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

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 = 0.0032333, 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, Rules used = {12, 3770} \[ -\frac{3}{4} \tanh ^{-1}\left (\cos \left (\frac{x}{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x/3]/4,x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{4} \csc \left (\frac{x}{3}\right ) \, dx &=\frac{1}{4} \int \csc \left (\frac{x}{3}\right ) \, dx\\ &=-\frac{3}{4} \tanh ^{-1}\left (\cos \left (\frac{x}{3}\right )\right )\\ \end{align*}

Mathematica [B]  time = 0.0062143, 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 = 0.006, size = 15, normalized size = 1.4 \begin{align*}{\frac{3}{4}\ln \left ( \csc \left ({\frac{x}{3}} \right ) -\cot \left ({\frac{x}{3}} \right ) \right ) } \end{align*}

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 [B]  time = 0.918564, size = 26, normalized size = 2.36 \begin{align*} -\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 ) \end{align*}

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 [B]  time = 2.08161, size = 88, normalized size = 8. \begin{align*} -\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 ) \end{align*}

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 [B]  time = 0.097576, size = 22, normalized size = 2. \begin{align*} \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} \end{align*}

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 [B]  time = 1.06149, size = 31, normalized size = 2.82 \begin{align*} -\frac{3}{8} \, \log \left (3 \, \cos \left (\frac{1}{3} \, x\right ) + 3\right ) + \frac{3}{8} \, \log \left (-3 \, \cos \left (\frac{1}{3} \, x\right ) + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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