Integrand size = 13, antiderivative size = 255 \[ \int \frac {\text {csch}^5(x)}{a+b \tanh (x)} \, dx=-\frac {b \arctan (\sinh (x))}{a^2}+\frac {b^3 \arctan (\sinh (x))}{a^4}+\frac {b \left (a^2-b^2\right ) \arctan (\sinh (x))}{a^4}-\frac {b \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{a^5}-\frac {3 \text {arctanh}(\cosh (x))}{8 a}+\frac {3 b^2 \text {arctanh}(\cosh (x))}{2 a^3}-\frac {b^4 \text {arctanh}(\cosh (x))}{a^5}-\frac {b \text {csch}(x)}{a^2}+\frac {3 b^3 \text {csch}(x)}{2 a^4}+\frac {3 \coth (x) \text {csch}(x)}{8 a}+\frac {b \text {csch}^3(x)}{3 a^2}-\frac {\coth (x) \text {csch}^3(x)}{4 a}-\frac {3 b^2 \text {sech}(x)}{2 a^3}+\frac {b^4 \text {sech}(x)}{a^5}+\frac {b^2 \left (a^2-b^2\right ) \text {sech}(x)}{a^5}-\frac {b^2 \text {csch}^2(x) \text {sech}(x)}{2 a^3}-\frac {b^3 \text {csch}(x) \text {sech}^2(x)}{2 a^4}-\frac {b^3 \text {sech}(x) \tanh (x)}{2 a^4} \]
-b*arctan(sinh(x))/a^2+b^3*arctan(sinh(x))/a^4+b*(a^2-b^2)*arctan(sinh(x)) /a^4-b*(a^2-b^2)^(3/2)*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/a^5-3 /8*arctanh(cosh(x))/a+3/2*b^2*arctanh(cosh(x))/a^3-b^4*arctanh(cosh(x))/a^ 5-b*csch(x)/a^2+3/2*b^3*csch(x)/a^4+3/8*coth(x)*csch(x)/a+1/3*b*csch(x)^3/ a^2-1/4*coth(x)*csch(x)^3/a-3/2*b^2*sech(x)/a^3+b^4*sech(x)/a^5+b^2*(a^2-b ^2)*sech(x)/a^5-1/2*b^2*csch(x)^2*sech(x)/a^3-1/2*b^3*csch(x)*sech(x)^2/a^ 4-1/2*b^3*sech(x)*tanh(x)/a^4
Time = 1.08 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.31 \[ \int \frac {\text {csch}^5(x)}{a+b \tanh (x)} \, dx=\frac {-384 a^2 \sqrt {a-b} b \sqrt {a+b} \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )+384 \sqrt {a-b} b^3 \sqrt {a+b} \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )-16 a b \left (7 a^2-6 b^2\right ) \coth \left (\frac {x}{2}\right )+6 a^2 \left (3 a^2-4 b^2\right ) \text {csch}^2\left (\frac {x}{2}\right )-72 a^4 \log \left (\cosh \left (\frac {x}{2}\right )\right )+288 a^2 b^2 \log \left (\cosh \left (\frac {x}{2}\right )\right )-192 b^4 \log \left (\cosh \left (\frac {x}{2}\right )\right )+72 a^4 \log \left (\sinh \left (\frac {x}{2}\right )\right )-288 a^2 b^2 \log \left (\sinh \left (\frac {x}{2}\right )\right )+192 b^4 \log \left (\sinh \left (\frac {x}{2}\right )\right )+18 a^4 \text {sech}^2\left (\frac {x}{2}\right )-24 a^2 b^2 \text {sech}^2\left (\frac {x}{2}\right )+3 a^4 \text {sech}^4\left (\frac {x}{2}\right )+64 a^3 b \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )+a^3 \text {csch}^4\left (\frac {x}{2}\right ) (-3 a+4 b \sinh (x))+112 a^3 b \tanh \left (\frac {x}{2}\right )-96 a b^3 \tanh \left (\frac {x}{2}\right )}{192 a^5} \]
(-384*a^2*Sqrt[a - b]*b*Sqrt[a + b]*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]* Sqrt[a + b])] + 384*Sqrt[a - b]*b^3*Sqrt[a + b]*ArcTan[(b + a*Tanh[x/2])/( Sqrt[a - b]*Sqrt[a + b])] - 16*a*b*(7*a^2 - 6*b^2)*Coth[x/2] + 6*a^2*(3*a^ 2 - 4*b^2)*Csch[x/2]^2 - 72*a^4*Log[Cosh[x/2]] + 288*a^2*b^2*Log[Cosh[x/2] ] - 192*b^4*Log[Cosh[x/2]] + 72*a^4*Log[Sinh[x/2]] - 288*a^2*b^2*Log[Sinh[ x/2]] + 192*b^4*Log[Sinh[x/2]] + 18*a^4*Sech[x/2]^2 - 24*a^2*b^2*Sech[x/2] ^2 + 3*a^4*Sech[x/2]^4 + 64*a^3*b*Csch[x]^3*Sinh[x/2]^4 + a^3*Csch[x/2]^4* (-3*a + 4*b*Sinh[x]) + 112*a^3*b*Tanh[x/2] - 96*a*b^3*Tanh[x/2])/(192*a^5)
Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.17, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3042, 26, 4001, 26, 3042, 26, 3589, 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.
| \(\displaystyle \int \frac {\text {csch}^5(x)}{a+b \tanh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sin (i x)^5 (a-i b \tan (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\sin (i x)^5 (a-i b \tan (i x))}dx\) |
| \(\Big \downarrow \) 4001 |
| \(\displaystyle i \int -\frac {i \coth (x) \text {csch}^4(x)}{a \cosh (x)+b \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\coth (x) \text {csch}^4(x)}{a \cosh (x)+b \sinh (x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \cos (i x)}{\sin (i x)^5 (a \cos (i x)-i b \sin (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\cos (i x)}{\sin (i x)^5 (a \cos (i x)-i b \sin (i x))}dx\) |
| \(\Big \downarrow \) 3589 |
| \(\displaystyle i \int \left (\frac {i \text {sech}^4(x) b^5}{a^5 (a \cosh (x)+b \sinh (x))}-\frac {i \text {csch}(x) \text {sech}^4(x) b^4}{a^5}+\frac {i \text {csch}^2(x) \text {sech}^3(x) b^3}{a^4}-\frac {i \text {csch}^3(x) \text {sech}^2(x) b^2}{a^3}+\frac {i \text {csch}^4(x) \text {sech}(x) b}{a^2}-\frac {i \text {csch}^5(x)}{a}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (\frac {i b^4 \text {arctanh}(\cosh (x))}{a^5}-\frac {i b^4 \text {sech}(x)}{a^5}-\frac {i b^3 \arctan (\sinh (x))}{a^4}-\frac {3 i b^3 \text {csch}(x)}{2 a^4}+\frac {i b^3 \text {csch}(x) \text {sech}^2(x)}{2 a^4}+\frac {i b^3 \tanh (x) \text {sech}(x)}{2 a^4}-\frac {3 i b^2 \text {arctanh}(\cosh (x))}{2 a^3}+\frac {3 i b^2 \text {sech}(x)}{2 a^3}+\frac {i b^2 \text {csch}^2(x) \text {sech}(x)}{2 a^3}+\frac {i b \arctan (\sinh (x))}{a^2}-\frac {i b \text {csch}^3(x)}{3 a^2}+\frac {i b \text {csch}(x)}{a^2}+\frac {i b \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{a^5}-\frac {i b^2 \left (a^2-b^2\right ) \text {sech}(x)}{a^5}-\frac {i b \left (a^2-b^2\right ) \arctan (\sinh (x))}{a^4}+\frac {3 i \text {arctanh}(\cosh (x))}{8 a}+\frac {i \coth (x) \text {csch}^3(x)}{4 a}-\frac {3 i \coth (x) \text {csch}(x)}{8 a}\right )\) |
I*((I*b*ArcTan[Sinh[x]])/a^2 - (I*b^3*ArcTan[Sinh[x]])/a^4 - (I*b*(a^2 - b ^2)*ArcTan[Sinh[x]])/a^4 + (I*b*(a^2 - b^2)^(3/2)*ArcTan[(b*Cosh[x] + a*Si nh[x])/Sqrt[a^2 - b^2]])/a^5 + (((3*I)/8)*ArcTanh[Cosh[x]])/a - (((3*I)/2) *b^2*ArcTanh[Cosh[x]])/a^3 + (I*b^4*ArcTanh[Cosh[x]])/a^5 + (I*b*Csch[x])/ a^2 - (((3*I)/2)*b^3*Csch[x])/a^4 - (((3*I)/8)*Coth[x]*Csch[x])/a - ((I/3) *b*Csch[x]^3)/a^2 + ((I/4)*Coth[x]*Csch[x]^3)/a + (((3*I)/2)*b^2*Sech[x])/ a^3 - (I*b^4*Sech[x])/a^5 - (I*b^2*(a^2 - b^2)*Sech[x])/a^5 + ((I/2)*b^2*C sch[x]^2*Sech[x])/a^3 + ((I/2)*b^3*Csch[x]*Sech[x]^2)/a^4 + ((I/2)*b^3*Sec h[x]*Tanh[x])/a^4)
3.1.88.3.1 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[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_. ) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[Ex pandTrig[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]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Int[Sin[e + f*x]^m*((a*Cos[e + f*x] + b*Sin[e + f*x])^n/C os[e + f*x]^n), x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && ILtQ [n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))
Time = 3.25 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.89
| method | result | size |
| default | \(\frac {\frac {a^{3} \tanh \left (\frac {x}{2}\right )^{4}}{4}-\frac {2 b \tanh \left (\frac {x}{2}\right )^{3} a^{2}}{3}-2 a^{3} \tanh \left (\frac {x}{2}\right )^{2}+2 a \,b^{2} \tanh \left (\frac {x}{2}\right )^{2}+10 a^{2} b \tanh \left (\frac {x}{2}\right )-8 b^{3} \tanh \left (\frac {x}{2}\right )}{16 a^{4}}-\frac {2 b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{5} \sqrt {a^{2}-b^{2}}}-\frac {1}{64 a \tanh \left (\frac {x}{2}\right )^{4}}-\frac {-4 a^{2}+4 b^{2}}{32 a^{3} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (6 a^{4}-24 a^{2} b^{2}+16 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{16 a^{5}}+\frac {b}{24 a^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {b \left (5 a^{2}-4 b^{2}\right )}{8 a^{4} \tanh \left (\frac {x}{2}\right )}\) | \(227\) |
| risch | \(\frac {{\mathrm e}^{x} \left (9 a^{3} {\mathrm e}^{6 x}-24 \,{\mathrm e}^{6 x} a^{2} b -12 \,{\mathrm e}^{6 x} a \,b^{2}+24 b^{3} {\mathrm e}^{6 x}-33 a^{3} {\mathrm e}^{4 x}+104 a^{2} b \,{\mathrm e}^{4 x}+12 a \,b^{2} {\mathrm e}^{4 x}-72 b^{3} {\mathrm e}^{4 x}-33 a^{3} {\mathrm e}^{2 x}-104 \,{\mathrm e}^{2 x} a^{2} b +12 \,{\mathrm e}^{2 x} a \,b^{2}+72 b^{3} {\mathrm e}^{2 x}+9 a^{3}+24 a^{2} b -12 a \,b^{2}-24 b^{3}\right )}{12 a^{4} \left ({\mathrm e}^{2 x}-1\right )^{4}}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{8 a}+\frac {3 \ln \left ({\mathrm e}^{x}+1\right ) b^{2}}{2 a^{3}}-\frac {\ln \left ({\mathrm e}^{x}+1\right ) b^{4}}{a^{5}}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{8 a}-\frac {3 \ln \left ({\mathrm e}^{x}-1\right ) b^{2}}{2 a^{3}}+\frac {\ln \left ({\mathrm e}^{x}-1\right ) b^{4}}{a^{5}}+\frac {\sqrt {-a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{x}-\frac {\sqrt {-a^{2}+b^{2}}}{a +b}\right )}{a^{3}}-\frac {\sqrt {-a^{2}+b^{2}}\, b^{3} \ln \left ({\mathrm e}^{x}-\frac {\sqrt {-a^{2}+b^{2}}}{a +b}\right )}{a^{5}}-\frac {\sqrt {-a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{x}+\frac {\sqrt {-a^{2}+b^{2}}}{a +b}\right )}{a^{3}}+\frac {\sqrt {-a^{2}+b^{2}}\, b^{3} \ln \left ({\mathrm e}^{x}+\frac {\sqrt {-a^{2}+b^{2}}}{a +b}\right )}{a^{5}}\) | \(381\) |
1/16/a^4*(1/4*a^3*tanh(1/2*x)^4-2/3*b*tanh(1/2*x)^3*a^2-2*a^3*tanh(1/2*x)^ 2+2*a*b^2*tanh(1/2*x)^2+10*a^2*b*tanh(1/2*x)-8*b^3*tanh(1/2*x))-2*b*(a^4-2 *a^2*b^2+b^4)/a^5/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^ 2)^(1/2))-1/64/a/tanh(1/2*x)^4-1/32*(-4*a^2+4*b^2)/a^3/tanh(1/2*x)^2+1/16/ a^5*(6*a^4-24*a^2*b^2+16*b^4)*ln(tanh(1/2*x))+1/24/a^2*b/tanh(1/2*x)^3-1/8 *b*(5*a^2-4*b^2)/a^4/tanh(1/2*x)
Leaf count of result is larger than twice the leaf count of optimal. 2646 vs. \(2 (231) = 462\).
Time = 0.35 (sec) , antiderivative size = 5347, normalized size of antiderivative = 20.97 \[ \int \frac {\text {csch}^5(x)}{a+b \tanh (x)} \, dx=\text {Too large to display} \]
\[ \int \frac {\text {csch}^5(x)}{a+b \tanh (x)} \, dx=\int \frac {\operatorname {csch}^{5}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {\text {csch}^5(x)}{a+b \tanh (x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.28 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.07 \[ \int \frac {\text {csch}^5(x)}{a+b \tanh (x)} \, dx=-\frac {{\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left (e^{x} + 1\right )}{8 \, a^{5}} + \frac {{\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{8 \, a^{5}} - \frac {2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{5}} + \frac {9 \, a^{3} e^{\left (7 \, x\right )} - 24 \, a^{2} b e^{\left (7 \, x\right )} - 12 \, a b^{2} e^{\left (7 \, x\right )} + 24 \, b^{3} e^{\left (7 \, x\right )} - 33 \, a^{3} e^{\left (5 \, x\right )} + 104 \, a^{2} b e^{\left (5 \, x\right )} + 12 \, a b^{2} e^{\left (5 \, x\right )} - 72 \, b^{3} e^{\left (5 \, x\right )} - 33 \, a^{3} e^{\left (3 \, x\right )} - 104 \, a^{2} b e^{\left (3 \, x\right )} + 12 \, a b^{2} e^{\left (3 \, x\right )} + 72 \, b^{3} e^{\left (3 \, x\right )} + 9 \, a^{3} e^{x} + 24 \, a^{2} b e^{x} - 12 \, a b^{2} e^{x} - 24 \, b^{3} e^{x}}{12 \, a^{4} {\left (e^{\left (2 \, x\right )} - 1\right )}^{4}} \]
-1/8*(3*a^4 - 12*a^2*b^2 + 8*b^4)*log(e^x + 1)/a^5 + 1/8*(3*a^4 - 12*a^2*b ^2 + 8*b^4)*log(abs(e^x - 1))/a^5 - 2*(a^4*b - 2*a^2*b^3 + b^5)*arctan((a* e^x + b*e^x)/sqrt(a^2 - b^2))/(sqrt(a^2 - b^2)*a^5) + 1/12*(9*a^3*e^(7*x) - 24*a^2*b*e^(7*x) - 12*a*b^2*e^(7*x) + 24*b^3*e^(7*x) - 33*a^3*e^(5*x) + 104*a^2*b*e^(5*x) + 12*a*b^2*e^(5*x) - 72*b^3*e^(5*x) - 33*a^3*e^(3*x) - 1 04*a^2*b*e^(3*x) + 12*a*b^2*e^(3*x) + 72*b^3*e^(3*x) + 9*a^3*e^x + 24*a^2* b*e^x - 12*a*b^2*e^x - 24*b^3*e^x)/(a^4*(e^(2*x) - 1)^4)
Time = 3.75 (sec) , antiderivative size = 753, normalized size of antiderivative = 2.95 \[ \int \frac {\text {csch}^5(x)}{a+b \tanh (x)} \, dx=\frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (3\,a^4-12\,a^2\,b^2+8\,b^4\right )}{8\,a^5}-\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (3\,a^4-12\,a^2\,b^2+8\,b^4\right )}{8\,a^5}-\frac {4\,{\mathrm {e}}^x}{a\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {2\,{\mathrm {e}}^x\,\left (9\,a-4\,b\right )}{3\,a^2\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {{\mathrm {e}}^x\,\left (3\,a^2-16\,a\,b+12\,b^2\right )}{6\,a^3\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {{\mathrm {e}}^x\,\left (-3\,a^3+8\,a^2\,b+4\,a\,b^2-8\,b^3\right )}{4\,a^4\,\left ({\mathrm {e}}^{2\,x}-1\right )}+\frac {b\,\ln \left (\frac {b\,{\mathrm {e}}^x\,{\left (a-b\right )}^2\,\left (-9\,a^7-24\,a^6\,b+144\,a^5\,b^2+24\,a^4\,b^3-456\,a^3\,b^4+224\,a^2\,b^5+288\,a\,b^6-192\,b^7\right )}{2\,a^{12}\,\left (a+b\right )}-\frac {b\,\left (a-b\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (8\,a^5\,b^3-9\,a^8-9\,a^7\,b+8\,a^6\,b^2+192\,b^5\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}-224\,a^2\,b^3\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}-88\,a^3\,b^2\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}+96\,a\,b^4\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}+24\,a^4\,b\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}\right )}{2\,a^{12}\,{\left (a+b\right )}^4}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^5}-\frac {b\,\ln \left (\frac {b\,{\mathrm {e}}^x\,{\left (a-b\right )}^2\,\left (-9\,a^7-24\,a^6\,b+144\,a^5\,b^2+24\,a^4\,b^3-456\,a^3\,b^4+224\,a^2\,b^5+288\,a\,b^6-192\,b^7\right )}{2\,a^{12}\,\left (a+b\right )}-\frac {b\,\left (a-b\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (9\,a^7\,b+9\,a^8-8\,a^5\,b^3-8\,a^6\,b^2+192\,b^5\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}-224\,a^2\,b^3\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}-88\,a^3\,b^2\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}+96\,a\,b^4\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}+24\,a^4\,b\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}\right )}{2\,a^{12}\,{\left (a+b\right )}^4}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^5} \]
(log(exp(x) - 1)*(3*a^4 + 8*b^4 - 12*a^2*b^2))/(8*a^5) - (log(exp(x) + 1)* (3*a^4 + 8*b^4 - 12*a^2*b^2))/(8*a^5) - (4*exp(x))/(a*(6*exp(4*x) - 4*exp( 2*x) - 4*exp(6*x) + exp(8*x) + 1)) - (2*exp(x)*(9*a - 4*b))/(3*a^2*(3*exp( 2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (exp(x)*(3*a^2 - 16*a*b + 12*b^2))/(6 *a^3*(exp(4*x) - 2*exp(2*x) + 1)) - (exp(x)*(4*a*b^2 + 8*a^2*b - 3*a^3 - 8 *b^3))/(4*a^4*(exp(2*x) - 1)) + (b*log((b*exp(x)*(a - b)^2*(288*a*b^6 - 24 *a^6*b - 9*a^7 - 192*b^7 + 224*a^2*b^5 - 456*a^3*b^4 + 24*a^4*b^3 + 144*a^ 5*b^2))/(2*a^12*(a + b)) - (b*(a - b)*(-(a + b)^3*(a - b)^3)^(1/2)*(8*a^5* b^3 - 9*a^8 - 9*a^7*b + 8*a^6*b^2 + 192*b^5*exp(x)*(-(a^2 - b^2)^3)^(1/2) - 224*a^2*b^3*exp(x)*(-(a^2 - b^2)^3)^(1/2) - 88*a^3*b^2*exp(x)*(-(a^2 - b ^2)^3)^(1/2) + 96*a*b^4*exp(x)*(-(a^2 - b^2)^3)^(1/2) + 24*a^4*b*exp(x)*(- (a^2 - b^2)^3)^(1/2)))/(2*a^12*(a + b)^4))*(-(a + b)^3*(a - b)^3)^(1/2))/a ^5 - (b*log((b*exp(x)*(a - b)^2*(288*a*b^6 - 24*a^6*b - 9*a^7 - 192*b^7 + 224*a^2*b^5 - 456*a^3*b^4 + 24*a^4*b^3 + 144*a^5*b^2))/(2*a^12*(a + b)) - (b*(a - b)*(-(a + b)^3*(a - b)^3)^(1/2)*(9*a^7*b + 9*a^8 - 8*a^5*b^3 - 8*a ^6*b^2 + 192*b^5*exp(x)*(-(a^2 - b^2)^3)^(1/2) - 224*a^2*b^3*exp(x)*(-(a^2 - b^2)^3)^(1/2) - 88*a^3*b^2*exp(x)*(-(a^2 - b^2)^3)^(1/2) + 96*a*b^4*exp (x)*(-(a^2 - b^2)^3)^(1/2) + 24*a^4*b*exp(x)*(-(a^2 - b^2)^3)^(1/2)))/(2*a ^12*(a + b)^4))*(-(a + b)^3*(a - b)^3)^(1/2))/a^5