3.103 \(\int \csc ^3(x) \, dx\)

Optimal. Leaf size=16 \[ -\frac{1}{2} \tanh ^{-1}(\cos (x))-\frac{1}{2} \cot (x) \csc (x) \]

[Out]

-ArcTanh[Cos[x]]/2 - (Cot[x]*Csc[x])/2

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Rubi [A]  time = 0.0121642, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{1}{2} \tanh ^{-1}(\cos (x))-\frac{1}{2} \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]  Int[Csc[x]^3,x]

[Out]

-ArcTanh[Cos[x]]/2 - (Cot[x]*Csc[x])/2

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Rubi in Sympy [A]  time = 0.515742, size = 17, normalized size = 1.06 \[ - \frac{\operatorname{atanh}{\left (\cos{\left (x \right )} \right )}}{2} - \frac{\cos{\left (x \right )}}{2 \sin ^{2}{\left (x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(csc(x)**3,x)

[Out]

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

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Mathematica [B]  time = 0.00674204, size = 47, normalized size = 2.94 \[ -\frac{1}{8} \csc ^2\left (\frac{x}{2}\right )+\frac{1}{8} \sec ^2\left (\frac{x}{2}\right )+\frac{1}{2} \log \left (\sin \left (\frac{x}{2}\right )\right )-\frac{1}{2} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Csc[x]^3,x]

[Out]

-Csc[x/2]^2/8 - Log[Cos[x/2]]/2 + Log[Sin[x/2]]/2 + Sec[x/2]^2/8

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Maple [A]  time = 0.047, size = 18, normalized size = 1.1 \[ -{\frac{\cot \left ( x \right ) \csc \left ( x \right ) }{2}}+{\frac{\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(csc(x)^3,x)

[Out]

-1/2*cot(x)*csc(x)+1/2*ln(csc(x)-cot(x))

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Maxima [A]  time = 1.41989, size = 36, normalized size = 2.25 \[ \frac{\cos \left (x\right )}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} - \frac{1}{4} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\cos \left (x\right ) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.264926, size = 59, normalized size = 3.69 \[ -\frac{{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 2 \, \cos \left (x\right )}{4 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.129703, size = 27, normalized size = 1.69 \[ \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{4} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{4} + \frac{\cos{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(csc(x)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.204533, size = 73, normalized size = 4.56 \[ -\frac{{\left (\frac{2 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 1\right )}{\left (\cos \left (x\right ) + 1\right )}}{8 \,{\left (\cos \left (x\right ) - 1\right )}} - \frac{\cos \left (x\right ) - 1}{8 \,{\left (\cos \left (x\right ) + 1\right )}} + \frac{1}{4} \,{\rm ln}\left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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