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

Optimal. Leaf size=11 \[ \csc (x)-\frac{\csc ^3(x)}{3} \]

[Out]

Csc[x] - Csc[x]^3/3

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Rubi [A]  time = 0.0247094, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \csc (x)-\frac{\csc ^3(x)}{3} \]

Antiderivative was successfully verified.

[In]  Int[Cot[x]^3*Csc[x],x]

[Out]

Csc[x] - Csc[x]^3/3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 \int \frac{1}{\sin{\left (x \right )} \tan{\left (x \right )}}\, dx}{3} - \frac{1}{3 \sin{\left (x \right )} \tan ^{2}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(x)**3*csc(x),x)

[Out]

-2*Integral(1/(sin(x)*tan(x)), x)/3 - 1/(3*sin(x)*tan(x)**2)

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Mathematica [B]  time = 0.00745944, size = 57, normalized size = 5.18 \[ \frac{5}{12} \tan \left (\frac{x}{2}\right )+\frac{5}{12} \cot \left (\frac{x}{2}\right )-\frac{1}{24} \cot \left (\frac{x}{2}\right ) \csc ^2\left (\frac{x}{2}\right )-\frac{1}{24} \tan \left (\frac{x}{2}\right ) \sec ^2\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x]^3*Csc[x],x]

[Out]

(5*Cot[x/2])/12 - (Cot[x/2]*Csc[x/2]^2)/24 + (5*Tan[x/2])/12 - (Sec[x/2]^2*Tan[x
/2])/24

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Maple [B]  time = 0.013, size = 32, normalized size = 2.9 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{3\, \left ( \sin \left ( x \right ) \right ) ^{3}}}+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{3\,\sin \left ( x \right ) }}+{\frac{ \left ( 2+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(x)^3*csc(x),x)

[Out]

-1/3/sin(x)^3*cos(x)^4+1/3/sin(x)*cos(x)^4+1/3*(2+cos(x)^2)*sin(x)

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Maxima [A]  time = 1.45372, size = 19, normalized size = 1.73 \[ \frac{3 \, \sin \left (x\right )^{2} - 1}{3 \, \sin \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^3*csc(x),x, algorithm="maxima")

[Out]

1/3*(3*sin(x)^2 - 1)/sin(x)^3

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Fricas [A]  time = 0.212615, size = 30, normalized size = 2.73 \[ \frac{3 \, \cos \left (x\right )^{2} - 2}{3 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^3*csc(x),x, algorithm="fricas")

[Out]

1/3*(3*cos(x)^2 - 2)/((cos(x)^2 - 1)*sin(x))

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Sympy [A]  time = 0.093162, size = 14, normalized size = 1.27 \[ \frac{3 \sin ^{2}{\left (x \right )} - 1}{3 \sin ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)**3*csc(x),x)

[Out]

(3*sin(x)**2 - 1)/(3*sin(x)**3)

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GIAC/XCAS [A]  time = 0.199707, size = 19, normalized size = 1.73 \[ \frac{3 \, \sin \left (x\right )^{2} - 1}{3 \, \sin \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^3*csc(x),x, algorithm="giac")

[Out]

1/3*(3*sin(x)^2 - 1)/sin(x)^3