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

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 = 0.0099821, 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, 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 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]

Rule 8

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

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*}

Mathematica [A]  time = 0.0030371, 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., size = 14, normalized size = 1.2 \begin{align*} -{\frac{ \left ( \cot \left ( x \right ) \right ) ^{3}}{3}}+\cot \left ( x \right ) -{\frac{\pi }{2}}+x \end{align*}

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 = 1.41448, size = 22, normalized size = 1.83 \begin{align*} x + \frac{3 \, \tan \left (x\right )^{2} - 1}{3 \, \tan \left (x\right )^{3}} \end{align*}

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 [B]  time = 1.65477, size = 126, normalized size = 10.5 \begin{align*} \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 )} \end{align*}

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 = 0.065953, size = 19, normalized size = 1.58 \begin{align*} x + \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}} - \frac{\cos ^{3}{\left (x \right )}}{3 \sin ^{3}{\left (x \right )}} \end{align*}

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 [B]  time = 1.06264, size = 46, normalized size = 3.83 \begin{align*} \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 ) \end{align*}

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)