3.9 \(\int \cot (x) \csc ^3(x) \, dx\)

Optimal. Leaf size=8 \[ -\frac{1}{3} \csc ^3(x) \]

[Out]

-Csc[x]^3/3

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Rubi [A]  time = 0.0210645, 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 \[ -\frac{1}{3} \csc ^3(x) \]

Antiderivative was successfully verified.

[In]  Int[Cot[x]*Csc[x]^3,x]

[Out]

-Csc[x]^3/3

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Rubi in Sympy [A]  time = 1.11114, size = 8, normalized size = 1. \[ - \frac{1}{3 \sin ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)*csc(x)**2/sin(x)**2,x)

[Out]

-1/(3*sin(x)**3)

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Mathematica [A]  time = 0.00417226, size = 8, normalized size = 1. \[ -\frac{1}{3} \csc ^3(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x]*Csc[x]^3,x]

[Out]

-Csc[x]^3/3

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Maple [A]  time = 0.018, size = 7, normalized size = 0.9 \[ -{\frac{1}{3\, \left ( \sin \left ( x \right ) \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)*csc(x)^2/sin(x)^2,x)

[Out]

-1/3/sin(x)^3

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Maxima [A]  time = 1.35713, size = 8, normalized size = 1. \[ -\frac{1}{3 \, \sin \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)*csc(x)^2/sin(x)^2,x, algorithm="maxima")

[Out]

-1/3/sin(x)^3

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Fricas [A]  time = 0.219539, size = 19, normalized size = 2.38 \[ \frac{1}{3 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)*csc(x)^2/sin(x)^2,x, algorithm="fricas")

[Out]

1/3/((cos(x)^2 - 1)*sin(x))

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Sympy [A]  time = 0.050865, size = 8, normalized size = 1. \[ - \frac{1}{3 \sin ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)*csc(x)**2/sin(x)**2,x)

[Out]

-1/(3*sin(x)**3)

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GIAC/XCAS [A]  time = 0.20089, size = 8, normalized size = 1. \[ -\frac{1}{3 \, \sin \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)*csc(x)^2/sin(x)^2,x, algorithm="giac")

[Out]

-1/3/sin(x)^3