\(\int \csc (3 x) \sin (x) \, dx\) [91]

Optimal result

Integrand size = 7, antiderivative size = 45 \[ \int \csc (3 x) \sin (x) \, dx=-\frac {\log \left (\sqrt {3} \cos (x)-\sin (x)\right )}{2 \sqrt {3}}+\frac {\log \left (\sqrt {3} \cos (x)+\sin (x)\right )}{2 \sqrt {3}} \]

[Out]

-1/6*ln(-sin(x)+cos(x)*3^(1/2))*3^(1/2)+1/6*ln(sin(x)+cos(x)*3^(1/2))*3^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {212} \[ \int \csc (3 x) \sin (x) \, dx=\frac {\log \left (\sin (x)+\sqrt {3} \cos (x)\right )}{2 \sqrt {3}}-\frac {\log \left (\sqrt {3} \cos (x)-\sin (x)\right )}{2 \sqrt {3}} \]

[In]

Int[Csc[3*x]*Sin[x],x]

[Out]

-1/2*Log[Sqrt[3]*Cos[x] - Sin[x]]/Sqrt[3] + Log[Sqrt[3]*Cos[x] + Sin[x]]/(2*Sqrt[3])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{3-x^2} \, dx,x,\tan (x)\right ) \\ & = -\frac {\log \left (\sqrt {3} \cos (x)-\sin (x)\right )}{2 \sqrt {3}}+\frac {\log \left (\sqrt {3} \cos (x)+\sin (x)\right )}{2 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.33 \[ \int \csc (3 x) \sin (x) \, dx=\frac {\text {arctanh}\left (\frac {\tan (x)}{\sqrt {3}}\right )}{\sqrt {3}} \]

[In]

Integrate[Csc[3*x]*Sin[x],x]

[Out]

ArcTanh[Tan[x]/Sqrt[3]]/Sqrt[3]

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.31

method	result	size
default	\(\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\tan \left (x \right ) \sqrt {3}}{3}\right )}{3}\)	\(14\)
risch	\(\frac {\sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{6}-\frac {\sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{6}\)	\(40\)

[In]

int(csc(3*x)*sin(x),x,method=_RETURNVERBOSE)

[Out]

1/3*3^(1/2)*arctanh(1/3*tan(x)*3^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.29 \[ \int \csc (3 x) \sin (x) \, dx=\frac {1}{12} \, \sqrt {3} \log \left (-\frac {8 \, \cos \left (x\right )^{4} - 16 \, \cos \left (x\right )^{2} - 4 \, {\left (2 \, \sqrt {3} \cos \left (x\right )^{3} + \sqrt {3} \cos \left (x\right )\right )} \sin \left (x\right ) - 1}{16 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2} + 1}\right ) \]

[In]

integrate(csc(3*x)*sin(x),x, algorithm="fricas")

[Out]

1/12*sqrt(3)*log(-(8*cos(x)^4 - 16*cos(x)^2 - 4*(2*sqrt(3)*cos(x)^3 + sqrt(3)*cos(x))*sin(x) - 1)/(16*cos(x)^4
- 8*cos(x)^2 + 1))

Sympy [A] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.69 \[ \int \csc (3 x) \sin (x) \, dx=\frac {\sqrt {3} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {3} \right )}}{6} - \frac {\sqrt {3} \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {\sqrt {3}}{3} \right )}}{6} + \frac {\sqrt {3} \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {\sqrt {3}}{3} \right )}}{6} - \frac {\sqrt {3} \log {\left (\tan {\left (\frac {x}{2} \right )} + \sqrt {3} \right )}}{6} \]

[In]

integrate(csc(3*x)*sin(x),x)

[Out]

sqrt(3)*log(tan(x/2) - sqrt(3))/6 - sqrt(3)*log(tan(x/2) - sqrt(3)/3)/6 + sqrt(3)*log(tan(x/2) + sqrt(3)/3)/6
- sqrt(3)*log(tan(x/2) + sqrt(3))/6

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (33) = 66\).

Time = 0.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.78 \[ \int \csc (3 x) \sin (x) \, dx=-\frac {1}{12} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \left (x\right )^{2} + \frac {4}{3} \, \sin \left (x\right )^{2} + \frac {4}{3} \, \sqrt {3} \sin \left (x\right ) + \frac {4}{3} \, \cos \left (x\right ) + \frac {4}{3}\right ) + \frac {1}{12} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \left (x\right )^{2} + \frac {4}{3} \, \sin \left (x\right )^{2} + \frac {4}{3} \, \sqrt {3} \sin \left (x\right ) - \frac {4}{3} \, \cos \left (x\right ) + \frac {4}{3}\right ) + \frac {1}{12} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \left (x\right )^{2} + \frac {4}{3} \, \sin \left (x\right )^{2} - \frac {4}{3} \, \sqrt {3} \sin \left (x\right ) + \frac {4}{3} \, \cos \left (x\right ) + \frac {4}{3}\right ) - \frac {1}{12} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \left (x\right )^{2} + \frac {4}{3} \, \sin \left (x\right )^{2} - \frac {4}{3} \, \sqrt {3} \sin \left (x\right ) - \frac {4}{3} \, \cos \left (x\right ) + \frac {4}{3}\right ) \]

[In]

integrate(csc(3*x)*sin(x),x, algorithm="maxima")

[Out]

-1/12*sqrt(3)*log(4/3*cos(x)^2 + 4/3*sin(x)^2 + 4/3*sqrt(3)*sin(x) + 4/3*cos(x) + 4/3) + 1/12*sqrt(3)*log(4/3*
cos(x)^2 + 4/3*sin(x)^2 + 4/3*sqrt(3)*sin(x) - 4/3*cos(x) + 4/3) + 1/12*sqrt(3)*log(4/3*cos(x)^2 + 4/3*sin(x)^
2 - 4/3*sqrt(3)*sin(x) + 4/3*cos(x) + 4/3) - 1/12*sqrt(3)*log(4/3*cos(x)^2 + 4/3*sin(x)^2 - 4/3*sqrt(3)*sin(x)
- 4/3*cos(x) + 4/3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.69 \[ \int \csc (3 x) \sin (x) \, dx=-\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \tan \left (x\right ) \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \tan \left (x\right ) \right |}}\right ) \]

[In]

integrate(csc(3*x)*sin(x),x, algorithm="giac")

[Out]

-1/6*sqrt(3)*log(abs(-2*sqrt(3) + 2*tan(x))/abs(2*sqrt(3) + 2*tan(x)))

Mupad [B] (verification not implemented)

Time = 28.31 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.38 \[ \int \csc (3 x) \sin (x) \, dx=\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\sin \left (x\right )}{3\,\cos \left (x\right )}\right )}{3} \]

[In]

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

[Out]

(3^(1/2)*atanh((3^(1/2)*sin(x))/(3*cos(x))))/3