3.100 \(\int \cot ^4(x) \csc ^4(x) \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{7} \cot ^7(x)-\frac{\cot ^5(x)}{5} \]

[Out]

-Cot[x]^5/5 - Cot[x]^7/7

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Rubi [A]  time = 0.0395665, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{1}{7} \cot ^7(x)-\frac{\cot ^5(x)}{5} \]

Antiderivative was successfully verified.

[In]  Int[Cot[x]^4*Csc[x]^4,x]

[Out]

-Cot[x]^5/5 - Cot[x]^7/7

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Rubi in Sympy [A]  time = 2.84001, size = 44, normalized size = 2.59 \[ - \frac{2 \cos{\left (x \right )}}{35 \sin{\left (x \right )}} - \frac{\cos{\left (x \right )}}{35 \sin ^{3}{\left (x \right )}} + \frac{3}{35 \sin ^{4}{\left (x \right )} \tan{\left (x \right )}} - \frac{1}{7 \sin ^{4}{\left (x \right )} \tan ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(x)**4*csc(x)**4,x)

[Out]

-2*cos(x)/(35*sin(x)) - cos(x)/(35*sin(x)**3) + 3/(35*sin(x)**4*tan(x)) - 1/(7*s
in(x)**4*tan(x)**3)

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Mathematica [B]  time = 0.0199545, size = 37, normalized size = 2.18 \[ -\frac{2 \cot (x)}{35}-\frac{1}{7} \cot (x) \csc ^6(x)+\frac{8}{35} \cot (x) \csc ^4(x)-\frac{1}{35} \cot (x) \csc ^2(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x]^4*Csc[x]^4,x]

[Out]

(-2*Cot[x])/35 - (Cot[x]*Csc[x]^2)/35 + (8*Cot[x]*Csc[x]^4)/35 - (Cot[x]*Csc[x]^
6)/7

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Maple [A]  time = 0.019, size = 22, normalized size = 1.3 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}}{7\, \left ( \sin \left ( x \right ) \right ) ^{7}}}-{\frac{2\, \left ( \cos \left ( x \right ) \right ) ^{5}}{35\, \left ( \sin \left ( x \right ) \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(x)^4*csc(x)^4,x)

[Out]

-1/7/sin(x)^7*cos(x)^5-2/35/sin(x)^5*cos(x)^5

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Maxima [A]  time = 1.45815, size = 19, normalized size = 1.12 \[ -\frac{7 \, \tan \left (x\right )^{2} + 5}{35 \, \tan \left (x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^4*csc(x)^4,x, algorithm="maxima")

[Out]

-1/35*(7*tan(x)^2 + 5)/tan(x)^7

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Fricas [A]  time = 0.234645, size = 53, normalized size = 3.12 \[ -\frac{2 \, \cos \left (x\right )^{7} - 7 \, \cos \left (x\right )^{5}}{35 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^4*csc(x)^4,x, algorithm="fricas")

[Out]

-1/35*(2*cos(x)^7 - 7*cos(x)^5)/((cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*sin(x)
)

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Sympy [A]  time = 0.057872, size = 41, normalized size = 2.41 \[ - \frac{2 \cos{\left (x \right )}}{35 \sin{\left (x \right )}} - \frac{\cos{\left (x \right )}}{35 \sin ^{3}{\left (x \right )}} + \frac{8 \cos{\left (x \right )}}{35 \sin ^{5}{\left (x \right )}} - \frac{\cos{\left (x \right )}}{7 \sin ^{7}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)**4*csc(x)**4,x)

[Out]

-2*cos(x)/(35*sin(x)) - cos(x)/(35*sin(x)**3) + 8*cos(x)/(35*sin(x)**5) - cos(x)
/(7*sin(x)**7)

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GIAC/XCAS [A]  time = 0.213308, size = 19, normalized size = 1.12 \[ -\frac{7 \, \tan \left (x\right )^{2} + 5}{35 \, \tan \left (x\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^4*csc(x)^4,x, algorithm="giac")

[Out]

-1/35*(7*tan(x)^2 + 5)/tan(x)^7