3.81 \(\int \cot (3 x) \sin (x) \, dx\)

Optimal. Leaf size=20 \[ \sin (x)-\frac{\tanh ^{-1}\left (\frac{2 \sin (x)}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.026517, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {388, 206} \[ \sin (x)-\frac{\tanh ^{-1}\left (\frac{2 \sin (x)}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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 \cot (3 x) \sin (x) \, dx &=\operatorname{Subst}\left (\int \frac{1-4 x^2}{3-4 x^2} \, dx,x,\sin (x)\right )\\ &=\sin (x)-2 \operatorname{Subst}\left (\int \frac{1}{3-4 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac{\tanh ^{-1}\left (\frac{2 \sin (x)}{\sqrt{3}}\right )}{\sqrt{3}}+\sin (x)\\ \end{align*}

Mathematica [A]  time = 0.0168605, size = 20, normalized size = 1. \[ \sin (x)-\frac{\tanh ^{-1}\left (\frac{2 \sin (x)}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 17, normalized size = 0.9 \begin{align*} \sin \left ( x \right ) -{\frac{\sqrt{3}}{3}{\it Artanh} \left ({\frac{2\,\sin \left ( x \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)-1/3*arctanh(2/3*sin(x)*3^(1/2))*3^(1/2)

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Maxima [B]  time = 1.58707, size = 171, normalized size = 8.55 \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 ) + \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(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) + sin(x)

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Fricas [B]  time = 2.32617, size = 109, normalized size = 5.45 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (-\frac{4 \, \cos \left (x\right )^{2} + 4 \, \sqrt{3} \sin \left (x\right ) - 7}{4 \, \cos \left (x\right )^{2} - 1}\right ) + \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*log(-(4*cos(x)^2 + 4*sqrt(3)*sin(x) - 7)/(4*cos(x)^2 - 1)) + sin(x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (x \right )} \cot{\left (3 x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(x)*cot(3*x), x)

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Giac [B]  time = 1.15718, size = 46, normalized size = 2.3 \begin{align*} \frac{1}{6} \, \sqrt{3} \log \left (\frac{{\left | -4 \, \sqrt{3} + 8 \, \sin \left (x\right ) \right |}}{{\left | 4 \, \sqrt{3} + 8 \, \sin \left (x\right ) \right |}}\right ) + \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(3)*log(abs(-4*sqrt(3) + 8*sin(x))/abs(4*sqrt(3) + 8*sin(x))) + sin(x)