3.1.4 \(\int \frac {1}{4} \csc (\frac {x}{3}) \, dx\) [4]

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

[Out]

-3/4*arctanh(cos(1/3*x))

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 3855} \begin {gather*} -\frac {3}{4} \tanh ^{-1}\left (\cos \left (\frac {x}{3}\right )\right ) \end {gather*}

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 3855

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] Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(11)=22\).
time = 0.01, size = 23, normalized size = 2.09 \begin {gather*} \frac {1}{4} \left (-3 \log \left (\cos \left (\frac {x}{6}\right )\right )+3 \log \left (\sin \left (\frac {x}{6}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 15, normalized size = 1.36

method	result	size
norman	\(\frac {3 \ln \left (\tan \left (\frac {x}{6}\right )\right )}{4}\)	\(8\)
derivativedivides	\(\frac {3 \ln \left (\csc \left (\frac {x}{3}\right )-\cot \left (\frac {x}{3}\right )\right )}{4}\)	\(15\)
default	\(\frac {3 \ln \left (\csc \left (\frac {x}{3}\right )-\cot \left (\frac {x}{3}\right )\right )}{4}\)	\(15\)
risch	\(-\frac {3 \ln \left ({\mathrm e}^{\frac {i x}{3}}+1\right )}{4}+\frac {3 \ln \left ({\mathrm e}^{\frac {i x}{3}}-1\right )}{4}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4/sin(1/3*x),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (7) = 14\).
time = 1.58, size = 19, normalized size = 1.73 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (7) = 14\).
time = 0.59, size = 23, normalized size = 2.09 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (10) = 20\).
time = 0.05, size = 22, normalized size = 2.00 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (7) = 14\).
time = 1.16, size = 21, normalized size = 1.91 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.07, size = 7, normalized size = 0.64 \begin {gather*} \frac {3\,\ln \left (\mathrm {tan}\left (\frac {x}{6}\right )\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*sin(x/3)),x)

[Out]

(3*log(tan(x/6)))/4

________________________________________________________________________________________