3.37 \(\int \cot ^2(\frac {3 x}{4}) \, dx\)

Optimal. Leaf size=14 \[ -x-\frac {4}{3} \cot \left (\frac {3 x}{4}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3473, 8} \[ -x-\frac {4}{3} \cot \left (\frac {3 x}{4}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[(3*x)/4]^2,x]

[Out]

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

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Mathematica [C]  time = 0.01, size = 28, normalized size = 2.00 \[ -\frac {4}{3} \cot \left (\frac {3 x}{4}\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2\left (\frac {3 x}{4}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[(3*x)/4]^2,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot ^2\left (\frac {3 x}{4}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cot[(3*x)/4]^2,x]

[Out]

Could not integrate

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fricas [B]  time = 1.00, size = 23, normalized size = 1.64 \[ -\frac {3 \, x \sin \left (\frac {3}{2} \, x\right ) + 4 \, \cos \left (\frac {3}{2} \, x\right ) + 4}{3 \, \sin \left (\frac {3}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(3/4*x)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.19, size = 18, normalized size = 1.29 \[ -x - \frac {2}{3 \, \tan \left (\frac {3}{8} \, x\right )} + \frac {2}{3} \, \tan \left (\frac {3}{8} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(3/4*x)^2,x, algorithm="giac")

[Out]

-x - 2/3/tan(3/8*x) + 2/3*tan(3/8*x)

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maple [A]  time = 0.02, size = 17, normalized size = 1.21

method	result	size
norman	\(\frac {-\frac {4}{3}-x \tan \left (\frac {3 x}{4}\right )}{\tan \left (\frac {3 x}{4}\right )}\)	\(17\)
risch	\(-x -\frac {8 i}{3 \left ({\mathrm e}^{\frac {3 i x}{2}}-1\right )}\)	\(17\)
derivativedivides	\(-\frac {4 \cot \left (\frac {3 x}{4}\right )}{3}+\frac {2 \pi }{3}-\frac {4 \,\mathrm {arccot}\left (\cot \left (\frac {3 x}{4}\right )\right )}{3}\)	\(18\)
default	\(-\frac {4 \cot \left (\frac {3 x}{4}\right )}{3}+\frac {2 \pi }{3}-\frac {4 \,\mathrm {arccot}\left (\cot \left (\frac {3 x}{4}\right )\right )}{3}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(3/4*x)^2,x,method=_RETURNVERBOSE)

[Out]

(-4/3-x*tan(3/4*x))/tan(3/4*x)

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maxima [A]  time = 0.98, size = 12, normalized size = 0.86 \[ -x - \frac {4}{3 \, \tan \left (\frac {3}{4} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(3/4*x)^2,x, algorithm="maxima")

[Out]

-x - 4/3/tan(3/4*x)

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mupad [B]  time = 0.17, size = 10, normalized size = 0.71 \[ -x-\frac {4\,\mathrm {cot}\left (\frac {3\,x}{4}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot((3*x)/4)^2,x)

[Out]

- x - (4*cot((3*x)/4))/3

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sympy [A]  time = 0.07, size = 19, normalized size = 1.36 \[ - x - \frac {4 \cos {\left (\frac {3 x}{4} \right )}}{3 \sin {\left (\frac {3 x}{4} \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(3/4*x)**2,x)

[Out]

-x - 4*cos(3*x/4)/(3*sin(3*x/4))

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