3.91 \(\int \csc (3 x) \sin (x) \, dx\)

Optimal. Leaf size=45 \[ \frac{\log \left (\sin (x)+\sqrt{3} \cos (x)\right )}{2 \sqrt{3}}-\frac{\log \left (\sqrt{3} \cos (x)-\sin (x)\right )}{2 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0399368, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {206} \[ \frac{\log \left (\sin (x)+\sqrt{3} \cos (x)\right )}{2 \sqrt{3}}-\frac{\log \left (\sqrt{3} \cos (x)-\sin (x)\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

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

Rubi steps

\begin{align*} \int \csc (3 x) \sin (x) \, dx &=\operatorname{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]  time = 0.01982, size = 15, normalized size = 0.33 \[ \frac{\tanh ^{-1}\left (\frac{\tan (x)}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.064, size = 14, normalized size = 0.3 \begin{align*}{\frac{\sqrt{3}}{3}{\it Artanh} \left ({\frac{\tan \left ( x \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.57434, size = 169, normalized size = 3.76 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.34947, size = 177, normalized size = 3.93 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [A]  time = 4.89941, size = 76, normalized size = 1.69 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.20102, size = 42, normalized size = 0.93 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))