Integrand size = 17, antiderivative size = 74 \[ \int \frac {a+b \text {sech}^2(x)}{c+d \cosh (x)} \, dx=-\frac {b d \arctan (\sinh (x))}{c^2}+\frac {2 \left (a c^2+b d^2\right ) \text {arctanh}\left (\frac {\sqrt {c-d} \tanh \left (\frac {x}{2}\right )}{\sqrt {c+d}}\right )}{c^2 \sqrt {c-d} \sqrt {c+d}}+\frac {b \tanh (x)}{c} \]
-b*d*arctan(sinh(x))/c^2+2*(a*c^2+b*d^2)*arctanh((c-d)^(1/2)*tanh(1/2*x)/( c+d)^(1/2))/c^2/(c-d)^(1/2)/(c+d)^(1/2)+b*tanh(x)/c
Time = 0.63 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.72 \[ \int \frac {a+b \text {sech}^2(x)}{c+d \cosh (x)} \, dx=-\frac {2 \left (b+a \cosh ^2(x)\right ) \text {sech}(x) \left (2 \left (b d \sqrt {-c^2+d^2} \arctan \left (\tanh \left (\frac {x}{2}\right )\right )+\left (a c^2+b d^2\right ) \arctan \left (\frac {(c-d) \tanh \left (\frac {x}{2}\right )}{\sqrt {-c^2+d^2}}\right )\right ) \cosh (x)-b c \sqrt {-c^2+d^2} \sinh (x)\right )}{c^2 \sqrt {-c^2+d^2} (a+2 b+a \cosh (2 x))} \]
(-2*(b + a*Cosh[x]^2)*Sech[x]*(2*(b*d*Sqrt[-c^2 + d^2]*ArcTan[Tanh[x/2]] + (a*c^2 + b*d^2)*ArcTan[((c - d)*Tanh[x/2])/Sqrt[-c^2 + d^2]])*Cosh[x] - b *c*Sqrt[-c^2 + d^2]*Sinh[x]))/(c^2*Sqrt[-c^2 + d^2]*(a + 2*b + a*Cosh[2*x] ))
Time = 0.62 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {3042, 4722, 3042, 3535, 25, 3042, 3480, 3042, 3138, 221, 4257}
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.
| \(\displaystyle \int \frac {a+b \text {sech}^2(x)}{c+d \cosh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a+b \sec (i x)^2}{c+d \cos (i x)}dx\) |
| \(\Big \downarrow \) 4722 |
| \(\displaystyle \int \frac {\text {sech}^2(x) \left (a \cosh ^2(x)+b\right )}{c+d \cosh (x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {b+a \sin \left (\frac {\pi }{2}+i x\right )^2}{\sin \left (\frac {\pi }{2}+i x\right )^2 \left (c+d \sin \left (\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\int -\frac {(b d-a c \cosh (x)) \text {sech}(x)}{c+d \cosh (x)}dx}{c}+\frac {b \tanh (x)}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \tanh (x)}{c}-\frac {\int \frac {(b d-a c \cosh (x)) \text {sech}(x)}{c+d \cosh (x)}dx}{c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \tanh (x)}{c}-\frac {\int \frac {b d-a c \sin \left (i x+\frac {\pi }{2}\right )}{\sin \left (i x+\frac {\pi }{2}\right ) \left (c+d \sin \left (i x+\frac {\pi }{2}\right )\right )}dx}{c}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {b \tanh (x)}{c}-\frac {\frac {b d \int \text {sech}(x)dx}{c}-\frac {\left (a c^2+b d^2\right ) \int \frac {1}{c+d \cosh (x)}dx}{c}}{c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b \tanh (x)}{c}-\frac {\frac {b d \int \csc \left (i x+\frac {\pi }{2}\right )dx}{c}-\frac {\left (a c^2+b d^2\right ) \int \frac {1}{c+d \sin \left (i x+\frac {\pi }{2}\right )}dx}{c}}{c}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {b \tanh (x)}{c}-\frac {-\frac {2 \left (a c^2+b d^2\right ) \int \frac {1}{-\left ((c-d) \tanh ^2\left (\frac {x}{2}\right )\right )+c+d}d\tanh \left (\frac {x}{2}\right )}{c}+\frac {b d \int \csc \left (i x+\frac {\pi }{2}\right )dx}{c}}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {b \tanh (x)}{c}-\frac {-\frac {2 \left (a c^2+b d^2\right ) \text {arctanh}\left (\frac {\sqrt {c-d} \tanh \left (\frac {x}{2}\right )}{\sqrt {c+d}}\right )}{c \sqrt {c-d} \sqrt {c+d}}+\frac {b d \int \csc \left (i x+\frac {\pi }{2}\right )dx}{c}}{c}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {b \tanh (x)}{c}-\frac {\frac {b d \arctan (\sinh (x))}{c}-\frac {2 \left (a c^2+b d^2\right ) \text {arctanh}\left (\frac {\sqrt {c-d} \tanh \left (\frac {x}{2}\right )}{\sqrt {c+d}}\right )}{c \sqrt {c-d} \sqrt {c+d}}}{c}\) |
-(((b*d*ArcTan[Sinh[x]])/c - (2*(a*c^2 + b*d^2)*ArcTanh[(Sqrt[c - d]*Tanh[ x/2])/Sqrt[c + d]])/(c*Sqrt[c - d]*Sqrt[c + d]))/c) + (b*Tanh[x])/c
3.6.78.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S in[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin [e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d *(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(u_)*((A_) + (C_.)*sec[(a_.) + (b_.)*(x_)]^2), x_Symbol] :> Int[Activat eTrig[u]*((C + A*Cos[a + b*x]^2)/Cos[a + b*x]^2), x] /; FreeQ[{a, b, A, C}, x] && KnownSineIntegrandQ[u, x]
Time = 0.60 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.14
| method | result | size |
| default | \(-\frac {2 \left (-a \,c^{2}-b \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{c^{2} \sqrt {\left (c +d \right ) \left (c -d \right )}}-\frac {2 b \left (-\frac {c \tanh \left (\frac {x}{2}\right )}{1+\tanh \left (\frac {x}{2}\right )^{2}}+d \arctan \left (\tanh \left (\frac {x}{2}\right )\right )\right )}{c^{2}}\) | \(84\) |
| risch | \(-\frac {2 b}{c \left (1+{\mathrm e}^{2 x}\right )}+\frac {i b d \ln \left ({\mathrm e}^{x}-i\right )}{c^{2}}-\frac {i b d \ln \left ({\mathrm e}^{x}+i\right )}{c^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\sqrt {c^{2}-d^{2}}\, c -c^{2}+d^{2}}{\sqrt {c^{2}-d^{2}}\, d}\right ) a}{\sqrt {c^{2}-d^{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\sqrt {c^{2}-d^{2}}\, c -c^{2}+d^{2}}{\sqrt {c^{2}-d^{2}}\, d}\right ) b \,d^{2}}{\sqrt {c^{2}-d^{2}}\, c^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\sqrt {c^{2}-d^{2}}\, c +c^{2}-d^{2}}{\sqrt {c^{2}-d^{2}}\, d}\right ) a}{\sqrt {c^{2}-d^{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\sqrt {c^{2}-d^{2}}\, c +c^{2}-d^{2}}{\sqrt {c^{2}-d^{2}}\, d}\right ) b \,d^{2}}{\sqrt {c^{2}-d^{2}}\, c^{2}}\) | \(274\) |
-2*(-a*c^2-b*d^2)/c^2/((c+d)*(c-d))^(1/2)*arctanh((c-d)*tanh(1/2*x)/((c+d) *(c-d))^(1/2))-2*b/c^2*(-c*tanh(1/2*x)/(1+tanh(1/2*x)^2)+d*arctan(tanh(1/2 *x)))
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (64) = 128\).
Time = 0.48 (sec) , antiderivative size = 598, normalized size of antiderivative = 8.08 \[ \int \frac {a+b \text {sech}^2(x)}{c+d \cosh (x)} \, dx=\left [-\frac {2 \, b c^{3} - 2 \, b c d^{2} - {\left (a c^{2} + b d^{2} + {\left (a c^{2} + b d^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a c^{2} + b d^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a c^{2} + b d^{2}\right )} \sinh \left (x\right )^{2}\right )} \sqrt {c^{2} - d^{2}} \log \left (\frac {d^{2} \cosh \left (x\right )^{2} + d^{2} \sinh \left (x\right )^{2} + 2 \, c d \cosh \left (x\right ) + 2 \, c^{2} - d^{2} + 2 \, {\left (d^{2} \cosh \left (x\right ) + c d\right )} \sinh \left (x\right ) - 2 \, \sqrt {c^{2} - d^{2}} {\left (d \cosh \left (x\right ) + d \sinh \left (x\right ) + c\right )}}{d \cosh \left (x\right )^{2} + d \sinh \left (x\right )^{2} + 2 \, c \cosh \left (x\right ) + 2 \, {\left (d \cosh \left (x\right ) + c\right )} \sinh \left (x\right ) + d}\right ) + 2 \, {\left (b c^{2} d - b d^{3} + {\left (b c^{2} d - b d^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b c^{2} d - b d^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (b c^{2} d - b d^{3}\right )} \sinh \left (x\right )^{2}\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}{c^{4} - c^{2} d^{2} + {\left (c^{4} - c^{2} d^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (c^{4} - c^{2} d^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (c^{4} - c^{2} d^{2}\right )} \sinh \left (x\right )^{2}}, -\frac {2 \, {\left (b c^{3} - b c d^{2} + {\left (a c^{2} + b d^{2} + {\left (a c^{2} + b d^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a c^{2} + b d^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a c^{2} + b d^{2}\right )} \sinh \left (x\right )^{2}\right )} \sqrt {-c^{2} + d^{2}} \arctan \left (-\frac {\sqrt {-c^{2} + d^{2}} {\left (d \cosh \left (x\right ) + d \sinh \left (x\right ) + c\right )}}{c^{2} - d^{2}}\right ) + {\left (b c^{2} d - b d^{3} + {\left (b c^{2} d - b d^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (b c^{2} d - b d^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (b c^{2} d - b d^{3}\right )} \sinh \left (x\right )^{2}\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )\right )}}{c^{4} - c^{2} d^{2} + {\left (c^{4} - c^{2} d^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (c^{4} - c^{2} d^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (c^{4} - c^{2} d^{2}\right )} \sinh \left (x\right )^{2}}\right ] \]
[-(2*b*c^3 - 2*b*c*d^2 - (a*c^2 + b*d^2 + (a*c^2 + b*d^2)*cosh(x)^2 + 2*(a *c^2 + b*d^2)*cosh(x)*sinh(x) + (a*c^2 + b*d^2)*sinh(x)^2)*sqrt(c^2 - d^2) *log((d^2*cosh(x)^2 + d^2*sinh(x)^2 + 2*c*d*cosh(x) + 2*c^2 - d^2 + 2*(d^2 *cosh(x) + c*d)*sinh(x) - 2*sqrt(c^2 - d^2)*(d*cosh(x) + d*sinh(x) + c))/( d*cosh(x)^2 + d*sinh(x)^2 + 2*c*cosh(x) + 2*(d*cosh(x) + c)*sinh(x) + d)) + 2*(b*c^2*d - b*d^3 + (b*c^2*d - b*d^3)*cosh(x)^2 + 2*(b*c^2*d - b*d^3)*c osh(x)*sinh(x) + (b*c^2*d - b*d^3)*sinh(x)^2)*arctan(cosh(x) + sinh(x)))/( c^4 - c^2*d^2 + (c^4 - c^2*d^2)*cosh(x)^2 + 2*(c^4 - c^2*d^2)*cosh(x)*sinh (x) + (c^4 - c^2*d^2)*sinh(x)^2), -2*(b*c^3 - b*c*d^2 + (a*c^2 + b*d^2 + ( a*c^2 + b*d^2)*cosh(x)^2 + 2*(a*c^2 + b*d^2)*cosh(x)*sinh(x) + (a*c^2 + b* d^2)*sinh(x)^2)*sqrt(-c^2 + d^2)*arctan(-sqrt(-c^2 + d^2)*(d*cosh(x) + d*s inh(x) + c)/(c^2 - d^2)) + (b*c^2*d - b*d^3 + (b*c^2*d - b*d^3)*cosh(x)^2 + 2*(b*c^2*d - b*d^3)*cosh(x)*sinh(x) + (b*c^2*d - b*d^3)*sinh(x)^2)*arcta n(cosh(x) + sinh(x)))/(c^4 - c^2*d^2 + (c^4 - c^2*d^2)*cosh(x)^2 + 2*(c^4 - c^2*d^2)*cosh(x)*sinh(x) + (c^4 - c^2*d^2)*sinh(x)^2)]
\[ \int \frac {a+b \text {sech}^2(x)}{c+d \cosh (x)} \, dx=\int \frac {a + b \operatorname {sech}^{2}{\left (x \right )}}{c + d \cosh {\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {a+b \text {sech}^2(x)}{c+d \cosh (x)} \, dx=\text {Exception raised: ValueError} \]
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*c^2-4*d^2>0)', see `assume?` f or more de
Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \text {sech}^2(x)}{c+d \cosh (x)} \, dx=-\frac {2 \, b d \arctan \left (e^{x}\right )}{c^{2}} + \frac {2 \, {\left (a c^{2} + b d^{2}\right )} \arctan \left (\frac {d e^{x} + c}{\sqrt {-c^{2} + d^{2}}}\right )}{\sqrt {-c^{2} + d^{2}} c^{2}} - \frac {2 \, b}{c {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
-2*b*d*arctan(e^x)/c^2 + 2*(a*c^2 + b*d^2)*arctan((d*e^x + c)/sqrt(-c^2 + d^2))/(sqrt(-c^2 + d^2)*c^2) - 2*b/(c*(e^(2*x) + 1))
Time = 8.07 (sec) , antiderivative size = 704, normalized size of antiderivative = 9.51 \[ \int \frac {a+b \text {sech}^2(x)}{c+d \cosh (x)} \, dx=\frac {\ln \left (\frac {\sqrt {\left (c+d\right )\,\left (c-d\right )}\,\left (\frac {32\,\left (a^2\,c^4+2\,a\,b\,c^2\,d^2-4\,{\mathrm {e}}^x\,b^2\,c^3\,d-2\,b^2\,c^2\,d^2+3\,{\mathrm {e}}^x\,b^2\,c\,d^3+2\,b^2\,d^4\right )}{c^2\,d^4}-\frac {\sqrt {\left (c+d\right )\,\left (c-d\right )}\,\left (a\,c^2+b\,d^2\right )\,\left (\frac {32\,c\,\left (2\,b\,d^3+4\,a\,c^3\,{\mathrm {e}}^x+2\,a\,c^2\,d-a\,c\,d^2\,{\mathrm {e}}^x+3\,b\,c\,d^2\,{\mathrm {e}}^x\right )}{d^5}+\frac {32\,\sqrt {\left (c+d\right )\,\left (c-d\right )}\,\left (a\,c^2+b\,d^2\right )\,\left (4\,{\mathrm {e}}^x\,c^3+3\,c^2\,d-3\,{\mathrm {e}}^x\,c\,d^2-2\,d^3\right )}{d^5\,\left (c^2-d^2\right )}\right )}{c^2\,\left (c^2-d^2\right )}\right )\,\left (a\,c^2+b\,d^2\right )}{c^2\,\left (c^2-d^2\right )}-\frac {32\,b\,\left (a\,c^2+b\,d^2\right )\,\left (2\,b\,d+a\,c\,{\mathrm {e}}^x+4\,b\,c\,{\mathrm {e}}^x\right )}{c^3\,d^3}\right )\,\sqrt {\left (c+d\right )\,\left (c-d\right )}\,\left (a\,c^2+b\,d^2\right )}{c^4-c^2\,d^2}-\frac {2\,b}{c\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {\ln \left (-\frac {32\,b\,\left (a\,c^2+b\,d^2\right )\,\left (2\,b\,d+a\,c\,{\mathrm {e}}^x+4\,b\,c\,{\mathrm {e}}^x\right )}{c^3\,d^3}-\frac {\sqrt {\left (c+d\right )\,\left (c-d\right )}\,\left (\frac {32\,\left (a^2\,c^4+2\,a\,b\,c^2\,d^2-4\,{\mathrm {e}}^x\,b^2\,c^3\,d-2\,b^2\,c^2\,d^2+3\,{\mathrm {e}}^x\,b^2\,c\,d^3+2\,b^2\,d^4\right )}{c^2\,d^4}+\frac {\sqrt {\left (c+d\right )\,\left (c-d\right )}\,\left (a\,c^2+b\,d^2\right )\,\left (\frac {32\,c\,\left (2\,b\,d^3+4\,a\,c^3\,{\mathrm {e}}^x+2\,a\,c^2\,d-a\,c\,d^2\,{\mathrm {e}}^x+3\,b\,c\,d^2\,{\mathrm {e}}^x\right )}{d^5}-\frac {32\,\sqrt {\left (c+d\right )\,\left (c-d\right )}\,\left (a\,c^2+b\,d^2\right )\,\left (4\,{\mathrm {e}}^x\,c^3+3\,c^2\,d-3\,{\mathrm {e}}^x\,c\,d^2-2\,d^3\right )}{d^5\,\left (c^2-d^2\right )}\right )}{c^2\,\left (c^2-d^2\right )}\right )\,\left (a\,c^2+b\,d^2\right )}{c^2\,\left (c^2-d^2\right )}\right )\,\sqrt {\left (c+d\right )\,\left (c-d\right )}\,\left (a\,c^2+b\,d^2\right )}{c^4-c^2\,d^2}+\frac {b\,d\,\ln \left ({\mathrm {e}}^x-\mathrm {i}\right )\,1{}\mathrm {i}}{c^2}-\frac {b\,d\,\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^2} \]
(log((((c + d)*(c - d))^(1/2)*((32*(a^2*c^4 + 2*b^2*d^4 - 2*b^2*c^2*d^2 + 3*b^2*c*d^3*exp(x) - 4*b^2*c^3*d*exp(x) + 2*a*b*c^2*d^2))/(c^2*d^4) - (((c + d)*(c - d))^(1/2)*(a*c^2 + b*d^2)*((32*c*(2*b*d^3 + 4*a*c^3*exp(x) + 2* a*c^2*d - a*c*d^2*exp(x) + 3*b*c*d^2*exp(x)))/d^5 + (32*((c + d)*(c - d))^ (1/2)*(a*c^2 + b*d^2)*(3*c^2*d - 2*d^3 + 4*c^3*exp(x) - 3*c*d^2*exp(x)))/( d^5*(c^2 - d^2))))/(c^2*(c^2 - d^2)))*(a*c^2 + b*d^2))/(c^2*(c^2 - d^2)) - (32*b*(a*c^2 + b*d^2)*(2*b*d + a*c*exp(x) + 4*b*c*exp(x)))/(c^3*d^3))*((c + d)*(c - d))^(1/2)*(a*c^2 + b*d^2))/(c^4 - c^2*d^2) - (2*b)/(c*(exp(2*x) + 1)) - (log(- (32*b*(a*c^2 + b*d^2)*(2*b*d + a*c*exp(x) + 4*b*c*exp(x))) /(c^3*d^3) - (((c + d)*(c - d))^(1/2)*((32*(a^2*c^4 + 2*b^2*d^4 - 2*b^2*c^ 2*d^2 + 3*b^2*c*d^3*exp(x) - 4*b^2*c^3*d*exp(x) + 2*a*b*c^2*d^2))/(c^2*d^4 ) + (((c + d)*(c - d))^(1/2)*(a*c^2 + b*d^2)*((32*c*(2*b*d^3 + 4*a*c^3*exp (x) + 2*a*c^2*d - a*c*d^2*exp(x) + 3*b*c*d^2*exp(x)))/d^5 - (32*((c + d)*( c - d))^(1/2)*(a*c^2 + b*d^2)*(3*c^2*d - 2*d^3 + 4*c^3*exp(x) - 3*c*d^2*ex p(x)))/(d^5*(c^2 - d^2))))/(c^2*(c^2 - d^2)))*(a*c^2 + b*d^2))/(c^2*(c^2 - d^2)))*((c + d)*(c - d))^(1/2)*(a*c^2 + b*d^2))/(c^4 - c^2*d^2) + (b*d*lo g(exp(x) - 1i)*1i)/c^2 - (b*d*log(exp(x) + 1i)*1i)/c^2