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

Optimal. Leaf size=38 \[ -\frac{1}{16} \tanh ^{-1}(\cos (x))-\frac{1}{6} \cot ^3(x) \csc ^3(x)+\frac{1}{8} \cot (x) \csc ^3(x)-\frac{1}{16} \cot (x) \csc (x) \]

[Out]

-ArcTanh[Cos[x]]/16 - (Cot[x]*Csc[x])/16 + (Cot[x]*Csc[x]^3)/8 - (Cot[x]^3*Csc[x
]^3)/6

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Rubi [A]  time = 0.083856, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{1}{16} \tanh ^{-1}(\cos (x))-\frac{1}{6} \cot ^3(x) \csc ^3(x)+\frac{1}{8} \cot (x) \csc ^3(x)-\frac{1}{16} \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]  Int[Cot[x]^4*Csc[x]^3,x]

[Out]

-ArcTanh[Cos[x]]/16 - (Cot[x]*Csc[x])/16 + (Cot[x]*Csc[x]^3)/8 - (Cot[x]^3*Csc[x
]^3)/6

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Rubi in Sympy [A]  time = 4.07219, size = 46, normalized size = 1.21 \[ - \frac{\operatorname{atanh}{\left (\cos{\left (x \right )} \right )}}{16} - \frac{\cos{\left (x \right )}}{16 \left (- \cos ^{2}{\left (x \right )} + 1\right )} + \frac{\cos{\left (x \right )}}{8 \left (- \cos ^{2}{\left (x \right )} + 1\right )^{2}} - \frac{\cos ^{3}{\left (x \right )}}{6 \left (- \cos ^{2}{\left (x \right )} + 1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(cos(x))/16 - cos(x)/(16*(-cos(x)**2 + 1)) + cos(x)/(8*(-cos(x)**2 + 1)**2
) - cos(x)**3/(6*(-cos(x)**2 + 1)**3)

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Mathematica [B]  time = 0.0147586, size = 95, normalized size = 2.5 \[ -\frac{1}{384} \csc ^6\left (\frac{x}{2}\right )+\frac{1}{64} \csc ^4\left (\frac{x}{2}\right )-\frac{1}{64} \csc ^2\left (\frac{x}{2}\right )+\frac{1}{384} \sec ^6\left (\frac{x}{2}\right )-\frac{1}{64} \sec ^4\left (\frac{x}{2}\right )+\frac{1}{64} \sec ^2\left (\frac{x}{2}\right )+\frac{1}{16} \log \left (\sin \left (\frac{x}{2}\right )\right )-\frac{1}{16} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Cot[x]^4*Csc[x]^3,x]

[Out]

-Csc[x/2]^2/64 + Csc[x/2]^4/64 - Csc[x/2]^6/384 - Log[Cos[x/2]]/16 + Log[Sin[x/2
]]/16 + Sec[x/2]^2/64 - Sec[x/2]^4/64 + Sec[x/2]^6/384

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Maple [A]  time = 0.018, size = 52, normalized size = 1.4 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}}{6\, \left ( \sin \left ( x \right ) \right ) ^{6}}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}}{24\, \left ( \sin \left ( x \right ) \right ) ^{4}}}+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5}}{48\, \left ( \sin \left ( x \right ) \right ) ^{2}}}+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{48}}+{\frac{\cos \left ( x \right ) }{16}}+{\frac{\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cot(x)^4*csc(x)^3,x)

[Out]

-1/6/sin(x)^6*cos(x)^5-1/24/sin(x)^4*cos(x)^5+1/48/sin(x)^2*cos(x)^5+1/48*cos(x)
^3+1/16*cos(x)+1/16*ln(csc(x)-cot(x))

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Maxima [A]  time = 1.41991, size = 73, normalized size = 1.92 \[ \frac{3 \, \cos \left (x\right )^{5} + 8 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{48 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} - \frac{1}{32} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{32} \, \log \left (\cos \left (x\right ) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(3*cos(x)^5 + 8*cos(x)^3 - 3*cos(x))/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 -
1) - 1/32*log(cos(x) + 1) + 1/32*log(cos(x) - 1)

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Fricas [A]  time = 0.231298, size = 126, normalized size = 3.32 \[ \frac{6 \, \cos \left (x\right )^{5} + 16 \, \cos \left (x\right )^{3} - 3 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 6 \, \cos \left (x\right )}{96 \,{\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/96*(6*cos(x)^5 + 16*cos(x)^3 - 3*(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*log(
1/2*cos(x) + 1/2) + 3*(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*log(-1/2*cos(x) +
1/2) - 6*cos(x))/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)

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Sympy [A]  time = 0.209411, size = 56, normalized size = 1.47 \[ \frac{3 \cos ^{5}{\left (x \right )} + 8 \cos ^{3}{\left (x \right )} - 3 \cos{\left (x \right )}}{48 \cos ^{6}{\left (x \right )} - 144 \cos ^{4}{\left (x \right )} + 144 \cos ^{2}{\left (x \right )} - 48} + \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{32} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(3*cos(x)**5 + 8*cos(x)**3 - 3*cos(x))/(48*cos(x)**6 - 144*cos(x)**4 + 144*cos(x
)**2 - 48) + log(cos(x) - 1)/32 - log(cos(x) + 1)/32

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GIAC/XCAS [A]  time = 0.208855, size = 59, normalized size = 1.55 \[ \frac{3 \, \cos \left (x\right )^{5} + 8 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{48 \,{\left (\cos \left (x\right )^{2} - 1\right )}^{3}} - \frac{1}{32} \,{\rm ln}\left (\cos \left (x\right ) + 1\right ) + \frac{1}{32} \,{\rm ln}\left (-\cos \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(3*cos(x)^5 + 8*cos(x)^3 - 3*cos(x))/(cos(x)^2 - 1)^3 - 1/32*ln(cos(x) + 1)
+ 1/32*ln(-cos(x) + 1)