3.8 \(\int \csc ^2(x) \, dx\)

Optimal. Leaf size=4 \[ -\cot (x) \]

[Out]

-Cot[x]

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Rubi [A]  time = 0.00735385, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\cot (x) \]

Antiderivative was successfully verified.

[In]  Int[Csc[x]^2,x]

[Out]

-Cot[x]

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Rubi in Sympy [A]  time = 0.473958, size = 7, normalized size = 1.75 \[ - \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(csc(x)**2,x)

[Out]

-cos(x)/sin(x)

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Mathematica [A]  time = 0.00273265, size = 4, normalized size = 1. \[ -\cot (x) \]

Antiderivative was successfully verified.

[In]  Integrate[Csc[x]^2,x]

[Out]

-Cot[x]

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Maple [A]  time = 0.006, size = 5, normalized size = 1.3 \[ -\cot \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(csc(x)^2,x)

[Out]

-cot(x)

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Maxima [A]  time = 1.42099, size = 8, normalized size = 2. \[ -\frac{1}{\tan \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csc(x)^2,x, algorithm="maxima")

[Out]

-1/tan(x)

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Fricas [A]  time = 0.206007, size = 11, normalized size = 2.75 \[ -\frac{\cos \left (x\right )}{\sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csc(x)^2,x, algorithm="fricas")

[Out]

-cos(x)/sin(x)

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Sympy [A]  time = 0.043857, size = 7, normalized size = 1.75 \[ - \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csc(x)**2,x)

[Out]

-cos(x)/sin(x)

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GIAC/XCAS [A]  time = 0.210021, size = 8, normalized size = 2. \[ -\frac{1}{\tan \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csc(x)^2,x, algorithm="giac")

[Out]

-1/tan(x)