3.105 \(\int \csc ^4(x) \, dx\)

Optimal. Leaf size=13 \[ -\frac{1}{3} \cot ^3(x)-\cot (x) \]

[Out]

-Cot[x] - Cot[x]^3/3

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Rubi [A]  time = 0.0115975, antiderivative size = 13, 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 \[ -\frac{1}{3} \cot ^3(x)-\cot (x) \]

Antiderivative was successfully verified.

[In]  Int[Csc[x]^4,x]

[Out]

-Cot[x] - Cot[x]^3/3

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Rubi in Sympy [A]  time = 0.537123, size = 20, normalized size = 1.54 \[ - \frac{2 \cos{\left (x \right )}}{3 \sin{\left (x \right )}} - \frac{\cos{\left (x \right )}}{3 \sin ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/sin(x)**4,x)

[Out]

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

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Mathematica [A]  time = 0.00387051, size = 17, normalized size = 1.31 \[ -\frac{2 \cot (x)}{3}-\frac{1}{3} \cot (x) \csc ^2(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Csc[x]^4,x]

[Out]

(-2*Cot[x])/3 - (Cot[x]*Csc[x]^2)/3

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Maple [A]  time = 0.045, size = 12, normalized size = 0.9 \[ \left ( -{\frac{2}{3}}-{\frac{ \left ( \csc \left ( x \right ) \right ) ^{2}}{3}} \right ) \cot \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/sin(x)^4,x)

[Out]

(-2/3-1/3*csc(x)^2)*cot(x)

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Maxima [A]  time = 1.34744, size = 19, normalized size = 1.46 \[ -\frac{3 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^(-4),x, algorithm="maxima")

[Out]

-1/3*(3*tan(x)^2 + 1)/tan(x)^3

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Fricas [A]  time = 0.219669, size = 34, normalized size = 2.62 \[ -\frac{2 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{3 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^(-4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.043178, size = 20, normalized size = 1.54 \[ - \frac{2 \cos{\left (x \right )}}{3 \sin{\left (x \right )}} - \frac{\cos{\left (x \right )}}{3 \sin ^{3}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sin(x)**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.201392, size = 19, normalized size = 1.46 \[ -\frac{3 \, \tan \left (x\right )^{2} + 1}{3 \, \tan \left (x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^(-4),x, algorithm="giac")

[Out]

-1/3*(3*tan(x)^2 + 1)/tan(x)^3