### 3.787 $$\int \frac{\text{csch}(x)}{a+b \coth (x)+c \text{csch}(x)} \, dx$$

Optimal. Leaf size=50 $-\frac{2 \tanh ^{-1}\left (\frac{a+(b-c) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\sqrt{a^2-b^2+c^2}}$

[Out]

(-2*ArcTanh[(a + (b - c)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/Sqrt[a^2 - b^2 + c^2]

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Rubi [A]  time = 0.0900589, antiderivative size = 50, 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} $-\frac{2 \tanh ^{-1}\left (\frac{a+(b-c) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right )}{\sqrt{a^2-b^2+c^2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[x]/(a + b*Coth[x] + c*Csch[x]),x]

[Out]

(-2*ArcTanh[(a + (b - c)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/Sqrt[a^2 - b^2 + c^2]

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

Mathematica [A]  time = 0.0401134, size = 54, normalized size = 1.08 $\frac{2 \tan ^{-1}\left (\frac{a+(b-c) \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2+b^2-c^2}}\right )}{\sqrt{-a^2+b^2-c^2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csch[x]/(a + b*Coth[x] + c*Csch[x]),x]

[Out]

(2*ArcTan[(a + (b - c)*Tanh[x/2])/Sqrt[-a^2 + b^2 - c^2]])/Sqrt[-a^2 + b^2 - c^2]

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Maple [A]  time = 0.039, size = 53, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{\sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( b-c \right ) \tanh \left ( x/2 \right ) +2\,a}{\sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)/(a+b*coth(x)+c*csch(x)),x)

[Out]

2/(-a^2+b^2-c^2)^(1/2)*arctan(1/2*(2*(b-c)*tanh(1/2*x)+2*a)/(-a^2+b^2-c^2)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(a+b*coth(x)+c*csch(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.4323, size = 657, normalized size = 13.14 \begin{align*} \left [\frac{\log \left (\frac{2 \,{\left (a + b\right )} c \cosh \left (x\right ) +{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} +{\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2} + 2 \, c^{2} + 2 \,{\left ({\left (a + b\right )} c +{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 2 \, \sqrt{a^{2} - b^{2} + c^{2}}{\left ({\left (a + b\right )} \cosh \left (x\right ) +{\left (a + b\right )} \sinh \left (x\right ) + c\right )}}{{\left (a + b\right )} \cosh \left (x\right )^{2} +{\left (a + b\right )} \sinh \left (x\right )^{2} + 2 \, c \cosh \left (x\right ) + 2 \,{\left ({\left (a + b\right )} \cosh \left (x\right ) + c\right )} \sinh \left (x\right ) - a + b}\right )}{\sqrt{a^{2} - b^{2} + c^{2}}}, \frac{2 \, \sqrt{-a^{2} + b^{2} - c^{2}} \arctan \left (\frac{\sqrt{-a^{2} + b^{2} - c^{2}}{\left ({\left (a + b\right )} \cosh \left (x\right ) +{\left (a + b\right )} \sinh \left (x\right ) + c\right )}}{a^{2} - b^{2} + c^{2}}\right )}{a^{2} - b^{2} + c^{2}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(a+b*coth(x)+c*csch(x)),x, algorithm="fricas")

[Out]

[log((2*(a + b)*c*cosh(x) + (a^2 + 2*a*b + b^2)*cosh(x)^2 + (a^2 + 2*a*b + b^2)*sinh(x)^2 + a^2 - b^2 + 2*c^2
+ 2*((a + b)*c + (a^2 + 2*a*b + b^2)*cosh(x))*sinh(x) - 2*sqrt(a^2 - b^2 + c^2)*((a + b)*cosh(x) + (a + b)*sin
h(x) + c))/((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + 2*c*cosh(x) + 2*((a + b)*cosh(x) + c)*sinh(x) - a + b))/sq
rt(a^2 - b^2 + c^2), 2*sqrt(-a^2 + b^2 - c^2)*arctan(sqrt(-a^2 + b^2 - c^2)*((a + b)*cosh(x) + (a + b)*sinh(x)
+ c)/(a^2 - b^2 + c^2))/(a^2 - b^2 + c^2)]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(a+b*coth(x)+c*csch(x)),x)

[Out]

Integral(csch(x)/(a + b*coth(x) + c*csch(x)), x)

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Giac [A]  time = 1.1714, size = 62, normalized size = 1.24 \begin{align*} \frac{2 \, \arctan \left (\frac{a e^{x} + b e^{x} + c}{\sqrt{-a^{2} + b^{2} - c^{2}}}\right )}{\sqrt{-a^{2} + b^{2} - c^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)/(a+b*coth(x)+c*csch(x)),x, algorithm="giac")

[Out]

2*arctan((a*e^x + b*e^x + c)/sqrt(-a^2 + b^2 - c^2))/sqrt(-a^2 + b^2 - c^2)