3.695 \(\int \frac {\coth (x)}{b \cosh (x)+a \sinh (x)} \, dx\)

Optimal. Leaf size=51 \[ \frac {a \tanh ^{-1}\left (\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}\right )}{b \sqrt {a^2-b^2}}-\frac {\tanh ^{-1}(\cosh (x))}{b} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3110, 3770, 3074, 206} \[ \frac {a \tanh ^{-1}\left (\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}\right )}{b \sqrt {a^2-b^2}}-\frac {\tanh ^{-1}(\cosh (x))}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

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

Rule 3074

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

Rule 3110

Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(
c_.) + (d_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[(cos[c + d*x]^m*sin[c + d*x]^n)/(a*cos[c + d*x] + b*sin[c + d
*x]), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegersQ[m, n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 59, normalized size = 1.16 \[ \frac {\log \left (\tanh \left (\frac {x}{2}\right )\right )-\frac {2 a \tan ^{-1}\left (\frac {a+b \tanh \left (\frac {x}{2}\right )}{\sqrt {b-a} \sqrt {a+b}}\right )}{\sqrt {b-a} \sqrt {a+b}}}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 239, normalized size = 4.69 \[ \left [\frac {\sqrt {a^{2} - b^{2}} a \log \left (\frac {{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + a - b}{{\left (a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (a + b\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a + b\right )} \sinh \relax (x)^{2} - a + b}\right ) - {\left (a^{2} - b^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + {\left (a^{2} - b^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right )}{a^{2} b - b^{3}}, -\frac {2 \, \sqrt {-a^{2} + b^{2}} a \arctan \left (\frac {\sqrt {-a^{2} + b^{2}}}{{\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x)}\right ) + {\left (a^{2} - b^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) - {\left (a^{2} - b^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right )}{a^{2} b - b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(sqrt(a^2 - b^2)*a*log(((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + 2*sqrt(a^2 - b^2)
*(cosh(x) + sinh(x)) + a - b)/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 - a + b)) - (
a^2 - b^2)*log(cosh(x) + sinh(x) + 1) + (a^2 - b^2)*log(cosh(x) + sinh(x) - 1))/(a^2*b - b^3), -(2*sqrt(-a^2 +
b^2)*a*arctan(sqrt(-a^2 + b^2)/((a + b)*cosh(x) + (a + b)*sinh(x))) + (a^2 - b^2)*log(cosh(x) + sinh(x) + 1)
- (a^2 - b^2)*log(cosh(x) + sinh(x) - 1))/(a^2*b - b^3)]

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giac [A]  time = 0.12, size = 60, normalized size = 1.18 \[ -\frac {2 \, a \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b} - \frac {\log \left (e^{x} + 1\right )}{b} + \frac {\log \left ({\left | e^{x} - 1 \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.30, size = 53, normalized size = 1.04 \[ -\frac {2 a \arctan \left (\frac {2 \tanh \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{b \sqrt {-a^{2}+b^{2}}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more details)Is 4*a^2-4*b^2 positive or negative?

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mupad [B]  time = 1.78, size = 177, normalized size = 3.47 \[ \frac {\ln \left (32\,a\,b-32\,b^2+32\,b^2\,{\mathrm {e}}^x-32\,a\,b\,{\mathrm {e}}^x\right )}{b}-\frac {\ln \left (32\,a\,b-32\,b^2-32\,b^2\,{\mathrm {e}}^x+32\,a\,b\,{\mathrm {e}}^x\right )}{b}-\frac {a\,\ln \left (32\,a\,b^2\,{\mathrm {e}}^x+32\,a^2\,b\,{\mathrm {e}}^x-32\,a\,b\,\sqrt {a^2-b^2}\right )\,\sqrt {a^2-b^2}}{a^2\,b-b^3}+\frac {a\,\ln \left (32\,a\,b^2\,{\mathrm {e}}^x+32\,a^2\,b\,{\mathrm {e}}^x+32\,a\,b\,\sqrt {a^2-b^2}\right )\,\sqrt {a^2-b^2}}{a^2\,b-b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(32*a*b - 32*b^2 + 32*b^2*exp(x) - 32*a*b*exp(x))/b - log(32*a*b - 32*b^2 - 32*b^2*exp(x) + 32*a*b*exp(x))/
b - (a*log(32*a*b^2*exp(x) + 32*a^2*b*exp(x) - 32*a*b*(a^2 - b^2)^(1/2))*(a^2 - b^2)^(1/2))/(a^2*b - b^3) + (a
*log(32*a*b^2*exp(x) + 32*a^2*b*exp(x) + 32*a*b*(a^2 - b^2)^(1/2))*(a^2 - b^2)^(1/2))/(a^2*b - b^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth {\relax (x )}}{a \sinh {\relax (x )} + b \cosh {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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