3.10 \(\int \cot (x) \csc (x) \, dx\)

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

[Out]

-Csc[x]

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Rubi [A]  time = 0.0108525, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\csc (x) \]

Antiderivative was successfully verified.

[In]  Int[Cot[x]*Csc[x],x]

[Out]

-Csc[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\sin{\left (x \right )}}{\tan ^{2}{\left (x \right )}} - \int \frac{\sin{\left (x \right )}}{\tan{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(x)*csc(x),x)

[Out]

-sin(x)/tan(x)**2 - Integral(sin(x)/tan(x), x)

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Mathematica [A]  time = 0.00243347, size = 4, normalized size = 1. \[ -\csc (x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x]*Csc[x],x]

[Out]

-Csc[x]

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Maple [A]  time = 0.009, size = 5, normalized size = 1.3 \[ -\csc \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(x)*csc(x),x)

[Out]

-csc(x)

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Maxima [A]  time = 1.37498, size = 8, normalized size = 2. \[ -\frac{1}{\sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)*csc(x),x, algorithm="maxima")

[Out]

-1/sin(x)

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Fricas [A]  time = 0.20786, size = 8, normalized size = 2. \[ -\frac{1}{\sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)*csc(x),x, algorithm="fricas")

[Out]

-1/sin(x)

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Sympy [A]  time = 0.049476, size = 5, normalized size = 1.25 \[ - \frac{1}{\sin{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)*csc(x),x)

[Out]

-1/sin(x)

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GIAC/XCAS [A]  time = 0.203036, size = 8, normalized size = 2. \[ -\frac{1}{\sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)*csc(x),x, algorithm="giac")

[Out]

-1/sin(x)