3.9 \(\int \frac{2 \csc (4+6 x)}{-3 \cot (4+6 x)+\csc (4+6 x)} \, dx\)

Optimal. Leaf size=60 \[ \frac{\log \left (\cos (3 x+2)-\sqrt{2} \sin (3 x+2)\right )}{6 \sqrt{2}}-\frac{\log \left (\sqrt{2} \sin (3 x+2)+\cos (3 x+2)\right )}{6 \sqrt{2}} \]

[Out]

Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])

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Rubi [A]  time = 0.046477, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {12, 3166, 2659, 207} \[ \frac{\log \left (\cos (3 x+2)-\sqrt{2} \sin (3 x+2)\right )}{6 \sqrt{2}}-\frac{\log \left (\sqrt{2} \sin (3 x+2)+\cos (3 x+2)\right )}{6 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(2*Csc[4 + 6*x])/(-3*Cot[4 + 6*x] + Csc[4 + 6*x]),x]

[Out]

Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3166

Int[csc[(d_.) + (e_.)*(x_)]^(n_.)*((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(m_)
, x_Symbol] :> Int[1/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0]
&& IntegerQ[n]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 207

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

Rubi steps

\begin{align*} \int \frac{2 \csc (4+6 x)}{-3 \cot (4+6 x)+\csc (4+6 x)} \, dx &=2 \int \frac{\csc (4+6 x)}{-3 \cot (4+6 x)+\csc (4+6 x)} \, dx\\ &=2 \int \frac{1}{1-3 \cos (4+6 x)} \, dx\\ &=\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-2+4 x^2} \, dx,x,\tan \left (\frac{1}{2} (4+6 x)\right )\right )\\ &=\frac{\log \left (\cos (2+3 x)-\sqrt{2} \sin (2+3 x)\right )}{6 \sqrt{2}}-\frac{\log \left (\cos (2+3 x)+\sqrt{2} \sin (2+3 x)\right )}{6 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0390506, size = 22, normalized size = 0.37 \[ -\frac{\tanh ^{-1}\left (\sqrt{2} \tan (3 x+2)\right )}{3 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2*Csc[4 + 6*x])/(-3*Cot[4 + 6*x] + Csc[4 + 6*x]),x]

[Out]

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

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Maple [A]  time = 0.084, size = 17, normalized size = 0.3 \begin{align*} -{\frac{\sqrt{2}{\it Artanh} \left ( \tan \left ( 2+3\,x \right ) \sqrt{2} \right ) }{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*csc(4+6*x)/(-3*cot(4+6*x)+csc(4+6*x)),x)

[Out]

-1/6*2^(1/2)*arctanh(tan(2+3*x)*2^(1/2))

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Maxima [A]  time = 1.66518, size = 73, normalized size = 1.22 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{2 \, \sin \left (6 \, x + 4\right )}{\cos \left (6 \, x + 4\right ) + 1}}{\sqrt{2} + \frac{2 \, \sin \left (6 \, x + 4\right )}{\cos \left (6 \, x + 4\right ) + 1}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*csc(4+6*x)/(-3*cot(4+6*x)+csc(4+6*x)),x, algorithm="maxima")

[Out]

1/12*sqrt(2)*log(-(sqrt(2) - 2*sin(6*x + 4)/(cos(6*x + 4) + 1))/(sqrt(2) + 2*sin(6*x + 4)/(cos(6*x + 4) + 1)))

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Fricas [A]  time = 1.48476, size = 207, normalized size = 3.45 \begin{align*} \frac{1}{24} \, \sqrt{2} \log \left (-\frac{7 \, \cos \left (6 \, x + 4\right )^{2} - 4 \,{\left (\sqrt{2} \cos \left (6 \, x + 4\right ) - 3 \, \sqrt{2}\right )} \sin \left (6 \, x + 4\right ) + 6 \, \cos \left (6 \, x + 4\right ) - 17}{9 \, \cos \left (6 \, x + 4\right )^{2} - 6 \, \cos \left (6 \, x + 4\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*csc(4+6*x)/(-3*cot(4+6*x)+csc(4+6*x)),x, algorithm="fricas")

[Out]

1/24*sqrt(2)*log(-(7*cos(6*x + 4)^2 - 4*(sqrt(2)*cos(6*x + 4) - 3*sqrt(2))*sin(6*x + 4) + 6*cos(6*x + 4) - 17)
/(9*cos(6*x + 4)^2 - 6*cos(6*x + 4) + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - 2 \int \frac{\csc{\left (6 x + 4 \right )}}{3 \cot{\left (6 x + 4 \right )} - \csc{\left (6 x + 4 \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*csc(4+6*x)/(-3*cot(4+6*x)+csc(4+6*x)),x)

[Out]

-2*Integral(csc(6*x + 4)/(3*cot(6*x + 4) - csc(6*x + 4)), x)

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Giac [A]  time = 1.30167, size = 53, normalized size = 0.88 \begin{align*} \frac{1}{12} \, \sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 4 \, \tan \left (3 \, x + 2\right ) \right |}}{{\left | 2 \, \sqrt{2} + 4 \, \tan \left (3 \, x + 2\right ) \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*csc(4+6*x)/(-3*cot(4+6*x)+csc(4+6*x)),x, algorithm="giac")

[Out]

1/12*sqrt(2)*log(abs(-2*sqrt(2) + 4*tan(3*x + 2))/abs(2*sqrt(2) + 4*tan(3*x + 2)))