\(\int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx\) [119]

Optimal result

Integrand size = 13, antiderivative size = 57 \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a}-\frac {\text {arctanh}(\cosh (x))}{b}+\frac {2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3979, 4136, 3855, 4004, 3916, 2739, 632, 212} \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b}+\frac {x}{a}-\frac {\text {arctanh}(\cosh (x))}{b} \]

[In]

Int[Coth[x]^2/(a + b*Csch[x]),x]

[Out]

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

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 3855

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

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]

Rule 3979

Int[cot[(c_.) + (d_.)*(x_)]^2*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[(-1 + Csc[c + d*x]
^2)*(a + b*Csc[c + d*x])^n, x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

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

Rule 4136

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[C/b, I
nt[Csc[e + f*x], x], x] + Dist[1/b, Int[(A*b - a*C*Csc[e + f*x])/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b,
e, f, A, C}, x]

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.32 \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {b x+2 \sqrt {-a^2-b^2} \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )-a \log \left (\cosh \left (\frac {x}{2}\right )\right )+a \log \left (\sinh \left (\frac {x}{2}\right )\right )}{a b} \]

[In]

Integrate[Coth[x]^2/(a + b*Csch[x]),x]

[Out]

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.47

method	result	size
default	\(\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\left (2 a^{2}+2 b^{2}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a b \sqrt {a^{2}+b^{2}}}\)	\(84\)
risch	\(\frac {x}{a}+\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}+\frac {b +\sqrt {a^{2}+b^{2}}}{a}\right )}{b a}-\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{x}-\frac {-b +\sqrt {a^{2}+b^{2}}}{a}\right )}{b a}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{b}-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{b}\)	\(100\)

[In]

int(coth(x)^2/(a+b*csch(x)),x,method=_RETURNVERBOSE)

[Out]

1/a*ln(tanh(1/2*x)+1)+1/b*ln(tanh(1/2*x))-1/a*ln(tanh(1/2*x)-1)+(2*a^2+2*b^2)/a/b/(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. 141 vs. \(2 (53) = 106\).

Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.47 \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {b x - a \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + a \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + \sqrt {a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) - a}\right )}{a b} \]

[In]

integrate(coth(x)^2/(a+b*csch(x)),x, algorithm="fricas")

[Out]

(b*x - a*log(cosh(x) + sinh(x) + 1) + a*log(cosh(x) + sinh(x) - 1) + sqrt(a^2 + b^2)*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*sin
h(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*cosh(x) + b)*sinh(x) - a)))/(a*b)

Sympy [F]

\[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\coth ^{2}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]

[In]

integrate(coth(x)**2/(a+b*csch(x)),x)

[Out]

Integral(coth(x)**2/(a + b*csch(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.58 \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-x\right )} - 1\right )}{b} - \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{a b} \]

[In]

integrate(coth(x)^2/(a+b*csch(x)),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.56 \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a} - \frac {\log \left (e^{x} + 1\right )}{b} + \frac {\log \left ({\left | e^{x} - 1 \right |}\right )}{b} - \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{a b} \]

[In]

integrate(coth(x)^2/(a+b*csch(x)),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 316, normalized size of antiderivative = 5.54 \[ \int \frac {\coth ^2(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a}+\frac {\ln \left (32\,a^2\,b+32\,b^3-32\,b^3\,{\mathrm {e}}^x-32\,a^2\,b\,{\mathrm {e}}^x\right )}{b}-\frac {\ln \left (32\,a^2\,b+32\,b^3+32\,b^3\,{\mathrm {e}}^x+32\,a^2\,b\,{\mathrm {e}}^x\right )}{b}+\frac {\ln \left (128\,b^5\,{\mathrm {e}}^x-64\,a^3\,b^2-64\,a\,b^4-128\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+32\,a^4\,b\,{\mathrm {e}}^x+160\,a^2\,b^3\,{\mathrm {e}}^x+64\,a\,b^3\,\sqrt {a^2+b^2}+32\,a^3\,b\,\sqrt {a^2+b^2}-96\,a^2\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a\,b}-\frac {\ln \left (128\,b^5\,{\mathrm {e}}^x-64\,a^3\,b^2-64\,a\,b^4+128\,b^4\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+32\,a^4\,b\,{\mathrm {e}}^x+160\,a^2\,b^3\,{\mathrm {e}}^x-64\,a\,b^3\,\sqrt {a^2+b^2}-32\,a^3\,b\,\sqrt {a^2+b^2}+96\,a^2\,b^2\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a\,b} \]

[In]

int(coth(x)^2/(a + b/sinh(x)),x)

[Out]

x/a + log(32*a^2*b + 32*b^3 - 32*b^3*exp(x) - 32*a^2*b*exp(x))/b - log(32*a^2*b + 32*b^3 + 32*b^3*exp(x) + 32*
a^2*b*exp(x))/b + (log(128*b^5*exp(x) - 64*a^3*b^2 - 64*a*b^4 - 128*b^4*exp(x)*(a^2 + b^2)^(1/2) + 32*a^4*b*ex
p(x) + 160*a^2*b^3*exp(x) + 64*a*b^3*(a^2 + b^2)^(1/2) + 32*a^3*b*(a^2 + b^2)^(1/2) - 96*a^2*b^2*exp(x)*(a^2 +
b^2)^(1/2))*(a^2 + b^2)^(1/2))/(a*b) - (log(128*b^5*exp(x) - 64*a^3*b^2 - 64*a*b^4 + 128*b^4*exp(x)*(a^2 + b^
2)^(1/2) + 32*a^4*b*exp(x) + 160*a^2*b^3*exp(x) - 64*a*b^3*(a^2 + b^2)^(1/2) - 32*a^3*b*(a^2 + b^2)^(1/2) + 96
*a^2*b^2*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(a*b)