### 3.786 $$\int \frac{\text{csch}(x)}{2+2 \coth (x)+3 \text{csch}(x)} \, dx$$

Optimal. Leaf size=19 $-\frac{2}{3} \tanh ^{-1}\left (\frac{1}{3} \left (2-\tanh \left (\frac{x}{2}\right )\right )\right )$

[Out]

(-2*ArcTanh[(2 - Tanh[x/2])/3])/3

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Rubi [A]  time = 0.0502114, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.267, Rules used = {3166, 3124, 618, 204} $-\frac{2}{3} \tanh ^{-1}\left (\frac{1}{3} \left (2-\tanh \left (\frac{x}{2}\right )\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[x]/(2 + 2*Coth[x] + 3*Csch[x]),x]

[Out]

(-2*ArcTanh[(2 - Tanh[x/2])/3])/3

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 3124

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

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{\text{csch}(x)}{2+2 \coth (x)+3 \text{csch}(x)} \, dx &=i \int \frac{1}{3 i+2 i \cosh (x)+2 i \sinh (x)} \, dx\\ &=2 i \operatorname{Subst}\left (\int \frac{1}{5 i+4 i x-i x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )\\ &=-\left (4 i \operatorname{Subst}\left (\int \frac{1}{-36-x^2} \, dx,x,4 i-2 i \tanh \left (\frac{x}{2}\right )\right )\right )\\ &=-\frac{2}{3} \tanh ^{-1}\left (\frac{1}{3} \left (2-\tanh \left (\frac{x}{2}\right )\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0685395, size = 28, normalized size = 1.47 $\frac{x}{6}-\frac{1}{3} \log \left (5 \cosh \left (\frac{x}{2}\right )-\sinh \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csch[x]/(2 + 2*Coth[x] + 3*Csch[x]),x]

[Out]

x/6 - Log[5*Cosh[x/2] - Sinh[x/2]]/3

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Maple [A]  time = 0.037, size = 20, normalized size = 1.1 \begin{align*}{\frac{1}{3}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{3}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -5 \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)/(2+2*coth(x)+3*csch(x)),x)

[Out]

1/3*ln(tanh(1/2*x)+1)-1/3*ln(tanh(1/2*x)-5)

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Maxima [A]  time = 1.09324, size = 15, normalized size = 0.79 \begin{align*} -\frac{1}{3} \, \log \left (3 \, e^{\left (-x\right )} + 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(2+2*coth(x)+3*csch(x)),x, algorithm="maxima")

[Out]

-1/3*log(3*e^(-x) + 2)

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Fricas [A]  time = 2.31758, size = 59, normalized size = 3.11 \begin{align*} \frac{1}{3} \, x - \frac{1}{3} \, \log \left (2 \, \cosh \left (x\right ) + 2 \, \sinh \left (x\right ) + 3\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(2+2*coth(x)+3*csch(x)),x, algorithm="fricas")

[Out]

1/3*x - 1/3*log(2*cosh(x) + 2*sinh(x) + 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}{\left (x \right )}}{2 \coth{\left (x \right )} + 3 \operatorname{csch}{\left (x \right )} + 2}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(2+2*coth(x)+3*csch(x)),x)

[Out]

Integral(csch(x)/(2*coth(x) + 3*csch(x) + 2), x)

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Giac [A]  time = 1.15271, size = 18, normalized size = 0.95 \begin{align*} \frac{1}{3} \, x - \frac{1}{3} \, \log \left (2 \, e^{x} + 3\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(2+2*coth(x)+3*csch(x)),x, algorithm="giac")

[Out]

1/3*x - 1/3*log(2*e^x + 3)