**3)

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Mathematica [A]  time = 0.00525028, size = 18, normalized size = 1.5 \[ x+\frac{4 \cot (x)}{3}-\frac{1}{3} \cot (x) \csc ^2(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x]^4,x]

[Out]

x + (4*Cot[x])/3 - (Cot[x]*Csc[x]^2)/3

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Maple [A]  time = 0.004, size = 14, normalized size = 1.2 \[ -{\frac{ \left ( \cot \left ( x \right ) \right ) ^{3}}{3}}+\cot \left ( x \right ) -{\frac{\pi }{2}}+x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(x)^4,x)

[Out]

-1/3*cot(x)^3+cot(x)-1/2*Pi+x

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Maxima [A]  time = 1.55278, size = 22, normalized size = 1.83 \[ x + \frac{3 \, \tan \left (x\right )^{2} - 1}{3 \, \tan \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^4,x, algorithm="maxima")

[Out]

x + 1/3*(3*tan(x)^2 - 1)/tan(x)^3

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Fricas [A]  time = 0.246982, size = 65, normalized size = 5.42 \[ \frac{4 \, \cos \left (2 \, x\right )^{2} + 3 \,{\left (x \cos \left (2 \, x\right ) - x\right )} \sin \left (2 \, x\right ) + 2 \, \cos \left (2 \, x\right ) - 2}{3 \,{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.052207, size = 19, normalized size = 1.58 \[ x + \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}} - \frac{\cos ^{3}{\left (x \right )}}{3 \sin ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)**4,x)

[Out]

x + cos(x)/sin(x) - cos(x)**3/(3*sin(x)**3)

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GIAC/XCAS [A]  time = 0.207786, size = 46, normalized size = 3.83 \[ \frac{1}{24} \, \tan \left (\frac{1}{2} \, x\right )^{3} + x + \frac{15 \, \tan \left (\frac{1}{2} \, x\right )^{2} - 1}{24 \, \tan \left (\frac{1}{2} \, x\right )^{3}} - \frac{5}{8} \, \tan \left (\frac{1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^4,x, algorithm="giac")

[Out]

1/24*tan(1/2*x)^3 + x + 1/24*(15*tan(1/2*x)^2 - 1)/tan(1/2*x)^3 - 5/8*tan(1/2*x)