\(\int \frac {A+B \coth (x)}{a+b \sinh (x)} \, dx\) [250]

Optimal result

Integrand size = 15, antiderivative size = 60 \[ \int \frac {A+B \coth (x)}{a+b \sinh (x)} \, dx=-\frac {2 A \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}+\frac {B \log (\sinh (x))}{a}-\frac {B \log (a+b \sinh (x))}{a} \]

[Out]

B*ln(sinh(x))/a-B*ln(a+b*sinh(x))/a-2*A*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {4486, 2739, 632, 212, 2800, 36, 29, 31} \[ \int \frac {A+B \coth (x)}{a+b \sinh (x)} \, dx=-\frac {2 A \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2}}-\frac {B \log (a+b \sinh (x))}{a}+\frac {B \log (\sinh (x))}{a} \]

[In]

Int[(A + B*Coth[x])/(a + b*Sinh[x]),x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

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

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 2800

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rule 4486

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /;  !InertTrigFreeQ[u]

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

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.08 \[ \int \frac {A+B \coth (x)}{a+b \sinh (x)} \, dx=\frac {2 A \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\frac {B (\log (\sinh (x))-\log (a+b \sinh (x)))}{a} \]

[In]

Integrate[(A + B*Coth[x])/(a + b*Sinh[x]),x]

[Out]

(2*A*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + (B*(Log[Sinh[x]] - Log[a + b*Sinh[x]]))/a

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.22

method	result	size
parts	\(\frac {2 A \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}+\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a}-\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}{a}\)	\(73\)
default	\(\frac {-B \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )+\frac {2 A a \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}}{a}+\frac {B \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a}\)	\(76\)
risch	\(-\frac {2 x B}{a}-\frac {2 x \,a^{3} B}{-a^{4}-a^{2} b^{2}}-\frac {2 x B a \,b^{2}}{-a^{4}-a^{2} b^{2}}+\frac {B \ln \left ({\mathrm e}^{2 x}-1\right )}{a}-\frac {a \ln \left ({\mathrm e}^{x}+\frac {a^{2} A -\sqrt {A^{2} a^{4}+A^{2} a^{2} b^{2}}}{A a b}\right ) B}{a^{2}+b^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a^{2} A -\sqrt {A^{2} a^{4}+A^{2} a^{2} b^{2}}}{A a b}\right ) B \,b^{2}}{\left (a^{2}+b^{2}\right ) a}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a^{2} A -\sqrt {A^{2} a^{4}+A^{2} a^{2} b^{2}}}{A a b}\right ) \sqrt {A^{2} a^{4}+A^{2} a^{2} b^{2}}}{\left (a^{2}+b^{2}\right ) a}-\frac {a \ln \left ({\mathrm e}^{x}+\frac {a^{2} A +\sqrt {A^{2} a^{4}+A^{2} a^{2} b^{2}}}{A a b}\right ) B}{a^{2}+b^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a^{2} A +\sqrt {A^{2} a^{4}+A^{2} a^{2} b^{2}}}{A a b}\right ) B \,b^{2}}{\left (a^{2}+b^{2}\right ) a}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a^{2} A +\sqrt {A^{2} a^{4}+A^{2} a^{2} b^{2}}}{A a b}\right ) \sqrt {A^{2} a^{4}+A^{2} a^{2} b^{2}}}{\left (a^{2}+b^{2}\right ) a}\)	\(443\)

[In]

int((A+B*coth(x))/(a+b*sinh(x)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (56) = 112\).

Time = 0.29 (sec) , antiderivative size = 183, normalized size of antiderivative = 3.05 \[ \int \frac {A+B \coth (x)}{a+b \sinh (x)} \, dx=\frac {\sqrt {a^{2} + b^{2}} A a \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) - {\left (B a^{2} + B b^{2}\right )} \log \left (\frac {2 \, {\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + {\left (B a^{2} + B b^{2}\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a^{3} + a b^{2}} \]

[In]

integrate((A+B*coth(x))/(a+b*sinh(x)),x, algorithm="fricas")

[Out]

(sqrt(a^2 + b^2)*A*a*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*
sinh(x) - 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(
x) + a)*sinh(x) - b)) - (B*a^2 + B*b^2)*log(2*(b*sinh(x) + a)/(cosh(x) - sinh(x))) + (B*a^2 + B*b^2)*log(2*sin
h(x)/(cosh(x) - sinh(x))))/(a^3 + a*b^2)

Sympy [F]

\[ \int \frac {A+B \coth (x)}{a+b \sinh (x)} \, dx=\int \frac {A + B \coth {\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \]

[In]

integrate((A+B*coth(x))/(a+b*sinh(x)),x)

[Out]

Integral((A + B*coth(x))/(a + b*sinh(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.77 \[ \int \frac {A+B \coth (x)}{a+b \sinh (x)} \, dx=-B {\left (\frac {\log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{a} - \frac {\log \left (e^{\left (-x\right )} - 1\right )}{a}\right )} + \frac {A \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}}} \]

[In]

integrate((A+B*coth(x))/(a+b*sinh(x)),x, algorithm="maxima")

[Out]

-B*(log(-2*a*e^(-x) + b*e^(-2*x) - b)/a - log(e^(-x) + 1)/a - log(e^(-x) - 1)/a) + A*log((b*e^(-x) - a - sqrt(
a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/sqrt(a^2 + b^2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.70 \[ \int \frac {A+B \coth (x)}{a+b \sinh (x)} \, dx=\frac {A \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}}} + \frac {B \log \left (e^{x} + 1\right )}{a} - \frac {B \log \left ({\left | b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b \right |}\right )}{a} + \frac {B \log \left ({\left | e^{x} - 1 \right |}\right )}{a} \]

[In]

integrate((A+B*coth(x))/(a+b*sinh(x)),x, algorithm="giac")

[Out]

A*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/sqrt(a^2 + b^2) + B*log(e
^x + 1)/a - B*log(abs(b*e^(2*x) + 2*a*e^x - b))/a + B*log(abs(e^x - 1))/a

Mupad [B] (verification not implemented)

Time = 12.32 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.73 \[ \int \frac {A+B \coth (x)}{a+b \sinh (x)} \, dx=\frac {B\,\ln \left (16\,B^2\,a^2+16\,B^2\,b^2-16\,B^2\,a^2\,{\mathrm {e}}^{2\,x}-16\,B^2\,b^2\,{\mathrm {e}}^{2\,x}\right )}{a}-\frac {2\,\mathrm {atan}\left (\frac {A^2\,b^2\,{\mathrm {e}}^x\,\sqrt {-a^2-b^2}+A^2\,a\,b\,\sqrt {-a^2-b^2}}{A\,b\,\sqrt {A^2}\,\left (a^2+b^2\right )}\right )\,\sqrt {A^2}}{\sqrt {-a^2-b^2}}-\frac {B\,\ln \left (32\,B^2\,a\,{\mathrm {e}}^x-16\,B^2\,b+16\,B^2\,b\,{\mathrm {e}}^{2\,x}\right )}{a} \]

[In]

int((A + B*coth(x))/(a + b*sinh(x)),x)

[Out]

(B*log(16*B^2*a^2 + 16*B^2*b^2 - 16*B^2*a^2*exp(2*x) - 16*B^2*b^2*exp(2*x)))/a - (2*atan((A^2*b^2*exp(x)*(- a^
2 - b^2)^(1/2) + A^2*a*b*(- a^2 - b^2)^(1/2))/(A*b*(A^2)^(1/2)*(a^2 + b^2)))*(A^2)^(1/2))/(- a^2 - b^2)^(1/2)
- (B*log(32*B^2*a*exp(x) - 16*B^2*b + 16*B^2*b*exp(2*x)))/a