3.337 \(\int \csc ^6(x) \, dx\)

Optimal. Leaf size=21 \[ -\frac{1}{5} \cot ^5(x)-\frac{2 \cot ^3(x)}{3}-\cot (x) \]

[Out]

-Cot[x] - (2*Cot[x]^3)/3 - Cot[x]^5/5

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Rubi [A]  time = 0.016691, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{1}{5} \cot ^5(x)-\frac{2 \cot ^3(x)}{3}-\cot (x) \]

Antiderivative was successfully verified.

[In]  Int[Csc[x]^6,x]

[Out]

-Cot[x] - (2*Cot[x]^3)/3 - Cot[x]^5/5

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Rubi in Sympy [A]  time = 0.624918, size = 32, normalized size = 1.52 \[ - \frac{8 \cos{\left (x \right )}}{15 \sin{\left (x \right )}} - \frac{4 \cos{\left (x \right )}}{15 \sin ^{3}{\left (x \right )}} - \frac{\cos{\left (x \right )}}{5 \sin ^{5}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/sin(x)**6,x)

[Out]

-8*cos(x)/(15*sin(x)) - 4*cos(x)/(15*sin(x)**3) - cos(x)/(5*sin(x)**5)

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Mathematica [A]  time = 0.00484198, size = 27, normalized size = 1.29 \[ -\frac{8 \cot (x)}{15}-\frac{1}{5} \cot (x) \csc ^4(x)-\frac{4}{15} \cot (x) \csc ^2(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Csc[x]^6,x]

[Out]

(-8*Cot[x])/15 - (4*Cot[x]*Csc[x]^2)/15 - (Cot[x]*Csc[x]^4)/5

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Maple [A]  time = 0.043, size = 18, normalized size = 0.9 \[ \left ( -{\frac{8}{15}}-{\frac{ \left ( \csc \left ( x \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \csc \left ( x \right ) \right ) ^{2}}{15}} \right ) \cot \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/sin(x)^6,x)

[Out]

(-8/15-1/5*csc(x)^4-4/15*csc(x)^2)*cot(x)

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Maxima [A]  time = 1.3529, size = 27, normalized size = 1.29 \[ -\frac{15 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 3}{15 \, \tan \left (x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^(-6),x, algorithm="maxima")

[Out]

-1/15*(15*tan(x)^4 + 10*tan(x)^2 + 3)/tan(x)^5

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Fricas [A]  time = 0.21989, size = 50, normalized size = 2.38 \[ -\frac{8 \, \cos \left (x\right )^{5} - 20 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )}{15 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^(-6),x, algorithm="fricas")

[Out]

-1/15*(8*cos(x)^5 - 20*cos(x)^3 + 15*cos(x))/((cos(x)^4 - 2*cos(x)^2 + 1)*sin(x)
)

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Sympy [A]  time = 0.041453, size = 32, normalized size = 1.52 \[ - \frac{8 \cos{\left (x \right )}}{15 \sin{\left (x \right )}} - \frac{4 \cos{\left (x \right )}}{15 \sin ^{3}{\left (x \right )}} - \frac{\cos{\left (x \right )}}{5 \sin ^{5}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sin(x)**6,x)

[Out]

-8*cos(x)/(15*sin(x)) - 4*cos(x)/(15*sin(x)**3) - cos(x)/(5*sin(x)**5)

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GIAC/XCAS [A]  time = 0.200622, size = 27, normalized size = 1.29 \[ -\frac{15 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 3}{15 \, \tan \left (x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^(-6),x, algorithm="giac")

[Out]

-1/15*(15*tan(x)^4 + 10*tan(x)^2 + 3)/tan(x)^5