Integrand size = 11, antiderivative size = 85 \[ \int \frac {\tanh (x)}{(a+b \sinh (x))^2} \, dx=\frac {2 a b \arctan (\sinh (x))}{\left (a^2+b^2\right )^2}+\frac {\left (a^2-b^2\right ) \log (\cosh (x))}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-b^2\right ) \log (a+b \sinh (x))}{\left (a^2+b^2\right )^2}+\frac {a}{\left (a^2+b^2\right ) (a+b \sinh (x))} \] Output:
2*a*b*arctan(sinh(x))/(a^2+b^2)^2+(a^2-b^2)*ln(cosh(x))/(a^2+b^2)^2-(a^2-b ^2)*ln(a+b*sinh(x))/(a^2+b^2)^2+a/(a^2+b^2)/(a+b*sinh(x))
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.72 \[ \int \frac {\tanh (x)}{(a+b \sinh (x))^2} \, dx=\frac {a \left ((a-i b)^2 \log (i-\sinh (x))+(a+i b)^2 \log (i+\sinh (x))+2 \left (a^2+b^2+\left (-a^2+b^2\right ) \log (a+b \sinh (x))\right )\right )+b \left ((a-i b)^2 \log (i-\sinh (x))+(a+i b)^2 \log (i+\sinh (x))+2 \left (-a^2+b^2\right ) \log (a+b \sinh (x))\right ) \sinh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))} \] Input:
Integrate[Tanh[x]/(a + b*Sinh[x])^2,x]
Output:
(a*((a - I*b)^2*Log[I - Sinh[x]] + (a + I*b)^2*Log[I + Sinh[x]] + 2*(a^2 + b^2 + (-a^2 + b^2)*Log[a + b*Sinh[x]])) + b*((a - I*b)^2*Log[I - Sinh[x]] + (a + I*b)^2*Log[I + Sinh[x]] + 2*(-a^2 + b^2)*Log[a + b*Sinh[x]])*Sinh[ x])/(2*(a^2 + b^2)^2*(a + b*Sinh[x]))
Time = 0.38 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.28, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3042, 26, 3200, 25, 594, 25, 657, 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.
| \(\displaystyle \int \frac {\tanh (x)}{(a+b \sinh (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \tan (i x)}{(a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\tan (i x)}{(a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 3200 |
| \(\displaystyle -\int -\frac {b \sinh (x)}{(a+b \sinh (x))^2 \left (\sinh ^2(x) b^2+b^2\right )}d(b \sinh (x))\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {b \sinh (x)}{\left (b^2 \sinh ^2(x)+b^2\right ) (a+b \sinh (x))^2}d(b \sinh (x))\) |
| \(\Big \downarrow \) 594 |
| \(\displaystyle \frac {a}{\left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\int -\frac {b^2+a \sinh (x) b}{(a+b \sinh (x)) \left (\sinh ^2(x) b^2+b^2\right )}d(b \sinh (x))}{a^2+b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b^2+a \sinh (x) b}{(a+b \sinh (x)) \left (\sinh ^2(x) b^2+b^2\right )}d(b \sinh (x))}{a^2+b^2}+\frac {a}{\left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle \frac {\int \left (\frac {b^2-a^2}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {2 a b^2+\left (a^2-b^2\right ) \sinh (x) b}{\left (a^2+b^2\right ) \left (\sinh ^2(x) b^2+b^2\right )}\right )d(b \sinh (x))}{a^2+b^2}+\frac {a}{\left (a^2+b^2\right ) (a+b \sinh (x))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 a b \arctan (\sinh (x))}{a^2+b^2}+\frac {\left (a^2-b^2\right ) \log \left (b^2 \sinh ^2(x)+b^2\right )}{2 \left (a^2+b^2\right )}-\frac {\left (a^2-b^2\right ) \log (a+b \sinh (x))}{a^2+b^2}}{a^2+b^2}+\frac {a}{\left (a^2+b^2\right ) (a+b \sinh (x))}\) |
Input:
Int[Tanh[x]/(a + b*Sinh[x])^2,x]
Output:
((2*a*b*ArcTan[Sinh[x]])/(a^2 + b^2) - ((a^2 - b^2)*Log[a + b*Sinh[x]])/(a ^2 + b^2) + ((a^2 - b^2)*Log[b^2 + b^2*Sinh[x]^2])/(2*(a^2 + b^2)))/(a^2 + b^2) + a/((a^2 + b^2)*(a + b*Sinh[x]))
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[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))) , x] + Simp[1/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2) ^p*(a*d*(n + 1) + b*c*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[(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]
Time = 0.88 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.60
| method | result | size |
| default | \(-\frac {2 \left (\frac {\left (-a^{2} b -b^{3}\right ) \tanh \left (\frac {x}{2}\right )}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}+\frac {\left (a^{2}-b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}{2}\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {2 \left (a^{2}-b^{2}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )+8 a b \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}\) | \(136\) |
| risch | \(\frac {2 a \,{\mathrm e}^{x}}{\left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) a^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) b^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 i \ln \left ({\mathrm e}^{x}+i\right ) a b}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {\ln \left ({\mathrm e}^{x}+i\right ) a^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {\ln \left ({\mathrm e}^{x}+i\right ) b^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 i \ln \left ({\mathrm e}^{x}-i\right ) a b}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {\ln \left ({\mathrm e}^{x}-i\right ) a^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {\ln \left ({\mathrm e}^{x}-i\right ) b^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}\) | \(272\) |
Input:
int(tanh(x)/(a+b*sinh(x))^2,x,method=_RETURNVERBOSE)
Output:
-2/(a^2+b^2)^2*((-a^2*b-b^3)*tanh(1/2*x)/(tanh(1/2*x)^2*a-2*b*tanh(1/2*x)- a)+1/2*(a^2-b^2)*ln(tanh(1/2*x)^2*a-2*b*tanh(1/2*x)-a))+4/(2*a^4+4*a^2*b^2 +2*b^4)*(1/2*(a^2-b^2)*ln(1+tanh(1/2*x)^2)+2*a*b*arctan(tanh(1/2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (85) = 170\).
Time = 0.09 (sec) , antiderivative size = 423, normalized size of antiderivative = 4.98 \[ \int \frac {\tanh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {4 \, {\left (a b^{2} \cosh \left (x\right )^{2} + a b^{2} \sinh \left (x\right )^{2} + 2 \, a^{2} b \cosh \left (x\right ) - a b^{2} + 2 \, {\left (a b^{2} \cosh \left (x\right ) + a^{2} b\right )} \sinh \left (x\right )\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right ) + {\left (a^{2} b - b^{3} - {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{2} - {\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - {\left (a^{2} b - b^{3} - {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )^{2} - {\left (a^{2} b - b^{3}\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (a^{3} + a b^{2}\right )} \sinh \left (x\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )^{2} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )} \] Input:
integrate(tanh(x)/(a+b*sinh(x))^2,x, algorithm="fricas")
Output:
-(4*(a*b^2*cosh(x)^2 + a*b^2*sinh(x)^2 + 2*a^2*b*cosh(x) - a*b^2 + 2*(a*b^ 2*cosh(x) + a^2*b)*sinh(x))*arctan(cosh(x) + sinh(x)) + 2*(a^3 + a*b^2)*co sh(x) + (a^2*b - b^3 - (a^2*b - b^3)*cosh(x)^2 - (a^2*b - b^3)*sinh(x)^2 - 2*(a^3 - a*b^2)*cosh(x) - 2*(a^3 - a*b^2 + (a^2*b - b^3)*cosh(x))*sinh(x) )*log(2*(b*sinh(x) + a)/(cosh(x) - sinh(x))) - (a^2*b - b^3 - (a^2*b - b^3 )*cosh(x)^2 - (a^2*b - b^3)*sinh(x)^2 - 2*(a^3 - a*b^2)*cosh(x) - 2*(a^3 - a*b^2 + (a^2*b - b^3)*cosh(x))*sinh(x))*log(2*cosh(x)/(cosh(x) - sinh(x)) ) + 2*(a^3 + a*b^2)*sinh(x))/(a^4*b + 2*a^2*b^3 + b^5 - (a^4*b + 2*a^2*b^3 + b^5)*cosh(x)^2 - (a^4*b + 2*a^2*b^3 + b^5)*sinh(x)^2 - 2*(a^5 + 2*a^3*b ^2 + a*b^4)*cosh(x) - 2*(a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^ 5)*cosh(x))*sinh(x))
\[ \int \frac {\tanh (x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\tanh {\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \] Input:
integrate(tanh(x)/(a+b*sinh(x))**2,x)
Output:
Integral(tanh(x)/(a + b*sinh(x))**2, x)
Time = 0.13 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.82 \[ \int \frac {\tanh (x)}{(a+b \sinh (x))^2} \, dx=-\frac {4 \, a b \arctan \left (e^{\left (-x\right )}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, a e^{\left (-x\right )}}{a^{2} b + b^{3} + 2 \, {\left (a^{3} + a b^{2}\right )} e^{\left (-x\right )} - {\left (a^{2} b + b^{3}\right )} e^{\left (-2 \, x\right )}} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} \] Input:
integrate(tanh(x)/(a+b*sinh(x))^2,x, algorithm="maxima")
Output:
-4*a*b*arctan(e^(-x))/(a^4 + 2*a^2*b^2 + b^4) + 2*a*e^(-x)/(a^2*b + b^3 + 2*(a^3 + a*b^2)*e^(-x) - (a^2*b + b^3)*e^(-2*x)) - (a^2 - b^2)*log(-2*a*e^ (-x) + b*e^(-2*x) - b)/(a^4 + 2*a^2*b^2 + b^4) + (a^2 - b^2)*log(e^(-2*x) + 1)/(a^4 + 2*a^2*b^2 + b^4)
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (85) = 170\).
Time = 0.13 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.34 \[ \int \frac {\tanh (x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} a b}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {{\left (a^{2} b - b^{3}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} + \frac {a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )} - b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} - 4 \, a^{3}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b {\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )}} \] Input:
integrate(tanh(x)/(a+b*sinh(x))^2,x, algorithm="giac")
Output:
(pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))*a*b/(a^4 + 2*a^2*b^2 + b^4) + 1/ 2*(a^2 - b^2)*log((e^(-x) - e^x)^2 + 4)/(a^4 + 2*a^2*b^2 + b^4) - (a^2*b - b^3)*log(abs(-b*(e^(-x) - e^x) + 2*a))/(a^4*b + 2*a^2*b^3 + b^5) + (a^2*b *(e^(-x) - e^x) - b^3*(e^(-x) - e^x) - 4*a^3)/((a^4 + 2*a^2*b^2 + b^4)*(b* (e^(-x) - e^x) - 2*a))
Time = 2.63 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.24 \[ \int \frac {\tanh (x)}{(a+b \sinh (x))^2} \, dx=\frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}{a^2+a\,b\,2{}\mathrm {i}-b^2}-\frac {\ln \left (b^5\,{\mathrm {e}}^{2\,x}-a^4\,b-b^5+a^2\,b^3+2\,a^5\,{\mathrm {e}}^x-a^2\,b^3\,{\mathrm {e}}^{2\,x}+2\,a\,b^4\,{\mathrm {e}}^x+a^4\,b\,{\mathrm {e}}^{2\,x}-2\,a^3\,b^2\,{\mathrm {e}}^x\right )\,\left (a^2-b^2\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {2\,a\,b\,{\mathrm {e}}^x}{\left (a^2\,b+b^3\right )\,\left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )}+\frac {\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}} \] Input:
int(tanh(x)/(a + b*sinh(x))^2,x)
Output:
log(exp(x)*1i + 1)/(a*b*2i + a^2 - b^2) + (log(exp(x) + 1i)*1i)/(2*a*b + a ^2*1i - b^2*1i) - (log(b^5*exp(2*x) - a^4*b - b^5 + a^2*b^3 + 2*a^5*exp(x) - a^2*b^3*exp(2*x) + 2*a*b^4*exp(x) + a^4*b*exp(2*x) - 2*a^3*b^2*exp(x))* (a^2 - b^2))/(a^4 + b^4 + 2*a^2*b^2) + (2*a*b*exp(x))/((a^2*b + b^3)*(2*a* exp(x) - b + b*exp(2*x)))
Time = 0.15 (sec) , antiderivative size = 401, normalized size of antiderivative = 4.72 \[ \int \frac {\tanh (x)}{(a+b \sinh (x))^2} \, dx=\frac {4 e^{2 x} \mathit {atan} \left (e^{x}\right ) a \,b^{2}+8 e^{x} \mathit {atan} \left (e^{x}\right ) a^{2} b -4 \mathit {atan} \left (e^{x}\right ) a \,b^{2}+e^{2 x} \mathrm {log}\left (e^{2 x}+1\right ) a^{2} b -e^{2 x} \mathrm {log}\left (e^{2 x}+1\right ) b^{3}-e^{2 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) a^{2} b +e^{2 x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) b^{3}-e^{2 x} a^{2} b -e^{2 x} b^{3}+2 e^{x} \mathrm {log}\left (e^{2 x}+1\right ) a^{3}-2 e^{x} \mathrm {log}\left (e^{2 x}+1\right ) a \,b^{2}-2 e^{x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) a^{3}+2 e^{x} \mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) a \,b^{2}-\mathrm {log}\left (e^{2 x}+1\right ) a^{2} b +\mathrm {log}\left (e^{2 x}+1\right ) b^{3}+\mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) a^{2} b -\mathrm {log}\left (e^{2 x} b +2 e^{x} a -b \right ) b^{3}+a^{2} b +b^{3}}{e^{2 x} a^{4} b +2 e^{2 x} a^{2} b^{3}+e^{2 x} b^{5}+2 e^{x} a^{5}+4 e^{x} a^{3} b^{2}+2 e^{x} a \,b^{4}-a^{4} b -2 a^{2} b^{3}-b^{5}} \] Input:
int(tanh(x)/(a+b*sinh(x))^2,x)
Output:
(4*e**(2*x)*atan(e**x)*a*b**2 + 8*e**x*atan(e**x)*a**2*b - 4*atan(e**x)*a* b**2 + e**(2*x)*log(e**(2*x) + 1)*a**2*b - e**(2*x)*log(e**(2*x) + 1)*b**3 - e**(2*x)*log(e**(2*x)*b + 2*e**x*a - b)*a**2*b + e**(2*x)*log(e**(2*x)* b + 2*e**x*a - b)*b**3 - e**(2*x)*a**2*b - e**(2*x)*b**3 + 2*e**x*log(e**( 2*x) + 1)*a**3 - 2*e**x*log(e**(2*x) + 1)*a*b**2 - 2*e**x*log(e**(2*x)*b + 2*e**x*a - b)*a**3 + 2*e**x*log(e**(2*x)*b + 2*e**x*a - b)*a*b**2 - log(e **(2*x) + 1)*a**2*b + log(e**(2*x) + 1)*b**3 + log(e**(2*x)*b + 2*e**x*a - b)*a**2*b - log(e**(2*x)*b + 2*e**x*a - b)*b**3 + a**2*b + b**3)/(e**(2*x )*a**4*b + 2*e**(2*x)*a**2*b**3 + e**(2*x)*b**5 + 2*e**x*a**5 + 4*e**x*a** 3*b**2 + 2*e**x*a*b**4 - a**4*b - 2*a**2*b**3 - b**5)