Integrand size = 23, antiderivative size = 53 \[ \int \frac {\text {csch}^2(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2} d}-\frac {\coth (c+d x)}{(a+b) d} \] Output:
b^(1/2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/(a+b)^(3/2)/d-coth(d*x+c) /(a+b)/d
Leaf count is larger than twice the leaf count of optimal. \(179\) vs. \(2(53)=106\).
Time = 0.81 (sec) , antiderivative size = 179, normalized size of antiderivative = 3.38 \[ \int \frac {\text {csch}^2(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}^2(c+d x) \left (b \text {arctanh}\left (\frac {\text {sech}(d x) (\cosh (2 c)-\sinh (2 c)) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right ) (\cosh (2 c)-\sinh (2 c))+\sqrt {a+b} \text {csch}(c) \text {csch}(c+d x) \sqrt {b (\cosh (c)-\sinh (c))^4} \sinh (d x)\right )}{2 (a+b)^{3/2} d \left (a+b \text {sech}^2(c+d x)\right ) \sqrt {b (\cosh (c)-\sinh (c))^4}} \] Input:
Integrate[Csch[c + d*x]^2/(a + b*Sech[c + d*x]^2),x]
Output:
((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*(b*ArcTanh[(Sech[d*x]*(Co sh[2*c] - Sinh[2*c])*((a + 2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(Cosh[2*c] - Sinh[2*c]) + Sqrt[a + b] *Csch[c]*Csch[c + d*x]*Sqrt[b*(Cosh[c] - Sinh[c])^4]*Sinh[d*x]))/(2*(a + b )^(3/2)*d*(a + b*Sech[c + d*x]^2)*Sqrt[b*(Cosh[c] - Sinh[c])^4])
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 25, 4620, 264, 221}
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.
| \(\displaystyle \int \frac {\text {csch}^2(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\sin (i c+i d x)^2 \left (a+b \sec (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\left (b \sec (i c+i d x)^2+a\right ) \sin (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 4620 |
| \(\displaystyle \frac {\int \frac {\coth ^2(c+d x)}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\frac {b \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a+b}-\frac {\coth (c+d x)}{a+b}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {\coth (c+d x)}{a+b}}{d}\) |
Input:
Int[Csch[c + d*x]^2/(a + b*Sech[c + d*x]^2),x]
Output:
((Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(3/2) - Co th[c + d*x]/(a + b))/d
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1)/f Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1 + f f^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(119\) vs. \(2(45)=90\).
Time = 0.92 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.26
| method | result | size |
| risch | \(-\frac {2}{d \left (a +b \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )}+\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-a +2 \sqrt {b \left (a +b \right )}-2 b}{a}\right )}{2 \left (a +b \right )^{2} d}-\frac {\sqrt {b \left (a +b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a +2 \sqrt {b \left (a +b \right )}+2 b}{a}\right )}{2 \left (a +b \right )^{2} d}\) | \(120\) |
| derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right )}-\frac {1}{2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a +b}}{d}\) | \(145\) |
| default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right )}-\frac {1}{2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{a +b}}{d}\) | \(145\) |
Input:
int(csch(d*x+c)^2/(a+b*sech(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
-2/d/(a+b)/(exp(2*d*x+2*c)-1)+1/2*(b*(a+b))^(1/2)/(a+b)^2/d*ln(exp(2*d*x+2 *c)-(-a+2*(b*(a+b))^(1/2)-2*b)/a)-1/2*(b*(a+b))^(1/2)/(a+b)^2/d*ln(exp(2*d *x+2*c)+(a+2*(b*(a+b))^(1/2)+2*b)/a)
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (45) = 90\).
Time = 0.23 (sec) , antiderivative size = 588, normalized size of antiderivative = 11.09 \[ \int \frac {\text {csch}^2(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(csch(d*x+c)^2/(a+b*sech(d*x+c)^2),x, algorithm="fricas")
Output:
[1/2*((cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(b/(a + b))*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d* x + c)^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^ 2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x + c)^2 + a^2 + 8*a*b + 8*b^2 + 4 *(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((a ^2 + a*b)*cosh(d*x + c)^2 + 2*(a^2 + a*b)*cosh(d*x + c)*sinh(d*x + c) + (a ^2 + a*b)*sinh(d*x + c)^2 + a^2 + 3*a*b + 2*b^2)*sqrt(b/(a + b)))/(a*cosh( d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) - 4 )/((a + b)*d*cosh(d*x + c)^2 + 2*(a + b)*d*cosh(d*x + c)*sinh(d*x + c) + ( a + b)*d*sinh(d*x + c)^2 - (a + b)*d), ((cosh(d*x + c)^2 + 2*cosh(d*x + c) *sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(-b/(a + b))*arctan(1/2*(a*cosh( d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b )*sqrt(-b/(a + b))/b) - 2)/((a + b)*d*cosh(d*x + c)^2 + 2*(a + b)*d*cosh(d *x + c)*sinh(d*x + c) + (a + b)*d*sinh(d*x + c)^2 - (a + b)*d)]
\[ \int \frac {\text {csch}^2(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int \frac {\operatorname {csch}^{2}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \] Input:
integrate(csch(d*x+c)**2/(a+b*sech(d*x+c)**2),x)
Output:
Integral(csch(c + d*x)**2/(a + b*sech(c + d*x)**2), x)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (45) = 90\).
Time = 0.14 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.89 \[ \int \frac {\text {csch}^2(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {b \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{2 \, \sqrt {{\left (a + b\right )} b} {\left (a + b\right )} d} + \frac {2}{{\left ({\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} - a - b\right )} d} \] Input:
integrate(csch(d*x+c)^2/(a+b*sech(d*x+c)^2),x, algorithm="maxima")
Output:
-1/2*b*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/(sqrt((a + b)*b)*(a + b)*d) + 2/(( (a + b)*e^(-2*d*x - 2*c) - a - b)*d)
Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.42 \[ \int \frac {\text {csch}^2(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\frac {b \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} {\left (a + b\right )}} - \frac {2}{{\left (a + b\right )} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}}{d} \] Input:
integrate(csch(d*x+c)^2/(a+b*sech(d*x+c)^2),x, algorithm="giac")
Output:
(b*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*(a + b)) - 2/((a + b)*(e^(2*d*x + 2*c) - 1)))/d
Time = 3.42 (sec) , antiderivative size = 847, normalized size of antiderivative = 15.98 \[ \int \frac {\text {csch}^2(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {2}{\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )\,\left (a\,d+b\,d\right )}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (\frac {2\,\left (8\,b^{5/2}\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+8\,a\,b^{3/2}\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+a^2\,\sqrt {b}\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}\right )\,\left (a^2+8\,a\,b+8\,b^2\right )}{a^5\,d\,{\left (a+b\right )}^3\,\left (a^2+2\,a\,b+b^2\right )\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}}+\frac {4\,\sqrt {b}\,\left (2\,a+4\,b\right )\,\left (4\,d\,a^3\,b+16\,d\,a^2\,b^2+20\,d\,a\,b^3+8\,d\,b^4\right )}{a^5\,\left (a+b\right )\,\sqrt {-d^2\,{\left (a+b\right )}^3}\,\left (a^2+2\,a\,b+b^2\right )\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}}\right )+\frac {2\,\left (2\,a\,b^{3/2}\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+a^2\,\sqrt {b}\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}\right )\,\left (a^2+8\,a\,b+8\,b^2\right )}{a^5\,d\,{\left (a+b\right )}^3\,\left (a^2+2\,a\,b+b^2\right )\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}}+\frac {4\,\sqrt {b}\,\left (2\,a+4\,b\right )\,\left (2\,d\,a^3\,b+4\,d\,a^2\,b^2+2\,d\,a\,b^3\right )}{a^5\,\left (a+b\right )\,\sqrt {-d^2\,{\left (a+b\right )}^3}\,\left (a^2+2\,a\,b+b^2\right )\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}}\right )\,\left (a^5\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+3\,a^4\,b\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+a^2\,b^3\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}+3\,a^3\,b^2\,\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}\right )}{4\,b}\right )}{\sqrt {-a^3\,d^2-3\,a^2\,b\,d^2-3\,a\,b^2\,d^2-b^3\,d^2}} \] Input:
int(1/(sinh(c + d*x)^2*(a + b/cosh(c + d*x)^2)),x)
Output:
- 2/((exp(2*c + 2*d*x) - 1)*(a*d + b*d)) - (b^(1/2)*atan(((exp(2*c)*exp(2* d*x)*((2*(8*b^(5/2)*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2 ) + 8*a*b^(3/2)*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2) + a^2*b^(1/2)*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2))*(8*a* b + a^2 + 8*b^2))/(a^5*d*(a + b)^3*(2*a*b + a^2 + b^2)*(- a^3*d^2 - b^3*d^ 2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2)) + (4*b^(1/2)*(2*a + 4*b)*(8*b^4*d + 16*a^2*b^2*d + 20*a*b^3*d + 4*a^3*b*d))/(a^5*(a + b)*(-d^2*(a + b)^3)^(1/2 )*(2*a*b + a^2 + b^2)*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1 /2))) + (2*(2*a*b^(3/2)*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^ (1/2) + a^2*b^(1/2)*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2 ))*(8*a*b + a^2 + 8*b^2))/(a^5*d*(a + b)^3*(2*a*b + a^2 + b^2)*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2)) + (4*b^(1/2)*(2*a + 4*b)*(4* a^2*b^2*d + 2*a*b^3*d + 2*a^3*b*d))/(a^5*(a + b)*(-d^2*(a + b)^3)^(1/2)*(2 *a*b + a^2 + b^2)*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2)) )*(a^5*(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2) + 3*a^4*b*( - a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2) + a^2*b^3*(- a^3*d^ 2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2) + 3*a^3*b^2*(- a^3*d^2 - b^ 3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2)))/(4*b)))/(- a^3*d^2 - b^3*d^2 - 3*a*b^2*d^2 - 3*a^2*b*d^2)^(1/2)
Time = 0.27 (sec) , antiderivative size = 334, normalized size of antiderivative = 6.30 \[ \int \frac {\text {csch}^2(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {e^{2 d x +2 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right )+e^{2 d x +2 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right )-e^{2 d x +2 c} \sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (2 \sqrt {b}\, \sqrt {a +b}+e^{2 d x +2 c} a +a +2 b \right )-\sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right )-\sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right )+\sqrt {b}\, \sqrt {a +b}\, \mathrm {log}\left (2 \sqrt {b}\, \sqrt {a +b}+e^{2 d x +2 c} a +a +2 b \right )-4 e^{2 d x +2 c} a -4 e^{2 d x +2 c} b}{2 d \left (e^{2 d x +2 c} a^{2}+2 e^{2 d x +2 c} a b +e^{2 d x +2 c} b^{2}-a^{2}-2 a b -b^{2}\right )} \] Input:
int(csch(d*x+c)^2/(a+b*sech(d*x+c)^2),x)
Output:
(e**(2*c + 2*d*x)*sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a)) + e**(2*c + 2*d*x)*sqrt(b)*sqrt(a + b)*lo g(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a)) - e**(2*c + 2*d*x)*sqrt(b)*sqrt(a + b)*log(2*sqrt(b)*sqrt(a + b) + e**(2*c + 2*d*x)* a + a + 2*b) - sqrt(b)*sqrt(a + b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a)) - sqrt(b)*sqrt(a + b)*log(sqrt(2*sqrt(b)*sqr t(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a)) + sqrt(b)*sqrt(a + b)*log(2*sq rt(b)*sqrt(a + b) + e**(2*c + 2*d*x)*a + a + 2*b) - 4*e**(2*c + 2*d*x)*a - 4*e**(2*c + 2*d*x)*b)/(2*d*(e**(2*c + 2*d*x)*a**2 + 2*e**(2*c + 2*d*x)*a* b + e**(2*c + 2*d*x)*b**2 - a**2 - 2*a*b - b**2))