3.101 \(\int \cot ^3(x) \csc ^4(x) \, dx\)

Optimal. Leaf size=17 \[ \frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6} \]

[Out]

Csc[x]^4/4 - Csc[x]^6/6

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Rubi [A]  time = 0.0403828, 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 \[ \frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]  Int[Cot[x]^3*Csc[x]^4,x]

[Out]

Csc[x]^4/4 - Csc[x]^6/6

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Rubi in Sympy [A]  time = 3.33918, size = 22, normalized size = 1.29 \[ \frac{1}{4 \left (- \cos ^{2}{\left (x \right )} + 1\right )^{2}} - \frac{1}{6 \left (- \cos ^{2}{\left (x \right )} + 1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(x)**3*csc(x)**4,x)

[Out]

1/(4*(-cos(x)**2 + 1)**2) - 1/(6*(-cos(x)**2 + 1)**3)

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Mathematica [A]  time = 0.008245, size = 17, normalized size = 1. \[ \frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x]^3*Csc[x]^4,x]

[Out]

Csc[x]^4/4 - Csc[x]^6/6

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Maple [A]  time = 0.019, size = 22, normalized size = 1.3 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{6\, \left ( \sin \left ( x \right ) \right ) ^{6}}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}}{12\, \left ( \sin \left ( x \right ) \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(x)^3*csc(x)^4,x)

[Out]

-1/6/sin(x)^6*cos(x)^4-1/12/sin(x)^4*cos(x)^4

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Maxima [A]  time = 1.33638, size = 19, normalized size = 1.12 \[ \frac{3 \, \sin \left (x\right )^{2} - 2}{12 \, \sin \left (x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^3*csc(x)^4,x, algorithm="maxima")

[Out]

1/12*(3*sin(x)^2 - 2)/sin(x)^6

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Fricas [A]  time = 0.265129, size = 41, normalized size = 2.41 \[ \frac{3 \, \cos \left (x\right )^{2} - 1}{12 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^3*csc(x)^4,x, algorithm="fricas")

[Out]

1/12*(3*cos(x)^2 - 1)/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)

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Sympy [A]  time = 0.119313, size = 14, normalized size = 0.82 \[ \frac{3 \sin ^{2}{\left (x \right )} - 2}{12 \sin ^{6}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)**3*csc(x)**4,x)

[Out]

(3*sin(x)**2 - 2)/(12*sin(x)**6)

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GIAC/XCAS [A]  time = 0.197925, size = 24, normalized size = 1.41 \[ \frac{3 \, \cos \left (x\right )^{2} - 1}{12 \,{\left (\cos \left (x\right )^{2} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^3*csc(x)^4,x, algorithm="giac")

[Out]

1/12*(3*cos(x)^2 - 1)/(cos(x)^2 - 1)^3