3.76 \(\int \csc ^5(x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0227575, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{3}{8} \tanh ^{-1}(\cos (x))-\frac{1}{4} \cot (x) \csc ^3(x)-\frac{3}{8} \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]  Int[Csc[x]^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 0.597512, size = 31, normalized size = 1.19 \[ - \frac{3 \operatorname{atanh}{\left (\cos{\left (x \right )} \right )}}{8} - \frac{3 \cos{\left (x \right )}}{8 \sin ^{2}{\left (x \right )}} - \frac{\cos{\left (x \right )}}{4 \sin ^{4}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/sin(x)**5,x)

[Out]

-3*atanh(cos(x))/8 - 3*cos(x)/(8*sin(x)**2) - cos(x)/(4*sin(x)**4)

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Mathematica [B]  time = 0.00863986, size = 71, normalized size = 2.73 \[ -\frac{1}{64} \csc ^4\left (\frac{x}{2}\right )-\frac{3}{32} \csc ^2\left (\frac{x}{2}\right )+\frac{1}{64} \sec ^4\left (\frac{x}{2}\right )+\frac{3}{32} \sec ^2\left (\frac{x}{2}\right )+\frac{3}{8} \log \left (\sin \left (\frac{x}{2}\right )\right )-\frac{3}{8} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Csc[x]^5,x]

[Out]

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

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Maple [A]  time = 0.056, size = 26, normalized size = 1. \[ \left ( -{\frac{ \left ( \csc \left ( x \right ) \right ) ^{3}}{4}}-{\frac{3\,\csc \left ( x \right ) }{8}} \right ) \cot \left ( x \right ) +{\frac{3\,\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/sin(x)^5,x)

[Out]

(-1/4*csc(x)^3-3/8*csc(x))*cot(x)+3/8*ln(csc(x)-cot(x))

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Maxima [A]  time = 1.34927, size = 57, normalized size = 2.19 \[ \frac{3 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )}{8 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} - \frac{3}{16} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{3}{16} \, \log \left (\cos \left (x\right ) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^(-5),x, algorithm="maxima")

[Out]

1/8*(3*cos(x)^3 - 5*cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1) - 3/16*log(cos(x) + 1) +
3/16*log(cos(x) - 1)

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Fricas [A]  time = 0.226835, size = 93, normalized size = 3.58 \[ \frac{6 \, \cos \left (x\right )^{3} - 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 3 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 10 \, \cos \left (x\right )}{16 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^(-5),x, algorithm="fricas")

[Out]

1/16*(6*cos(x)^3 - 3*(cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*cos(x) + 1/2) + 3*(cos(
x)^4 - 2*cos(x)^2 + 1)*log(-1/2*cos(x) + 1/2) - 10*cos(x))/(cos(x)^4 - 2*cos(x)^
2 + 1)

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Sympy [A]  time = 0.172121, size = 46, normalized size = 1.77 \[ \frac{3 \cos ^{3}{\left (x \right )} - 5 \cos{\left (x \right )}}{8 \cos ^{4}{\left (x \right )} - 16 \cos ^{2}{\left (x \right )} + 8} + \frac{3 \log{\left (\cos{\left (x \right )} - 1 \right )}}{16} - \frac{3 \log{\left (\cos{\left (x \right )} + 1 \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sin(x)**5,x)

[Out]

(3*cos(x)**3 - 5*cos(x))/(8*cos(x)**4 - 16*cos(x)**2 + 8) + 3*log(cos(x) - 1)/16
- 3*log(cos(x) + 1)/16

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GIAC/XCAS [A]  time = 0.216434, size = 51, normalized size = 1.96 \[ \frac{3 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )}{8 \,{\left (\cos \left (x\right )^{2} - 1\right )}^{2}} - \frac{3}{16} \,{\rm ln}\left (\cos \left (x\right ) + 1\right ) + \frac{3}{16} \,{\rm ln}\left (-\cos \left (x\right ) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sin(x)^(-5),x, algorithm="giac")

[Out]

1/8*(3*cos(x)^3 - 5*cos(x))/(cos(x)^2 - 1)^2 - 3/16*ln(cos(x) + 1) + 3/16*ln(-co
s(x) + 1)