3.357 \(\int \cot ^2(x) \csc ^3(x) \, dx\)

Optimal. Leaf size=26 \[ \frac{1}{8} \tanh ^{-1}(\cos (x))-\frac{1}{4} \cot (x) \csc ^3(x)+\frac{1}{8} \cot (x) \csc (x) \]

[Out]

ArcTanh[Cos[x]]/8 + (Cot[x]*Csc[x])/8 - (Cot[x]*Csc[x]^3)/4

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Rubi [A]  time = 0.0489788, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{8} \tanh ^{-1}(\cos (x))-\frac{1}{4} \cot (x) \csc ^3(x)+\frac{1}{8} \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]  Int[Cot[x]^2*Csc[x]^3,x]

[Out]

ArcTanh[Cos[x]]/8 + (Cot[x]*Csc[x])/8 - (Cot[x]*Csc[x]^3)/4

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Rubi in Sympy [A]  time = 3.02223, size = 31, normalized size = 1.19 \[ \frac{\operatorname{atanh}{\left (\cos{\left (x \right )} \right )}}{8} + \frac{\cos{\left (x \right )}}{8 \left (- \cos ^{2}{\left (x \right )} + 1\right )} - \frac{\cos{\left (x \right )}}{4 \left (- \cos ^{2}{\left (x \right )} + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(x)**2*csc(x)**3,x)

[Out]

atanh(cos(x))/8 + cos(x)/(8*(-cos(x)**2 + 1)) - cos(x)/(4*(-cos(x)**2 + 1)**2)

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Mathematica [B]  time = 0.0130118, size = 71, normalized size = 2.73 \[ -\frac{1}{64} \csc ^4\left (\frac{x}{2}\right )+\frac{1}{32} \csc ^2\left (\frac{x}{2}\right )+\frac{1}{64} \sec ^4\left (\frac{x}{2}\right )-\frac{1}{32} \sec ^2\left (\frac{x}{2}\right )-\frac{1}{8} \log \left (\sin \left (\frac{x}{2}\right )\right )+\frac{1}{8} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x]^2*Csc[x]^3,x]

[Out]

Csc[x/2]^2/32 - Csc[x/2]^4/64 + Log[Cos[x/2]]/8 - Log[Sin[x/2]]/8 - Sec[x/2]^2/3
2 + Sec[x/2]^4/64

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Maple [A]  time = 0.017, size = 36, normalized size = 1.4 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{4\, \left ( \sin \left ( x \right ) \right ) ^{4}}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{8\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{\cos \left ( x \right ) }{8}}-{\frac{\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(x)^2*csc(x)^3,x)

[Out]

-1/4*cos(x)^3/sin(x)^4-1/8/sin(x)^2*cos(x)^3-1/8*cos(x)-1/8*ln(csc(x)-cot(x))

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Maxima [A]  time = 1.43799, size = 51, normalized size = 1.96 \[ -\frac{\cos \left (x\right )^{3} + \cos \left (x\right )}{8 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} + \frac{1}{16} \, \log \left (\cos \left (x\right ) + 1\right ) - \frac{1}{16} \, \log \left (\cos \left (x\right ) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^2*csc(x)^3,x, algorithm="maxima")

[Out]

-1/8*(cos(x)^3 + cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1) + 1/16*log(cos(x) + 1) - 1/
16*log(cos(x) - 1)

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Fricas [A]  time = 0.223595, size = 92, normalized size = 3.54 \[ -\frac{2 \, \cos \left (x\right )^{3} -{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) +{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2 \, \cos \left (x\right )}{16 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^2*csc(x)^3,x, algorithm="fricas")

[Out]

-1/16*(2*cos(x)^3 - (cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*cos(x) + 1/2) + (cos(x)^
4 - 2*cos(x)^2 + 1)*log(-1/2*cos(x) + 1/2) + 2*cos(x))/(cos(x)^4 - 2*cos(x)^2 +
1)

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Sympy [A]  time = 0.173895, size = 39, normalized size = 1.5 \[ - \frac{\cos ^{3}{\left (x \right )} + \cos{\left (x \right )}}{8 \cos ^{4}{\left (x \right )} - 16 \cos ^{2}{\left (x \right )} + 8} - \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{16} + \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)**2*csc(x)**3,x)

[Out]

-(cos(x)**3 + cos(x))/(8*cos(x)**4 - 16*cos(x)**2 + 8) - log(cos(x) - 1)/16 + lo
g(cos(x) + 1)/16

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GIAC/XCAS [A]  time = 0.20536, size = 63, normalized size = 2.42 \[ -\frac{\frac{1}{\cos \left (x\right )} + \cos \left (x\right )}{8 \,{\left ({\left (\frac{1}{\cos \left (x\right )} + \cos \left (x\right )\right )}^{2} - 4\right )}} + \frac{1}{32} \,{\rm ln}\left ({\left | \frac{1}{\cos \left (x\right )} + \cos \left (x\right ) + 2 \right |}\right ) - \frac{1}{32} \,{\rm ln}\left ({\left | \frac{1}{\cos \left (x\right )} + \cos \left (x\right ) - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)^2*csc(x)^3,x, algorithm="giac")

[Out]

-1/8*(1/cos(x) + cos(x))/((1/cos(x) + cos(x))^2 - 4) + 1/32*ln(abs(1/cos(x) + co
s(x) + 2)) - 1/32*ln(abs(1/cos(x) + cos(x) - 2))