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

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

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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, 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*}

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Mathematica [B]  time = 0.01, 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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{4} \csc \left (\frac {x}{3}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Could not integrate

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fricas [B]  time = 0.96, size = 23, normalized size = 2.09 \[ -\frac {3}{8} \, \log \left (\frac {1}{2} \, \cos \left (\frac {1}{3} \, x\right ) + \frac {1}{2}\right ) + \frac {3}{8} \, \log \left (-\frac {1}{2} \, \cos \left (\frac {1}{3} \, x\right ) + \frac {1}{2}\right ) \]

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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giac [B]  time = 0.85, size = 23, normalized size = 2.09 \[ -\frac {3}{8} \, \log \left (3 \, \cos \left (\frac {1}{3} \, x\right ) + 3\right ) + \frac {3}{8} \, \log \left (-3 \, \cos \left (\frac {1}{3} \, x\right ) + 3\right ) \]

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)

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maple [A]  time = 0.08, size = 8, normalized size = 0.73

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(tan(1/6*x))

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maxima [B]  time = 0.43, size = 19, normalized size = 1.73 \[ -\frac {3}{8} \, \log \left (\cos \left (\frac {1}{3} \, x\right ) + 1\right ) + \frac {3}{8} \, \log \left (\cos \left (\frac {1}{3} \, x\right ) - 1\right ) \]

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

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

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sympy [B]  time = 0.11, size = 22, normalized size = 2.00 \[ \frac {3 \log {\left (\cos {\left (\frac {x}{3} \right )} - 1 \right )}}{8} - \frac {3 \log {\left (\cos {\left (\frac {x}{3} \right )} + 1 \right )}}{8} \]

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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