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

Optimal. Leaf size=8 \[ -\frac{1}{2} \csc ^2(x) \]

[Out]

-Csc[x]^2/2

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Rubi [A]  time = 0.0205464, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{1}{2} \csc ^2(x) \]

Antiderivative was successfully verified.

[In]  Int[Cot[x]*Csc[x]^2,x]

[Out]

-Csc[x]^2/2

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Rubi in Sympy [A]  time = 1.15365, size = 12, normalized size = 1.5 \[ - \frac{\cos ^{2}{\left (x \right )}}{2 \sin ^{2}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)/sin(x)**3,x)

[Out]

-cos(x)**2/(2*sin(x)**2)

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Mathematica [A]  time = 0.00307728, size = 8, normalized size = 1. \[ -\frac{1}{2} \cot ^2(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x]*Csc[x]^2,x]

[Out]

-Cot[x]^2/2

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Maple [A]  time = 0.021, size = 7, normalized size = 0.9 \[ -{\frac{1}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)/sin(x)^3,x)

[Out]

-1/2/sin(x)^2

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Maxima [A]  time = 1.35531, size = 8, normalized size = 1. \[ -\frac{1}{2 \, \sin \left (x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)/sin(x)^3,x, algorithm="maxima")

[Out]

-1/2/sin(x)^2

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Fricas [A]  time = 0.206378, size = 14, normalized size = 1.75 \[ \frac{1}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)/sin(x)^3,x, algorithm="fricas")

[Out]

1/2/(cos(x)^2 - 1)

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Sympy [A]  time = 0.04039, size = 8, normalized size = 1. \[ - \frac{1}{2 \sin ^{2}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)/sin(x)**3,x)

[Out]

-1/(2*sin(x)**2)

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GIAC/XCAS [A]  time = 0.218373, size = 14, normalized size = 1.75 \[ \frac{1}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)/sin(x)^3,x, algorithm="giac")

[Out]

1/2/(cos(x)^2 - 1)