3.4 \(\int \frac{1}{4} \csc \left (\frac{x}{3}\right ) \, dx\)

Optimal. Leaf size=11 \[ -\frac{3}{4} \tanh ^{-1}\left (\cos \left (\frac{x}{3}\right )\right ) \]

[Out]

(-3*ArcTanh[Cos[x/3]])/4

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Rubi [A]  time = 0.00796438, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3}{4} \tanh ^{-1}\left (\cos \left (\frac{x}{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Int[Csc[x/3]/4,x]

[Out]

(-3*ArcTanh[Cos[x/3]])/4

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Rubi in Sympy [A]  time = 0.659701, size = 10, normalized size = 0.91 \[ - \frac{3 \operatorname{atanh}{\left (\cos{\left (\frac{x}{3} \right )} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/4/sin(1/3*x),x)

[Out]

-3*atanh(cos(x/3))/4

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Mathematica [B]  time = 0.00549987, size = 23, normalized size = 2.09 \[ \frac{1}{4} \left (3 \log \left (\sin \left (\frac{x}{6}\right )\right )-3 \log \left (\cos \left (\frac{x}{6}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Csc[x/3]/4,x]

[Out]

(-3*Log[Cos[x/6]] + 3*Log[Sin[x/6]])/4

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Maple [A]  time = 0.01, size = 15, normalized size = 1.4 \[{\frac{3}{4}\ln \left ( \csc \left ({\frac{x}{3}} \right ) -\cot \left ({\frac{x}{3}} \right ) \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/4/sin(1/3*x),x)

[Out]

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

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Maxima [A]  time = 1.37099, size = 26, normalized size = 2.36 \[ -\frac{3}{8} \, \log \left (\cos \left (\frac{1}{3} \, x\right ) + 1\right ) + \frac{3}{8} \, \log \left (\cos \left (\frac{1}{3} \, x\right ) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/4/sin(1/3*x),x, algorithm="maxima")

[Out]

-3/8*log(cos(1/3*x) + 1) + 3/8*log(cos(1/3*x) - 1)

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Fricas [A]  time = 0.24232, size = 31, normalized size = 2.82 \[ -\frac{3}{8} \, \log \left (\frac{1}{2} \, \cos \left (\frac{1}{3} \, x\right ) + \frac{1}{2}\right ) + \frac{3}{8} \, \log \left (-\frac{1}{2} \, \cos \left (\frac{1}{3} \, x\right ) + \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/4/sin(1/3*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.099579, size = 22, normalized size = 2. \[ \frac{3 \log{\left (\cos{\left (\frac{x}{3} \right )} - 1 \right )}}{8} - \frac{3 \log{\left (\cos{\left (\frac{x}{3} \right )} + 1 \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/4/sin(1/3*x),x)

[Out]

3*log(cos(x/3) - 1)/8 - 3*log(cos(x/3) + 1)/8

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GIAC/XCAS [A]  time = 0.208553, size = 31, normalized size = 2.82 \[ -\frac{3}{8} \,{\rm ln}\left (3 \, \cos \left (\frac{1}{3} \, x\right ) + 3\right ) + \frac{3}{8} \,{\rm ln}\left (-3 \, \cos \left (\frac{1}{3} \, x\right ) + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/4/sin(1/3*x),x, algorithm="giac")

[Out]

-3/8*ln(3*cos(1/3*x) + 3) + 3/8*ln(-3*cos(1/3*x) + 3)