\(\int \cot ^4(\frac {\pi }{4}+\frac {x}{3}) \, dx\) [343]

Optimal result

Integrand size = 14, antiderivative size = 32 \[ \int \cot ^4\left (\frac {\pi }{4}+\frac {x}{3}\right ) \, dx=x+3 \cot \left (\frac {\pi }{4}+\frac {x}{3}\right )-\cot ^3\left (\frac {\pi }{4}+\frac {x}{3}\right ) \]

[Out]

x+3*cot(1/4*Pi+1/3*x)-cot(1/4*Pi+1/3*x)^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3554, 8} \[ \int \cot ^4\left (\frac {\pi }{4}+\frac {x}{3}\right ) \, dx=x-\cot ^3\left (\frac {x}{3}+\frac {\pi }{4}\right )+3 \cot \left (\frac {x}{3}+\frac {\pi }{4}\right ) \]

[In]

Int[Cot[Pi/4 + x/3]^4,x]

[Out]

x + 3*Cot[Pi/4 + x/3] - Cot[Pi/4 + 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*} \text {integral}& = -\cot ^3\left (\frac {\pi }{4}+\frac {x}{3}\right )-\int \tan ^2\left (\frac {\pi }{4}-\frac {x}{3}\right ) \, dx \\ & = 3 \cot \left (\frac {\pi }{4}+\frac {x}{3}\right )-\cot ^3\left (\frac {\pi }{4}+\frac {x}{3}\right )+\int 1 \, dx \\ & = x+3 \cot \left (\frac {\pi }{4}+\frac {x}{3}\right )-\cot ^3\left (\frac {\pi }{4}+\frac {x}{3}\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \cot ^4\left (\frac {\pi }{4}+\frac {x}{3}\right ) \, dx=-\cot ^3\left (\frac {\pi }{4}+\frac {x}{3}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\tan ^2\left (\frac {\pi }{4}+\frac {x}{3}\right )\right ) \]

[In]

Integrate[Cot[Pi/4 + x/3]^4,x]

[Out]

-(Cot[Pi/4 + x/3]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[Pi/4 + x/3]^2])

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78

method	result	size
parallelrisch	\(x +3 \cot \left (\frac {\pi }{4}+\frac {x}{3}\right )-\left (\cot ^{3}\left (\frac {\pi }{4}+\frac {x}{3}\right )\right )\)	\(25\)
derivativedivides	\(-\left (\cot ^{3}\left (\frac {\pi }{4}+\frac {x}{3}\right )\right )+3 \cot \left (\frac {\pi }{4}+\frac {x}{3}\right )-\frac {3 \pi }{2}+3 \,\operatorname {arccot}\left (\cot \left (\frac {\pi }{4}+\frac {x}{3}\right )\right )\)	\(38\)
default	\(-\left (\cot ^{3}\left (\frac {\pi }{4}+\frac {x}{3}\right )\right )+3 \cot \left (\frac {\pi }{4}+\frac {x}{3}\right )-\frac {3 \pi }{2}+3 \,\operatorname {arccot}\left (\cot \left (\frac {\pi }{4}+\frac {x}{3}\right )\right )\)	\(38\)
norman	\(\frac {-1+x \left (\tan ^{3}\left (\frac {\pi }{4}+\frac {x}{3}\right )\right )+3 \left (\tan ^{2}\left (\frac {\pi }{4}+\frac {x}{3}\right )\right )}{\tan \left (\frac {\pi }{4}+\frac {x}{3}\right )^{3}}\)	\(38\)
risch	\(x +\frac {4 i \left (-3 \,{\mathrm e}^{\frac {4 i x}{3}}-3 i {\mathrm e}^{\frac {2 i x}{3}}+2\right )}{\left ({\mathrm e}^{\frac {i \left (3 \pi +4 x \right )}{6}}-1\right )^{3}}\)	\(38\)

[In]

int(cot(1/4*Pi+1/3*x)^4,x,method=_RETURNVERBOSE)

[Out]

x+3*cot(1/4*Pi+1/3*x)-cot(1/4*Pi+1/3*x)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).

Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \[ \int \cot ^4\left (\frac {\pi }{4}+\frac {x}{3}\right ) \, dx=\frac {4 \, \cos \left (\frac {1}{2} \, \pi + \frac {2}{3} \, x\right )^{2} + {\left (x \cos \left (\frac {1}{2} \, \pi + \frac {2}{3} \, x\right ) - x\right )} \sin \left (\frac {1}{2} \, \pi + \frac {2}{3} \, x\right ) + 2 \, \cos \left (\frac {1}{2} \, \pi + \frac {2}{3} \, x\right ) - 2}{{\left (\cos \left (\frac {1}{2} \, \pi + \frac {2}{3} \, x\right ) - 1\right )} \sin \left (\frac {1}{2} \, \pi + \frac {2}{3} \, x\right )} \]

[In]

integrate(cot(1/4*pi+1/3*x)^4,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \cot ^4\left (\frac {\pi }{4}+\frac {x}{3}\right ) \, dx=x - \cot ^{3}{\left (\frac {x}{3} + \frac {\pi }{4} \right )} + 3 \cot {\left (\frac {x}{3} + \frac {\pi }{4} \right )} \]

[In]

integrate(cot(1/4*pi+1/3*x)**4,x)

[Out]

x - cot(x/3 + pi/4)**3 + 3*cot(x/3 + pi/4)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \cot ^4\left (\frac {\pi }{4}+\frac {x}{3}\right ) \, dx=\frac {3}{4} \, \pi + x + \frac {3 \, \tan \left (\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )^{2} - 1}{\tan \left (\frac {1}{4} \, \pi + \frac {1}{3} \, x\right )^{3}} \]

[In]

integrate(cot(1/4*pi+1/3*x)^4,x, algorithm="maxima")

[Out]

3/4*pi + x + (3*tan(1/4*pi + 1/3*x)^2 - 1)/tan(1/4*pi + 1/3*x)^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).

Time = 0.30 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.66 \[ \int \cot ^4\left (\frac {\pi }{4}+\frac {x}{3}\right ) \, dx=\frac {3}{4} \, \pi + \frac {1}{8} \, \tan \left (\frac {1}{8} \, \pi + \frac {1}{6} \, x\right )^{3} + x + \frac {15 \, \tan \left (\frac {1}{8} \, \pi + \frac {1}{6} \, x\right )^{2} - 1}{8 \, \tan \left (\frac {1}{8} \, \pi + \frac {1}{6} \, x\right )^{3}} - \frac {15}{8} \, \tan \left (\frac {1}{8} \, \pi + \frac {1}{6} \, x\right ) \]

[In]

integrate(cot(1/4*pi+1/3*x)^4,x, algorithm="giac")

[Out]

3/4*pi + 1/8*tan(1/8*pi + 1/6*x)^3 + x + 1/8*(15*tan(1/8*pi + 1/6*x)^2 - 1)/tan(1/8*pi + 1/6*x)^3 - 15/8*tan(1
/8*pi + 1/6*x)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \cot ^4\left (\frac {\pi }{4}+\frac {x}{3}\right ) \, dx=-{\mathrm {cot}\left (\frac {\Pi }{4}+\frac {x}{3}\right )}^3+3\,\mathrm {cot}\left (\frac {\Pi }{4}+\frac {x}{3}\right )+x \]

[In]

int(cot(Pi/4 + x/3)^4,x)

[Out]

x + 3*cot(Pi/4 + x/3) - cot(Pi/4 + x/3)^3