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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 8} \[ x-\frac {1}{3} \cot ^3(x)+\cot (x) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^4,x]

[Out]

x + Cot[x] - Cot[x]^3/3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rubi steps

\begin {align*} \int \cot ^4(x) \, dx &=-\frac {1}{3} \cot ^3(x)-\int \cot ^2(x) \, dx\\ &=\cot (x)-\frac {\cot ^3(x)}{3}+\int 1 \, dx\\ &=x+\cot (x)-\frac {\cot ^3(x)}{3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.50 \[ x+\frac {4 \cot (x)}{3}-\frac {1}{3} \cot (x) \csc ^2(x) \]

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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fricas [B]  time = 0.41, size = 48, normalized size = 4.00 \[ \frac {4 \, \cos \left (2 \, x\right )^{2} + 3 \, {\left (x \cos \left (2 \, x\right ) - x\right )} \sin \left (2 \, x\right ) + 2 \, \cos \left (2 \, x\right ) - 2}{3 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )} \]

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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giac [B]  time = 1.15, size = 34, normalized size = 2.83 \[ \frac {1}{24} \, \tan \left (\frac {1}{2} \, x\right )^{3} + x + \frac {15 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 1}{24 \, \tan \left (\frac {1}{2} \, x\right )^{3}} - \frac {5}{8} \, \tan \left (\frac {1}{2} \, x\right ) \]

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)

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maple [A]  time = 0.00, size = 14, normalized size = 1.17 \[ -\frac {\left (\cot ^{3}\relax (x )\right )}{3}+x +\cot \relax (x )-\frac {\pi }{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^4,x)

[Out]

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

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maxima [A]  time = 0.97, size = 16, normalized size = 1.33 \[ x + \frac {3 \, \tan \relax (x)^{2} - 1}{3 \, \tan \relax (x)^{3}} \]

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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mupad [B]  time = 0.03, size = 10, normalized size = 0.83 \[ -\frac {{\mathrm {cot}\relax (x)}^3}{3}+\mathrm {cot}\relax (x)+x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^4,x)

[Out]

x + cot(x) - cot(x)^3/3

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sympy [A]  time = 0.07, size = 19, normalized size = 1.58 \[ x + \frac {\cos {\relax (x )}}{\sin {\relax (x )}} - \frac {\cos ^{3}{\relax (x )}}{3 \sin ^{3}{\relax (x )}} \]

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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