Integrand size = 17, antiderivative size = 118 \[ \int \frac {\text {csch}^2(x)}{a+b \coth (x)+c \text {csch}(x)} \, dx=-\frac {2 a c \text {arctanh}\left (\frac {a+(b-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (b^2-c^2\right ) \sqrt {a^2-b^2+c^2}}+\frac {\log \left (\tanh \left (\frac {x}{2}\right )\right )}{b+c}-\frac {b \log \left (b+c+2 a \tanh \left (\frac {x}{2}\right )+(b-c) \tanh ^2\left (\frac {x}{2}\right )\right )}{b^2-c^2} \]
ln(tanh(1/2*x))/(b+c)-b*ln(b+c+2*a*tanh(1/2*x)+(b-c)*tanh(1/2*x)^2)/(b^2-c ^2)-2*a*c*arctanh((a+(b-c)*tanh(1/2*x))/(a^2-b^2+c^2)^(1/2))/(b^2-c^2)/(a^ 2-b^2+c^2)^(1/2)
Time = 1.57 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.92 \[ \int \frac {\text {csch}^2(x)}{a+b \coth (x)+c \text {csch}(x)} \, dx=\frac {\frac {2 a c \arctan \left (\frac {a+(b-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2-c^2}}\right )}{\sqrt {-a^2+b^2-c^2}}+(b+c) \log \left (\cosh \left (\frac {x}{2}\right )\right )+(b-c) \log \left (\sinh \left (\frac {x}{2}\right )\right )-b \log (c+b \cosh (x)+a \sinh (x))}{(b-c) (b+c)} \]
((2*a*c*ArcTan[(a + (b - c)*Tanh[x/2])/Sqrt[-a^2 + b^2 - c^2]])/Sqrt[-a^2 + b^2 - c^2] + (b + c)*Log[Cosh[x/2]] + (b - c)*Log[Sinh[x/2]] - b*Log[c + b*Cosh[x] + a*Sinh[x]])/((b - c)*(b + c))
Time = 0.63 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {3042, 25, 4897, 26, 26, 3042, 26, 4902, 27, 2159, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\text {csch}^2(x)}{a+b \coth (x)+c \text {csch}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\csc (i x)^2}{a+i b \cot (i x)+i c \csc (i x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\csc (i x)^2}{a+i b \cot (i x)+i c \csc (i x)}dx\) |
| \(\Big \downarrow \) 4897 |
| \(\displaystyle -\int -\frac {i \text {csch}(x)}{i c+i b \cosh (x)+i a \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int -\frac {i \text {csch}(x)}{c+b \cosh (x)+a \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\text {csch}(x)}{a \sinh (x)+b \cosh (x)+c}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \csc (i x)}{-i a \sin (i x)+b \cos (i x)+c}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\csc (i x)}{c+b \cos (i x)-i a \sin (i x)}dx\) |
| \(\Big \downarrow \) 4902 |
| \(\displaystyle 2 i \int -\frac {i \coth \left (\frac {x}{2}\right ) \left (1-\tanh ^2\left (\frac {x}{2}\right )\right )}{2 \left ((b-c) \tanh ^2\left (\frac {x}{2}\right )+2 a \tanh \left (\frac {x}{2}\right )+b+c\right )}d\tanh \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {\left (1-\tanh ^2\left (\frac {x}{2}\right )\right ) \coth \left (\frac {x}{2}\right )}{2 a \tanh \left (\frac {x}{2}\right )+(b-c) \tanh ^2\left (\frac {x}{2}\right )+b+c}d\tanh \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2159 |
| \(\displaystyle \int \left (\frac {2 \left (-a-b \tanh \left (\frac {x}{2}\right )\right )}{(b+c) \left (2 a \tanh \left (\frac {x}{2}\right )+(b-c) \tanh ^2\left (\frac {x}{2}\right )+b+c\right )}+\frac {\coth \left (\frac {x}{2}\right )}{b+c}\right )d\tanh \left (\frac {x}{2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 a c \text {arctanh}\left (\frac {a+(b-c) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (b^2-c^2\right ) \sqrt {a^2-b^2+c^2}}-\frac {b \log \left (2 a \tanh \left (\frac {x}{2}\right )+(b-c) \tanh ^2\left (\frac {x}{2}\right )+b+c\right )}{b^2-c^2}+\frac {\log \left (\tanh \left (\frac {x}{2}\right )\right )}{b+c}\) |
(-2*a*c*ArcTanh[(a + (b - c)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/((b^2 - c^ 2)*Sqrt[a^2 - b^2 + c^2]) + Log[Tanh[x/2]]/(b + c) - (b*Log[b + c + 2*a*Ta nh[x/2] + (b - c)*Tanh[x/2]^2])/(b^2 - c^2)
3.8.88.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[u_, x_Symbol] :> With[{w = Block[{$ShowSteps = False, $StepCounter = Nu ll}, Int[SubstFor[1/(1 + FreeFactors[Tan[FunctionOfTrig[u, x]/2], x]^2*x^2) , Tan[FunctionOfTrig[u, x]/2]/FreeFactors[Tan[FunctionOfTrig[u, x]/2], x], u, x], x]]}, Module[{v = FunctionOfTrig[u, x], d}, Simp[d = FreeFactors[Tan [v/2], x]; 2*(d/Coefficient[v, x, 1]) Subst[Int[SubstFor[1/(1 + d^2*x^2), Tan[v/2]/d, u, x], x], x, Tan[v/2]/d], x]] /; CalculusFreeQ[w, x]] /; Inve rseFunctionFreeQ[u, x] && !FalseQ[FunctionOfTrig[u, x]]
Time = 0.43 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {-\frac {b \ln \left (\tanh \left (\frac {x}{2}\right )^{2} b -c \tanh \left (\frac {x}{2}\right )^{2}+2 a \tanh \left (\frac {x}{2}\right )+b +c \right )}{b -c}+\frac {\left (-2 a +\frac {2 b a}{b -c}\right ) \arctan \left (\frac {2 \left (b -c \right ) \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right )}{\sqrt {-a^{2}+b^{2}-c^{2}}}}{b +c}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b +c}\) | \(124\) |
| risch | \(-\frac {x}{b -c}-\frac {x}{b +c}+\frac {2 x \,a^{2} b}{a^{2} b^{2}-a^{2} c^{2}-b^{4}+2 b^{2} c^{2}-c^{4}}-\frac {2 x \,b^{3}}{a^{2} b^{2}-a^{2} c^{2}-b^{4}+2 b^{2} c^{2}-c^{4}}+\frac {2 x b \,c^{2}}{a^{2} b^{2}-a^{2} c^{2}-b^{4}+2 b^{2} c^{2}-c^{4}}+\frac {\ln \left (1+{\mathrm e}^{x}\right )}{b -c}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{b +c}-\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) a^{2} b}{a^{2} b^{2}-a^{2} c^{2}-b^{4}+2 b^{2} c^{2}-c^{4}}+\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) b^{3}}{a^{2} b^{2}-a^{2} c^{2}-b^{4}+2 b^{2} c^{2}-c^{4}}-\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) b \,c^{2}}{a^{2} b^{2}-a^{2} c^{2}-b^{4}+2 b^{2} c^{2}-c^{4}}+\frac {\ln \left ({\mathrm e}^{x}-\frac {-a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) \sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{a^{2} b^{2}-a^{2} c^{2}-b^{4}+2 b^{2} c^{2}-c^{4}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) a^{2} b}{a^{2} b^{2}-a^{2} c^{2}-b^{4}+2 b^{2} c^{2}-c^{4}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) b^{3}}{a^{2} b^{2}-a^{2} c^{2}-b^{4}+2 b^{2} c^{2}-c^{4}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) b \,c^{2}}{a^{2} b^{2}-a^{2} c^{2}-b^{4}+2 b^{2} c^{2}-c^{4}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \,c^{2}+\sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{\left (a +b \right ) a c}\right ) \sqrt {a^{4} c^{2}-a^{2} b^{2} c^{2}+a^{2} c^{4}}}{a^{2} b^{2}-a^{2} c^{2}-b^{4}+2 b^{2} c^{2}-c^{4}}\) | \(959\) |
1/(b+c)*(-b/(b-c)*ln(tanh(1/2*x)^2*b-c*tanh(1/2*x)^2+2*a*tanh(1/2*x)+b+c)+ (-2*a+2*b*a/(b-c))/(-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)))+ln(tanh(1/2*x))/(b+c)
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (106) = 212\).
Time = 0.90 (sec) , antiderivative size = 546, normalized size of antiderivative = 4.63 \[ \int \frac {\text {csch}^2(x)}{a+b \coth (x)+c \text {csch}(x)} \, dx=\left [-\frac {\sqrt {a^{2} - b^{2} + c^{2}} a c \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 ) + {\left (a^{2} b - b^{3} + b c^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a \sinh \left (x\right ) + c\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - {\left (a^{2} b - b^{3} + b c^{2} + c^{3} + {\left (a^{2} - b^{2}\right )} c\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (a^{2} b - b^{3} + b c^{2} - c^{3} - {\left (a^{2} - b^{2}\right )} c\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{2} b^{2} - b^{4} - c^{4} - {\left (a^{2} - 2 \, b^{2}\right )} c^{2}}, \frac {2 \, \sqrt {-a^{2} + b^{2} - c^{2}} a c \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 ) - {\left (a^{2} b - b^{3} + b c^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (x\right ) + a \sinh \left (x\right ) + c\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + {\left (a^{2} b - b^{3} + b c^{2} + c^{3} + {\left (a^{2} - b^{2}\right )} c\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (a^{2} b - b^{3} + b c^{2} - c^{3} - {\left (a^{2} - b^{2}\right )} c\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right )}{a^{2} b^{2} - b^{4} - c^{4} - {\left (a^{2} - 2 \, b^{2}\right )} c^{2}}\right ] \]
[-(sqrt(a^2 - b^2 + c^2)*a*c*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)*sinh(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)) + (a^2*b - b^3 + b*c^2)*log(2*(b*cosh(x) + a*sinh(x) + c)/(cosh(x) - sinh(x))) - (a^2*b - b^3 + b*c^2 + c^3 + (a^2 - b^2)*c)*log(cosh(x) + sinh(x) + 1) - (a^2*b - b^3 + b*c^2 - c^3 - (a^2 - b^2)*c)*log(cosh(x) + sinh(x) - 1))/(a^2*b^2 - b^4 - c^4 - (a^2 - 2*b^2)*c^2), (2*sqrt(-a^2 + b^2 - c^2)*a*c*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 - b^3 + b*c^2)*log(2*(b*cosh(x) + a*sinh(x) + c)/(cosh(x) - s inh(x))) + (a^2*b - b^3 + b*c^2 + c^3 + (a^2 - b^2)*c)*log(cosh(x) + sinh( x) + 1) + (a^2*b - b^3 + b*c^2 - c^3 - (a^2 - b^2)*c)*log(cosh(x) + sinh(x ) - 1))/(a^2*b^2 - b^4 - c^4 - (a^2 - 2*b^2)*c^2)]
\[ \int \frac {\text {csch}^2(x)}{a+b \coth (x)+c \text {csch}(x)} \, dx=\int \frac {\operatorname {csch}^{2}{\left (x \right )}}{a + b \coth {\left (x \right )} + c \operatorname {csch}{\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {\text {csch}^2(x)}{a+b \coth (x)+c \text {csch}(x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c^2-b^2+a^2>0)', see `assume?` f or more de
Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.03 \[ \int \frac {\text {csch}^2(x)}{a+b \coth (x)+c \text {csch}(x)} \, dx=\frac {2 \, a c \arctan \left (\frac {a e^{x} + b e^{x} + c}{\sqrt {-a^{2} + b^{2} - c^{2}}}\right )}{\sqrt {-a^{2} + b^{2} - c^{2}} {\left (b^{2} - c^{2}\right )}} - \frac {b \log \left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + 2 \, c e^{x} - a + b\right )}{b^{2} - c^{2}} + \frac {\log \left (e^{x} + 1\right )}{b - c} + \frac {\log \left ({\left | e^{x} - 1 \right |}\right )}{b + c} \]
2*a*c*arctan((a*e^x + b*e^x + c)/sqrt(-a^2 + b^2 - c^2))/(sqrt(-a^2 + b^2 - c^2)*(b^2 - c^2)) - b*log(a*e^(2*x) + b*e^(2*x) + 2*c*e^x - a + b)/(b^2 - c^2) + log(e^x + 1)/(b - c) + log(abs(e^x - 1))/(b + c)
Time = 8.18 (sec) , antiderivative size = 1069, normalized size of antiderivative = 9.06 \[ \int \frac {\text {csch}^2(x)}{a+b \coth (x)+c \text {csch}(x)} \, dx=\frac {\ln \left ({\mathrm {e}}^x-1\right )}{b+c}+\frac {\ln \left ({\mathrm {e}}^x+1\right )}{b-c}+\frac {\ln \left (-\frac {64\,\left (b-a+2\,c\,{\mathrm {e}}^x\right )}{{\left (a+b\right )}^4}-\frac {\left (\frac {32\,\left (-2\,a^3+3\,{\mathrm {e}}^x\,a^2\,c+2\,a\,b^2+6\,{\mathrm {e}}^x\,a\,b\,c-2\,a\,c^2+3\,{\mathrm {e}}^x\,b^2\,c+2\,b\,c^2+4\,{\mathrm {e}}^x\,c^3\right )}{{\left (a+b\right )}^5}+\frac {\left (\frac {32\,\left (a-b\right )\,\left (2\,b^3+6\,{\mathrm {e}}^x\,b^2\,c+2\,a\,b^2+b\,c^2+6\,a\,{\mathrm {e}}^x\,b\,c-3\,{\mathrm {e}}^x\,c^3+2\,a\,c^2\right )}{{\left (a+b\right )}^5}-\frac {32\,\left (a^2\,b+b\,c^2-b^3+a\,c\,\sqrt {a^2-b^2+c^2}\right )\,\left (-2\,a^3\,b^2-4\,{\mathrm {e}}^x\,a^3\,b\,c-2\,a^3\,c^2+{\mathrm {e}}^x\,a^2\,b^2\,c+4\,a^2\,b\,c^2+3\,{\mathrm {e}}^x\,a^2\,c^3+2\,a\,b^4+6\,{\mathrm {e}}^x\,a\,b^3\,c+a\,b^2\,c^2-6\,{\mathrm {e}}^x\,a\,b\,c^3-3\,a\,c^4+{\mathrm {e}}^x\,b^4\,c-3\,b^3\,c^2-5\,{\mathrm {e}}^x\,b^2\,c^3+3\,b\,c^4+4\,{\mathrm {e}}^x\,c^5\right )}{{\left (a+b\right )}^5\,\left (b^2-c^2\right )\,\left (a^2-b^2+c^2\right )}\right )\,\left (a^2\,b+b\,c^2-b^3+a\,c\,\sqrt {a^2-b^2+c^2}\right )}{\left (b^2-c^2\right )\,\left (a^2-b^2+c^2\right )}\right )\,\left (a^2\,b+b\,c^2-b^3+a\,c\,\sqrt {a^2-b^2+c^2}\right )}{\left (b^2-c^2\right )\,\left (a^2-b^2+c^2\right )}\right )\,\left (a^2\,b+b\,c^2-b^3+a\,c\,\sqrt {a^2-b^2+c^2}\right )}{-a^2\,b^2+a^2\,c^2+b^4-2\,b^2\,c^2+c^4}+\frac {\ln \left (-\frac {64\,\left (b-a+2\,c\,{\mathrm {e}}^x\right )}{{\left (a+b\right )}^4}-\frac {\left (\frac {32\,\left (-2\,a^3+3\,{\mathrm {e}}^x\,a^2\,c+2\,a\,b^2+6\,{\mathrm {e}}^x\,a\,b\,c-2\,a\,c^2+3\,{\mathrm {e}}^x\,b^2\,c+2\,b\,c^2+4\,{\mathrm {e}}^x\,c^3\right )}{{\left (a+b\right )}^5}+\frac {\left (\frac {32\,\left (a-b\right )\,\left (2\,b^3+6\,{\mathrm {e}}^x\,b^2\,c+2\,a\,b^2+b\,c^2+6\,a\,{\mathrm {e}}^x\,b\,c-3\,{\mathrm {e}}^x\,c^3+2\,a\,c^2\right )}{{\left (a+b\right )}^5}-\frac {32\,\left (a^2\,b+b\,c^2-b^3-a\,c\,\sqrt {a^2-b^2+c^2}\right )\,\left (-2\,a^3\,b^2-4\,{\mathrm {e}}^x\,a^3\,b\,c-2\,a^3\,c^2+{\mathrm {e}}^x\,a^2\,b^2\,c+4\,a^2\,b\,c^2+3\,{\mathrm {e}}^x\,a^2\,c^3+2\,a\,b^4+6\,{\mathrm {e}}^x\,a\,b^3\,c+a\,b^2\,c^2-6\,{\mathrm {e}}^x\,a\,b\,c^3-3\,a\,c^4+{\mathrm {e}}^x\,b^4\,c-3\,b^3\,c^2-5\,{\mathrm {e}}^x\,b^2\,c^3+3\,b\,c^4+4\,{\mathrm {e}}^x\,c^5\right )}{{\left (a+b\right )}^5\,\left (b^2-c^2\right )\,\left (a^2-b^2+c^2\right )}\right )\,\left (a^2\,b+b\,c^2-b^3-a\,c\,\sqrt {a^2-b^2+c^2}\right )}{\left (b^2-c^2\right )\,\left (a^2-b^2+c^2\right )}\right )\,\left (a^2\,b+b\,c^2-b^3-a\,c\,\sqrt {a^2-b^2+c^2}\right )}{\left (b^2-c^2\right )\,\left (a^2-b^2+c^2\right )}\right )\,\left (a^2\,b+b\,c^2-b^3-a\,c\,\sqrt {a^2-b^2+c^2}\right )}{-a^2\,b^2+a^2\,c^2+b^4-2\,b^2\,c^2+c^4} \]
log(exp(x) - 1)/(b + c) + log(exp(x) + 1)/(b - c) + (log(- (64*(b - a + 2* c*exp(x)))/(a + b)^4 - (((32*(2*a*b^2 - 2*a*c^2 + 2*b*c^2 - 2*a^3 + 4*c^3* exp(x) + 3*a^2*c*exp(x) + 3*b^2*c*exp(x) + 6*a*b*c*exp(x)))/(a + b)^5 + (( (32*(a - b)*(2*a*b^2 + 2*a*c^2 + b*c^2 + 2*b^3 - 3*c^3*exp(x) + 6*b^2*c*ex p(x) + 6*a*b*c*exp(x)))/(a + b)^5 - (32*(a^2*b + b*c^2 - b^3 + a*c*(a^2 - b^2 + c^2)^(1/2))*(2*a*b^4 - 3*a*c^4 + 3*b*c^4 - 2*a^3*b^2 - 2*a^3*c^2 - 3 *b^3*c^2 + 4*c^5*exp(x) + a*b^2*c^2 + 4*a^2*b*c^2 + b^4*c*exp(x) + 3*a^2*c ^3*exp(x) - 5*b^2*c^3*exp(x) + a^2*b^2*c*exp(x) - 6*a*b*c^3*exp(x) + 6*a*b ^3*c*exp(x) - 4*a^3*b*c*exp(x)))/((a + b)^5*(b^2 - c^2)*(a^2 - b^2 + c^2)) )*(a^2*b + b*c^2 - b^3 + a*c*(a^2 - b^2 + c^2)^(1/2)))/((b^2 - c^2)*(a^2 - b^2 + c^2)))*(a^2*b + b*c^2 - b^3 + a*c*(a^2 - b^2 + c^2)^(1/2)))/((b^2 - c^2)*(a^2 - b^2 + c^2)))*(a^2*b + b*c^2 - b^3 + a*c*(a^2 - b^2 + c^2)^(1/ 2)))/(b^4 + c^4 - a^2*b^2 + a^2*c^2 - 2*b^2*c^2) + (log(- (64*(b - a + 2*c *exp(x)))/(a + b)^4 - (((32*(2*a*b^2 - 2*a*c^2 + 2*b*c^2 - 2*a^3 + 4*c^3*e xp(x) + 3*a^2*c*exp(x) + 3*b^2*c*exp(x) + 6*a*b*c*exp(x)))/(a + b)^5 + ((( 32*(a - b)*(2*a*b^2 + 2*a*c^2 + b*c^2 + 2*b^3 - 3*c^3*exp(x) + 6*b^2*c*exp (x) + 6*a*b*c*exp(x)))/(a + b)^5 - (32*(a^2*b + b*c^2 - b^3 - a*c*(a^2 - b ^2 + c^2)^(1/2))*(2*a*b^4 - 3*a*c^4 + 3*b*c^4 - 2*a^3*b^2 - 2*a^3*c^2 - 3* b^3*c^2 + 4*c^5*exp(x) + a*b^2*c^2 + 4*a^2*b*c^2 + b^4*c*exp(x) + 3*a^2*c^ 3*exp(x) - 5*b^2*c^3*exp(x) + a^2*b^2*c*exp(x) - 6*a*b*c^3*exp(x) + 6*a...