Integrand size = 27, antiderivative size = 121 \[ \int \frac {\text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a \left (a^2-b^2\right ) \arctan (\sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d}-\frac {a^2 b \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a^2 b \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d} \]
1/2*a*(a^2-b^2)*arctan(sinh(d*x+c))/(a^2+b^2)^2/d-a^2*b*ln(cosh(d*x+c))/(a ^2+b^2)^2/d+a^2*b*ln(a+b*sinh(d*x+c))/(a^2+b^2)^2/d-1/2*sech(d*x+c)^2*(b+a *sinh(d*x+c))/(a^2+b^2)/d
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07 \[ \int \frac {\text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a \left (\left (a^2+b^2\right ) \arctan (\sinh (c+d x))+a ((i a+b) \log (i-\sinh (c+d x))+(-i a+b) \log (i+\sinh (c+d x))-2 b \log (a+b \sinh (c+d x)))\right )+b \left (a^2+b^2\right ) \text {sech}^2(c+d x)+a \left (a^2+b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right )^2 d} \]
-1/2*(a*((a^2 + b^2)*ArcTan[Sinh[c + d*x]] + a*((I*a + b)*Log[I - Sinh[c + d*x]] + ((-I)*a + b)*Log[I + Sinh[c + d*x]] - 2*b*Log[a + b*Sinh[c + d*x] ])) + b*(a^2 + b^2)*Sech[c + d*x]^2 + a*(a^2 + b^2)*Sech[c + d*x]*Tanh[c + d*x])/((a^2 + b^2)^2*d)
Time = 0.41 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.21, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {3042, 25, 3316, 25, 27, 601, 25, 27, 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.
\(\displaystyle \int \frac {\tanh ^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\sin (i c+i d x)^2}{\cos (i c+i d x)^3 (a-i b \sin (i c+i d x))}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\sin (i c+i d x)^2}{\cos (i c+i d x)^3 (a-i b \sin (i c+i d x))}dx\) |
\(\Big \downarrow \) 3316 |
\(\displaystyle -\frac {b^3 \int -\frac {\sinh ^2(c+d x)}{(a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {b^3 \int \frac {\sinh ^2(c+d x)}{(a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b \int \frac {b^2 \sinh ^2(c+d x)}{(a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
\(\Big \downarrow \) 601 |
\(\displaystyle \frac {b \left (-\frac {\int -\frac {a b^2 (a-b \sinh (c+d x))}{\left (a^2+b^2\right ) (a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )}d(b \sinh (c+d x))}{2 b^2}-\frac {a b \sinh (c+d x)+b^2}{2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}\right )}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {b \left (\frac {\int \frac {a b^2 (a-b \sinh (c+d x))}{\left (a^2+b^2\right ) (a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )}d(b \sinh (c+d x))}{2 b^2}-\frac {a b \sinh (c+d x)+b^2}{2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}\right )}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b \left (\frac {a \int \frac {a-b \sinh (c+d x)}{(a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )}d(b \sinh (c+d x))}{2 \left (a^2+b^2\right )}-\frac {a b \sinh (c+d x)+b^2}{2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}\right )}{d}\) |
\(\Big \downarrow \) 657 |
\(\displaystyle \frac {b \left (\frac {a \int \left (\frac {2 a}{\left (a^2+b^2\right ) (a+b \sinh (c+d x))}+\frac {a^2-2 b \sinh (c+d x) a-b^2}{\left (a^2+b^2\right ) \left (\sinh ^2(c+d x) b^2+b^2\right )}\right )d(b \sinh (c+d x))}{2 \left (a^2+b^2\right )}-\frac {a b \sinh (c+d x)+b^2}{2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}\right )}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \left (\frac {a \left (\frac {\left (a^2-b^2\right ) \arctan (\sinh (c+d x))}{b \left (a^2+b^2\right )}-\frac {a \log \left (b^2 \sinh ^2(c+d x)+b^2\right )}{a^2+b^2}+\frac {2 a \log (a+b \sinh (c+d x))}{a^2+b^2}\right )}{2 \left (a^2+b^2\right )}-\frac {a b \sinh (c+d x)+b^2}{2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}\right )}{d}\) |
(b*((a*(((a^2 - b^2)*ArcTan[Sinh[c + d*x]])/(b*(a^2 + b^2)) + (2*a*Log[a + b*Sinh[c + d*x]])/(a^2 + b^2) - (a*Log[b^2 + b^2*Sinh[c + d*x]^2])/(a^2 + b^2)))/(2*(a^2 + b^2)) - (b^2 + a*b*Sinh[c + d*x])/(2*(a^2 + b^2)*(b^2 + b^2*Sinh[c + d*x]^2))))/d
3.4.89.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* (2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] && LtQ[p, -1] && ILtQ[n, 0] && 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[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Time = 2.86 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.73
method | result | size |
derivativedivides | \(\frac {\frac {\frac {2 \left (\left (\frac {1}{2} a^{3}+\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{2} b +b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {1}{2} a^{3}-\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+a \left (-a b \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {4 a^{2} b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{4 a^{4}+8 a^{2} b^{2}+4 b^{4}}}{d}\) | \(209\) |
default | \(\frac {\frac {\frac {2 \left (\left (\frac {1}{2} a^{3}+\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{2} b +b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {1}{2} a^{3}-\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+a \left (-a b \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {4 a^{2} b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{4 a^{4}+8 a^{2} b^{2}+4 b^{4}}}{d}\) | \(209\) |
risch | \(\frac {2 a^{2} b \,d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {2 a^{2} b d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {2 a^{2} b x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 a^{2} b c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {{\mathrm e}^{d x +c} \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}-a \right )}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}-\frac {i a^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {i a \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) b}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {i a \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) b}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {a^{2} b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(451\) |
1/d*(2/(a^4+2*a^2*b^2+b^4)*(((1/2*a^3+1/2*a*b^2)*tanh(1/2*d*x+1/2*c)^3+(a^ 2*b+b^3)*tanh(1/2*d*x+1/2*c)^2+(-1/2*a^3-1/2*a*b^2)*tanh(1/2*d*x+1/2*c))/( 1+tanh(1/2*d*x+1/2*c)^2)^2+1/2*a*(-a*b*ln(1+tanh(1/2*d*x+1/2*c)^2)+(a^2-b^ 2)*arctan(tanh(1/2*d*x+1/2*c))))+4*a^2*b/(4*a^4+8*a^2*b^2+4*b^4)*ln(tanh(1 /2*d*x+1/2*c)^2*a-2*b*tanh(1/2*d*x+1/2*c)-a))
Leaf count of result is larger than twice the leaf count of optimal. 917 vs. \(2 (117) = 234\).
Time = 0.28 (sec) , antiderivative size = 917, normalized size of antiderivative = 7.58 \[ \int \frac {\text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
-((a^3 + a*b^2)*cosh(d*x + c)^3 + (a^3 + a*b^2)*sinh(d*x + c)^3 + 2*(a^2*b + b^3)*cosh(d*x + c)^2 + (2*a^2*b + 2*b^3 + 3*(a^3 + a*b^2)*cosh(d*x + c) )*sinh(d*x + c)^2 - ((a^3 - a*b^2)*cosh(d*x + c)^4 + 4*(a^3 - a*b^2)*cosh( d*x + c)*sinh(d*x + c)^3 + (a^3 - a*b^2)*sinh(d*x + c)^4 + a^3 - a*b^2 + 2 *(a^3 - a*b^2)*cosh(d*x + c)^2 + 2*(a^3 - a*b^2 + 3*(a^3 - a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^3 - a*b^2)*cosh(d*x + c)^3 + (a^3 - a*b^2 )*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - (a ^3 + a*b^2)*cosh(d*x + c) - (a^2*b*cosh(d*x + c)^4 + 4*a^2*b*cosh(d*x + c) *sinh(d*x + c)^3 + a^2*b*sinh(d*x + c)^4 + 2*a^2*b*cosh(d*x + c)^2 + a^2*b + 2*(3*a^2*b*cosh(d*x + c)^2 + a^2*b)*sinh(d*x + c)^2 + 4*(a^2*b*cosh(d*x + c)^3 + a^2*b*cosh(d*x + c))*sinh(d*x + c))*log(2*(b*sinh(d*x + c) + a)/ (cosh(d*x + c) - sinh(d*x + c))) + (a^2*b*cosh(d*x + c)^4 + 4*a^2*b*cosh(d *x + c)*sinh(d*x + c)^3 + a^2*b*sinh(d*x + c)^4 + 2*a^2*b*cosh(d*x + c)^2 + a^2*b + 2*(3*a^2*b*cosh(d*x + c)^2 + a^2*b)*sinh(d*x + c)^2 + 4*(a^2*b*c osh(d*x + c)^3 + a^2*b*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/( cosh(d*x + c) - sinh(d*x + c))) - (a^3 + a*b^2 - 3*(a^3 + a*b^2)*cosh(d*x + c)^2 - 4*(a^2*b + b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d *x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*d*sinh(d*x + c)^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2 + 2*(3*(a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c...
\[ \int \frac {\text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\tanh ^{2}{\left (c + d x \right )} \operatorname {sech}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Time = 0.32 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.81 \[ \int \frac {\text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a^{2} b \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {a^{2} b \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {{\left (a^{3} - a b^{2}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {a e^{\left (-d x - c\right )} + 2 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{2} + b^{2} + 2 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} \]
a^2*b*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + 2*a^2*b^2 + b^4)*d) - a^2*b*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) - (a ^3 - a*b^2)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) - (a*e^(-d*x - c) + 2*b*e^(-2*d*x - 2*c) - a*e^(-3*d*x - 3*c))/((a^2 + b^2 + 2*(a^2 + b ^2)*e^(-2*d*x - 2*c) + (a^2 + b^2)*e^(-4*d*x - 4*c))*d)
Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (117) = 234\).
Time = 0.32 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.32 \[ \int \frac {\text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {4 \, a^{2} b^{2} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac {2 \, a^{2} b \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (a^{3} - a b^{2}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 2 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 4 \, b^{3}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}}}{4 \, d} \]
1/4*(4*a^2*b^2*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(a^4*b + 2*a ^2*b^3 + b^5) - 2*a^2*b*log((e^(d*x + c) - e^(-d*x - c))^2 + 4)/(a^4 + 2*a ^2*b^2 + b^4) + (pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(a ^3 - a*b^2)/(a^4 + 2*a^2*b^2 + b^4) + 2*(a^2*b*(e^(d*x + c) - e^(-d*x - c) )^2 - 2*a^3*(e^(d*x + c) - e^(-d*x - c)) - 2*a*b^2*(e^(d*x + c) - e^(-d*x - c)) - 4*b^3)/((a^4 + 2*a^2*b^2 + b^4)*((e^(d*x + c) - e^(-d*x - c))^2 + 4)))/d
Time = 2.74 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.80 \[ \int \frac {\text {sech}(c+d x) \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {2\,b}{d\,\left (a^2+b^2\right )}+\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2+b^2\right )}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {2\,\left (a^2\,b+b^3\right )}{d\,{\left (a^2+b^2\right )}^2}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3+a\,b^2\right )}{d\,{\left (a^2+b^2\right )}^2}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {a\,\ln \left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}{2\,\left (-1{}\mathrm {i}\,d\,a^2+2\,d\,a\,b+1{}\mathrm {i}\,d\,b^2\right )}+\frac {a^2\,b\,\ln \left (2\,a^7\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-a^2\,b^5-14\,a^4\,b^3-a^6\,b+a^6\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+2\,a^3\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+28\,a^5\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+a^2\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+14\,a^4\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )}{d\,a^4+2\,d\,a^2\,b^2+d\,b^4}-\frac {a\,\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,\left (-d\,a^2+2{}\mathrm {i}\,d\,a\,b+d\,b^2\right )} \]
((2*b)/(d*(a^2 + b^2)) + (2*a*exp(c + d*x))/(d*(a^2 + b^2)))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) - ((2*(a^2*b + b^3))/(d*(a^2 + b^2)^2) + (e xp(c + d*x)*(a*b^2 + a^3))/(d*(a^2 + b^2)^2))/(exp(2*c + 2*d*x) + 1) - (a* log(exp(c + d*x) + 1i)*1i)/(2*(b^2*d - a^2*d + a*b*d*2i)) - (a*log(exp(c + d*x)*1i + 1))/(2*(b^2*d*1i - a^2*d*1i + 2*a*b*d)) + (a^2*b*log(2*a^7*exp( d*x)*exp(c) - a^2*b^5 - 14*a^4*b^3 - a^6*b + a^6*b*exp(2*c)*exp(2*d*x) + 2 *a^3*b^4*exp(d*x)*exp(c) + 28*a^5*b^2*exp(d*x)*exp(c) + a^2*b^5*exp(2*c)*e xp(2*d*x) + 14*a^4*b^3*exp(2*c)*exp(2*d*x)))/(a^4*d + b^4*d + 2*a^2*b^2*d)