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

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

[Out]

-x - (4*Cot[(3*x)/4])/3

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Rubi [A]  time = 0.0122771, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -x-\frac{4}{3} \cot \left (\frac{3 x}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Cot[(3*x)/4]^2,x]

[Out]

-x - (4*Cot[(3*x)/4])/3

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Rubi in Sympy [A]  time = 0.515926, size = 12, normalized size = 0.86 \[ - x - \frac{4}{3 \tan{\left (\frac{3 x}{4} \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(3/4*x)**2,x)

[Out]

-x - 4/(3*tan(3*x/4))

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Mathematica [A]  time = 0.00976332, size = 14, normalized size = 1. \[ -x-\frac{4}{3} \cot \left (\frac{3 x}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[(3*x)/4]^2,x]

[Out]

-x - (4*Cot[(3*x)/4])/3

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Maple [A]  time = 0.004, size = 14, normalized size = 1. \[ -{\frac{4}{3}\cot \left ({\frac{3\,x}{4}} \right ) }+{\frac{2\,\pi }{3}}-x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(3/4*x)^2,x)

[Out]

-4/3*cot(3/4*x)+2/3*Pi-x

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Maxima [A]  time = 1.50453, size = 16, normalized size = 1.14 \[ -x - \frac{4}{3 \, \tan \left (\frac{3}{4} \, x\right )} \]

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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Fricas [A]  time = 0.84907, size = 31, normalized size = 2.21 \[ -\frac{3 \, x \sin \left (\frac{3}{2} \, x\right ) + 4 \, \cos \left (\frac{3}{2} \, x\right ) + 4}{3 \, \sin \left (\frac{3}{2} \, x\right )} \]

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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Sympy [A]  time = 0.051077, 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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GIAC/XCAS [A]  time = 0.21649, size = 24, normalized size = 1.71 \[ -x - \frac{2}{3 \, \tan \left (\frac{3}{8} \, x\right )} + \frac{2}{3} \, \tan \left (\frac{3}{8} \, x\right ) \]

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)