\(\int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx\) [80]

Optimal result

Integrand size = 11, antiderivative size = 37 \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=-\frac {2 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3916, 2739, 632, 212} \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=-\frac {2 \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \]

[In]

Int[Csch[x]/(a + b*Csch[x]),x]

[Out]

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

Rule 212

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

Rule 632

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 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*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 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{1+\frac {a \sinh (x)}{b}} \, dx}{b} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{1+\frac {2 a x}{b}-x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b} \\ & = -\frac {4 \text {Subst}\left (\int \frac {1}{4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,\frac {2 a}{b}-2 \tanh \left (\frac {x}{2}\right )\right )}{b} \\ & = -\frac {2 \text {arctanh}\left (\frac {b \left (\frac {a}{b}-\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=\frac {2 \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}} \]

[In]

Integrate[Csch[x]/(a + b*Csch[x]),x]

[Out]

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95

method	result	size
default	\(-\frac {2 \,\operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\)	\(35\)
risch	\(\frac {\ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {b \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, a}\right )}{\sqrt {a^{2}+b^{2}}}\)	\(97\)

[In]

int(csch(x)/(a+b*csch(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (33) = 66\).

Time = 0.26 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.00 \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=\frac {\log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right )}{\sqrt {a^{2} + b^{2}}} \]

[In]

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

[Out]

log((a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*b*cosh(x) + a^2 + 2*b^2 + 2*(a^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(a^2
+ b^2)*(a*cosh(x) + a*sinh(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*cosh(x) + b)*sinh(x) - a))
/sqrt(a^2 + b^2)

Sympy [F]

\[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\operatorname {csch}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]

[In]

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

[Out]

Integral(csch(x)/(a + b*csch(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.46 \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=\frac {\log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.51 \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=\frac {\log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}}} \]

[In]

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

[Out]

log(abs(2*a*e^x + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^x + 2*b + 2*sqrt(a^2 + b^2)))/sqrt(a^2 + b^2)

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.32 \[ \int \frac {\text {csch}(x)}{a+b \text {csch}(x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {b}{\sqrt {-a^2-b^2}}+\frac {a\,{\mathrm {e}}^x}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}} \]

[In]

int(1/(sinh(x)*(a + b/sinh(x))),x)

[Out]

(2*atan(b/(- a^2 - b^2)^(1/2) + (a*exp(x))/(- a^2 - b^2)^(1/2)))/(- a^2 - b^2)^(1/2)