Integrand size = 23, antiderivative size = 87 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {\sqrt {a} \sqrt {b} \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{(a+b)^2 d}+\frac {(a-b) \text {arctanh}(\cosh (c+d x))}{2 (a+b)^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 (a+b) d} \] Output:
-a^(1/2)*b^(1/2)*arctan(a^(1/2)*cosh(d*x+c)/b^(1/2))/(a+b)^2/d+1/2*(a-b)*a rctanh(cosh(d*x+c))/(a+b)^2/d-1/2*coth(d*x+c)*csch(d*x+c)/(a+b)/d
Result contains complex when optimal does not.
Time = 2.00 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.89 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {(a+2 b+a \cosh (2 (c+d x))) \left (8 \sqrt {a} \sqrt {b} \arctan \left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )+8 \sqrt {a} \sqrt {b} \arctan \left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )+(a+b) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-4 a \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+4 b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+4 a \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )-4 b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+(a+b) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )\right ) \text {sech}^2(c+d x)}{16 (a+b)^2 d \left (a+b \text {sech}^2(c+d x)\right )} \] Input:
Integrate[Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2),x]
Output:
-1/16*((a + 2*b + a*Cosh[2*(c + d*x)])*(8*Sqrt[a]*Sqrt[b]*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh [c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/S qrt[b]] + 8*Sqrt[a]*Sqrt[b]*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*S qrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] + (a + b)*Csch[(c + d* x)/2]^2 - 4*a*Log[Cosh[(c + d*x)/2]] + 4*b*Log[Cosh[(c + d*x)/2]] + 4*a*Lo g[Sinh[(c + d*x)/2]] - 4*b*Log[Sinh[(c + d*x)/2]] + (a + b)*Sech[(c + d*x) /2]^2)*Sech[c + d*x]^2)/((a + b)^2*d*(a + b*Sech[c + d*x]^2))
Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 4621, 373, 397, 218, 219}
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.
| \(\displaystyle \int \frac {\text {csch}^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i c+i d x)^3 \left (a+b \sec (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\left (b \sec (i c+i d x)^2+a\right ) \sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 4621 |
| \(\displaystyle \frac {\int \frac {\cosh ^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right )^2 \left (a \cosh ^2(c+d x)+b\right )}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle \frac {\frac {\cosh (c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right )}-\frac {\int \frac {b-a \cosh ^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}d\cosh (c+d x)}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\cosh (c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right )}-\frac {\frac {2 a b \int \frac {1}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{a+b}-\frac {(a-b) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a+b}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\cosh (c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right )}-\frac {\frac {2 \sqrt {a} \sqrt {b} \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{a+b}-\frac {(a-b) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a+b}}{2 (a+b)}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\cosh (c+d x)}{2 (a+b) \left (1-\cosh ^2(c+d x)\right )}-\frac {\frac {2 \sqrt {a} \sqrt {b} \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{a+b}-\frac {(a-b) \text {arctanh}(\cosh (c+d x))}{a+b}}{2 (a+b)}}{d}\) |
Input:
Int[Csch[c + d*x]^3/(a + b*Sech[c + d*x]^2),x]
Output:
(-1/2*((2*Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(a + b) - ((a - b)*ArcTanh[Cosh[c + d*x]])/(a + b))/(a + b) + Cosh[c + d*x]/(2*(a + b)*(1 - Cosh[c + d*x]^2)))/d
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_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/ 2] && IntegerQ[n] && IntegerQ[p]
Time = 1.46 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.28
| method | result | size |
| derivativedivides | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a +8 b}-\frac {1}{8 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a +2 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2}}-\frac {a b \arctan \left (\frac {2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a -2 b}{4 \sqrt {a b}}\right )}{\left (a +b \right )^{2} \sqrt {a b}}}{d}\) | \(111\) |
| default | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a +8 b}-\frac {1}{8 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a +2 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 \left (a +b \right )^{2}}-\frac {a b \arctan \left (\frac {2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a -2 b}{4 \sqrt {a b}}\right )}{\left (a +b \right )^{2} \sqrt {a b}}}{d}\) | \(111\) |
| risch | \(-\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d \left (a +b \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) a}{2 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b}{2 d \left (a^{2}+2 a b +b^{2}\right )}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) a}{2 d \left (a^{2}+2 a b +b^{2}\right )}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b}{2 d \left (a^{2}+2 a b +b^{2}\right )}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{2 \left (a +b \right )^{2} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{2 \left (a +b \right )^{2} d}\) | \(244\) |
Input:
int(csch(d*x+c)^3/(a+b*sech(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/(a+b)-1/8/(a+b)/tanh(1/2*d*x+1/2*c)^2+1/4/( a+b)^2*(-2*a+2*b)*ln(tanh(1/2*d*x+1/2*c))-a*b/(a+b)^2/(a*b)^(1/2)*arctan(1 /4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (75) = 150\).
Time = 0.33 (sec) , antiderivative size = 1884, normalized size of antiderivative = 21.66 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^3/(a+b*sech(d*x+c)^2),x, algorithm="fricas")
Output:
[-1/2*(2*(a + b)*cosh(d*x + c)^3 + 6*(a + b)*cosh(d*x + c)*sinh(d*x + c)^2 + 2*(a + b)*sinh(d*x + c)^3 - (cosh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + 2*(3*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - 2* cosh(d*x + c)^2 + 4*(cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) + 1)*s qrt(-a*b)*log((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*s inh(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))*s inh(d*x + c) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh (d*x + c)^3 + (3*cosh(d*x + c)^2 + 1)*sinh(d*x + c) + cosh(d*x + c))*sqrt( -a*b) + a)/(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)) + 2*(a + b)*cosh(d*x + c) - ((a - b)*cosh(d*x + c)^4 + 4*( a - b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a - b)*sinh(d*x + c)^4 - 2*(a - b) *cosh(d*x + c)^2 + 2*(3*(a - b)*cosh(d*x + c)^2 - a + b)*sinh(d*x + c)^2 + 4*((a - b)*cosh(d*x + c)^3 - (a - b)*cosh(d*x + c))*sinh(d*x + c) + a - b )*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((a - b)*cosh(d*x + c)^4 + 4*(a - b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a - b)*sinh(d*x + c)^4 - 2*(a - b)* cosh(d*x + c)^2 + 2*(3*(a - b)*cosh(d*x + c)^2 - a + b)*sinh(d*x + c)^2 + 4*((a - b)*cosh(d*x + c)^3 - (a - b)*cosh(d*x + c))*sinh(d*x + c) + a -...
\[ \int \frac {\text {csch}^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \] Input:
integrate(csch(d*x+c)**3/(a+b*sech(d*x+c)**2),x)
Output:
Integral(csch(c + d*x)**3/(a + b*sech(c + d*x)**2), x)
\[ \int \frac {\text {csch}^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{b \operatorname {sech}\left (d x + c\right )^{2} + a} \,d x } \] Input:
integrate(csch(d*x+c)^3/(a+b*sech(d*x+c)^2),x, algorithm="maxima")
Output:
1/2*(a - b)*log((e^(d*x + c) + 1)*e^(-c))/(a^2*d + 2*a*b*d + b^2*d) - 1/2* (a - b)*log((e^(d*x + c) - 1)*e^(-c))/(a^2*d + 2*a*b*d + b^2*d) - (e^(3*d* x + 3*c) + e^(d*x + c))/(a*d + b*d + (a*d*e^(4*c) + b*d*e^(4*c))*e^(4*d*x) - 2*(a*d*e^(2*c) + b*d*e^(2*c))*e^(2*d*x)) - 8*integrate(1/4*(a*b*e^(3*d* x + 3*c) - a*b*e^(d*x + c))/(a^3 + 2*a^2*b + a*b^2 + (a^3*e^(4*c) + 2*a^2* b*e^(4*c) + a*b^2*e^(4*c))*e^(4*d*x) + 2*(a^3*e^(2*c) + 4*a^2*b*e^(2*c) + 5*a*b^2*e^(2*c) + 2*b^3*e^(2*c))*e^(2*d*x)), x)
Exception generated. \[ \int \frac {\text {csch}^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(csch(d*x+c)^3/(a+b*sech(d*x+c)^2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Time = 3.86 (sec) , antiderivative size = 1586, normalized size of antiderivative = 18.23 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\text {Too large to display} \] Input:
int(1/(sinh(c + d*x)^3*(a + b/cosh(c + d*x)^2)),x)
Output:
((a*b)^(1/2)*(2*atan(((exp(d*x)*exp(c)*((64*(2*b^5*d*(a*b)^(1/2) + 2*a*b^4 *d*(a*b)^(1/2) + 2*a^4*b*d*(a*b)^(1/2) + 2*a^3*b^2*d*(a*b)^(1/2)))/(a^4*(a + b)^3*(d^2*(a + b)^4)^(1/2)*(2*a*b + a^2 + b^2)*(a^4*d^2 + b^4*d^2 + 4*a *b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^(1/2)) + (32*(a*b^3*(a^4*d^2 + b^4 *d^2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^(1/2) + a^3*b*(a^4*d^2 + b^4*d^2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^(1/2) - a^2*b^2*(a^4 *d^2 + b^4*d^2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^(1/2)))/(a^3*d *(a*b)^(1/2)*(a + b)^5*(2*a*b + a^2 + b^2)*(a^4*d^2 + b^4*d^2 + 4*a*b^3*d^ 2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^(1/2))) + (32*exp(3*c)*exp(3*d*x)*(a*b^3* (a^4*d^2 + b^4*d^2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^(1/2) + a^ 3*b*(a^4*d^2 + b^4*d^2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^(1/2) - a^2*b^2*(a^4*d^2 + b^4*d^2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^ (1/2)))/(a^3*d*(a*b)^(1/2)*(a + b)^5*(2*a*b + a^2 + b^2)*(a^4*d^2 + b^4*d^ 2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^(1/2)))*(a^8*(a^4*d^2 + b^4 *d^2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^(1/2) + 5*a^7*b*(a^4*d^2 + b^4*d^2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^(1/2) + a^3*b^5*(a ^4*d^2 + b^4*d^2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^(1/2) + 5*a^ 4*b^4*(a^4*d^2 + b^4*d^2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2*d^2)^(1/2 ) + 10*a^5*b^3*(a^4*d^2 + b^4*d^2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 + 6*a^2*b^2* d^2)^(1/2) + 10*a^6*b^2*(a^4*d^2 + b^4*d^2 + 4*a*b^3*d^2 + 4*a^3*b*d^2 ...
Time = 0.36 (sec) , antiderivative size = 1465, normalized size of antiderivative = 16.84 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^3/(a+b*sech(d*x+c)^2),x)
Output:
( - 2*e**(4*c + 4*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b))) + 4*e**(2*c + 2*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b )*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqr t(a + b) + a + 2*b))) - 2*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt( a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b ) + a + 2*b))) + 2*e**(4*c + 4*d*x)*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2* b)))*b - 4*e**(2*c + 2*d*x)*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)* atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*b + 2*sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sq rt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*b - e**(4*c + 4*d*x)*sqrt(b) *sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)*log( - sqrt(2*s qrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a)) + e**(4*c + 4*d*x)*s qrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)*log(sqrt( 2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a)) + 2*e**(2*c + 2*d *x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a)) - 2*e**(2 *c + 2*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) - a - 2 *b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a)) -...