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]

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

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Rubi [A]  time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 1.00 \[ \sin (x)-\frac {\tanh ^{-1}\left (\frac {2 \sin (x)}{\sqrt {3}}\right )}{\sqrt {3}} \]

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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fricas [B]  time = 0.98, size = 36, normalized size = 1.80 \[ \frac {1}{6} \, \sqrt {3} \log \left (-\frac {4 \, \cos \relax (x)^{2} + 4 \, \sqrt {3} \sin \relax (x) - 7}{4 \, \cos \relax (x)^{2} - 1}\right ) + \sin \relax (x) \]

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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giac [B]  time = 0.15, size = 34, normalized size = 1.70 \[ \frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | -4 \, \sqrt {3} + 8 \, \sin \relax (x) \right |}}{{\left | 4 \, \sqrt {3} + 8 \, \sin \relax (x) \right |}}\right ) + \sin \relax (x) \]

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)

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maple [A]  time = 0.12, size = 17, normalized size = 0.85 \[ \sin \relax (x )-\frac {\arctanh \left (\frac {2 \sin \relax (x ) \sqrt {3}}{3}\right ) \sqrt {3}}{3} \]

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 = 0.44, size = 127, normalized size = 6.35 \[ -\frac {1}{12} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \relax (x)^{2} + \frac {4}{3} \, \sin \relax (x)^{2} + \frac {4}{3} \, \sqrt {3} \sin \relax (x) + \frac {4}{3} \, \cos \relax (x) + \frac {4}{3}\right ) - \frac {1}{12} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \relax (x)^{2} + \frac {4}{3} \, \sin \relax (x)^{2} + \frac {4}{3} \, \sqrt {3} \sin \relax (x) - \frac {4}{3} \, \cos \relax (x) + \frac {4}{3}\right ) + \frac {1}{12} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \relax (x)^{2} + \frac {4}{3} \, \sin \relax (x)^{2} - \frac {4}{3} \, \sqrt {3} \sin \relax (x) + \frac {4}{3} \, \cos \relax (x) + \frac {4}{3}\right ) + \frac {1}{12} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \relax (x)^{2} + \frac {4}{3} \, \sin \relax (x)^{2} - \frac {4}{3} \, \sqrt {3} \sin \relax (x) - \frac {4}{3} \, \cos \relax (x) + \frac {4}{3}\right ) + \sin \relax (x) \]

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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mupad [B]  time = 2.37, size = 16, normalized size = 0.80 \[ \sin \relax (x)-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,\sin \relax (x)}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin {\relax (x )} \cot {\left (3 x \right )}\, dx \]

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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