3.1.45 \(\int \cot ^4(x) \, dx\) [45]

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

[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 = {3554, 8} \begin {gather*} x-\frac {1}{3} \cot ^3(x)+\cot (x) \end {gather*}

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 3554

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 \begin {gather*} x+\frac {4 \cot (x)}{3}-\frac {1}{3} \cot (x) \csc ^2(x) \end {gather*}

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 = 0.03, size = 16, normalized size = 1.33

method	result	size
derivativedivides	\(-\frac {\left (\cot ^{3}\left (x \right )\right )}{3}+\cot \left (x \right )-\frac {\pi }{2}+\mathrm {arccot}\left (\cot \left (x \right )\right )\)	\(16\)
default	\(-\frac {\left (\cot ^{3}\left (x \right )\right )}{3}+\cot \left (x \right )-\frac {\pi }{2}+\mathrm {arccot}\left (\cot \left (x \right )\right )\)	\(16\)
norman	\(\frac {-\frac {1}{3}+\tan ^{2}\left (x \right )+x \left (\tan ^{3}\left (x \right )\right )}{\tan \left (x \right )^{3}}\)	\(18\)
risch	\(x +\frac {4 i \left (3 \,{\mathrm e}^{4 i x}-3 \,{\mathrm e}^{2 i x}+2\right )}{3 \left ({\mathrm e}^{2 i x}-1\right )^{3}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.95, size = 16, normalized size = 1.33 \begin {gather*} x + \frac {3 \, \tan \left (x\right )^{2} - 1}{3 \, \tan \left (x\right )^{3}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (10) = 20\).
time = 1.72, size = 48, normalized size = 4.00 \begin {gather*} \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 {gather*}

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 = 0.02, size = 19, normalized size = 1.58 \begin {gather*} x + \frac {\cos {\left (x \right )}}{\sin {\left (x \right )}} - \frac {\cos ^{3}{\left (x \right )}}{3 \sin ^{3}{\left (x \right )}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (10) = 20\).
time = 0.49, size = 34, normalized size = 2.83 \begin {gather*} \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 {gather*}

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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Mupad [B]
time = 0.02, size = 10, normalized size = 0.83 \begin {gather*} -\frac {{\mathrm {cot}\left (x\right )}^3}{3}+\mathrm {cot}\left (x\right )+x \end {gather*}

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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