Integrand size = 8, antiderivative size = 176 \[ \int \frac {1}{1+\cosh ^4(x)} \, dx=-\frac {\arctan \left (\frac {\sqrt {1+\sqrt {2}}-2 \coth (x)}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\arctan \left (\frac {\sqrt {1+\sqrt {2}}+2 \coth (x)}{\sqrt {-1+\sqrt {2}}}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \coth (x)+2 \coth ^2(x)\right )+\frac {1}{8} \sqrt {1+\sqrt {2}} \log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \coth (x)+\sqrt {2} \coth ^2(x)\right ) \]
-1/4*arctan((-2*coth(x)+(1+2^(1/2))^(1/2))/(2^(1/2)-1)^(1/2))/(1+2^(1/2))^ (1/2)+1/4*arctan((2*coth(x)+(1+2^(1/2))^(1/2))/(2^(1/2)-1)^(1/2))/(1+2^(1/ 2))^(1/2)-1/8*ln(2*coth(x)^2+2^(1/2)-2*coth(x)*(1+2^(1/2))^(1/2))*(1+2^(1/ 2))^(1/2)+1/8*ln(1+coth(x)^2*2^(1/2)+coth(x)*(2+2*2^(1/2))^(1/2))*(1+2^(1/ 2))^(1/2)
Result contains complex when optimal does not.
Time = 4.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.26 \[ \int \frac {1}{1+\cosh ^4(x)} \, dx=\frac {\text {arctanh}\left (\frac {\tanh (x)}{\sqrt {1-i}}\right )}{2 \sqrt {1-i}}+\frac {\text {arctanh}\left (\frac {\tanh (x)}{\sqrt {1+i}}\right )}{2 \sqrt {1+i}} \]
Time = 0.49 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3688, 1483, 1142, 27, 1083, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{\cosh ^4(x)+1} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{1+\sin \left (\frac {\pi }{2}+i x\right )^4}dx\) |
| \(\Big \downarrow \) 3688 |
| \(\displaystyle \int \frac {1-\coth ^2(x)}{2 \coth ^4(x)-2 \coth ^2(x)+1}d\coth (x)\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\int \frac {2 \sqrt {1+\sqrt {2}}-\left (2+\sqrt {2}\right ) \coth (x)}{2 \coth ^2(x)-2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\int \frac {\left (2+\sqrt {2}\right ) \coth (x)+2 \sqrt {1+\sqrt {2}}}{2 \coth ^2(x)+2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \int \frac {1}{2 \coth ^2(x)-2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)-\frac {1}{4} \left (2+\sqrt {2}\right ) \int -\frac {2 \left (\sqrt {1+\sqrt {2}}-2 \coth (x)\right )}{2 \coth ^2(x)-2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \int \frac {1}{2 \coth ^2(x)+2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)+\frac {1}{4} \left (2+\sqrt {2}\right ) \int \frac {2 \left (2 \coth (x)+\sqrt {1+\sqrt {2}}\right )}{2 \coth ^2(x)+2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \int \frac {1}{2 \coth ^2(x)-2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)+\frac {1}{2} \left (2+\sqrt {2}\right ) \int \frac {\sqrt {1+\sqrt {2}}-2 \coth (x)}{2 \coth ^2(x)-2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \int \frac {1}{2 \coth ^2(x)+2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)+\frac {1}{2} \left (2+\sqrt {2}\right ) \int \frac {2 \coth (x)+\sqrt {1+\sqrt {2}}}{2 \coth ^2(x)+2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {1}{2} \left (2+\sqrt {2}\right ) \int \frac {\sqrt {1+\sqrt {2}}-2 \coth (x)}{2 \coth ^2(x)-2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)-\sqrt {2 \left (\sqrt {2}-1\right )} \int \frac {1}{4 \left (1-\sqrt {2}\right )-\left (4 \coth (x)-2 \sqrt {1+\sqrt {2}}\right )^2}d\left (4 \coth (x)-2 \sqrt {1+\sqrt {2}}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\frac {1}{2} \left (2+\sqrt {2}\right ) \int \frac {2 \coth (x)+\sqrt {1+\sqrt {2}}}{2 \coth ^2(x)+2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)-\sqrt {2 \left (\sqrt {2}-1\right )} \int \frac {1}{4 \left (1-\sqrt {2}\right )-\left (4 \coth (x)+2 \sqrt {1+\sqrt {2}}\right )^2}d\left (4 \coth (x)+2 \sqrt {1+\sqrt {2}}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{2} \left (2+\sqrt {2}\right ) \int \frac {\sqrt {1+\sqrt {2}}-2 \coth (x)}{2 \coth ^2(x)-2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)+\frac {\arctan \left (\frac {4 \coth (x)-2 \sqrt {1+\sqrt {2}}}{2 \sqrt {\sqrt {2}-1}}\right )}{\sqrt {2}}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\frac {1}{2} \left (2+\sqrt {2}\right ) \int \frac {2 \coth (x)+\sqrt {1+\sqrt {2}}}{2 \coth ^2(x)+2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}}d\coth (x)+\frac {\arctan \left (\frac {4 \coth (x)+2 \sqrt {1+\sqrt {2}}}{2 \sqrt {\sqrt {2}-1}}\right )}{\sqrt {2}}}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {4 \coth (x)-2 \sqrt {1+\sqrt {2}}}{2 \sqrt {\sqrt {2}-1}}\right )}{\sqrt {2}}-\frac {1}{4} \left (2+\sqrt {2}\right ) \log \left (2 \coth ^2(x)-2 \sqrt {1+\sqrt {2}} \coth (x)+\sqrt {2}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\frac {\arctan \left (\frac {4 \coth (x)+2 \sqrt {1+\sqrt {2}}}{2 \sqrt {\sqrt {2}-1}}\right )}{\sqrt {2}}+\frac {1}{4} \left (2+\sqrt {2}\right ) \log \left (\sqrt {2} \coth ^2(x)+\sqrt {2 \left (1+\sqrt {2}\right )} \coth (x)+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}\) |
(ArcTan[(-2*Sqrt[1 + Sqrt[2]] + 4*Coth[x])/(2*Sqrt[-1 + Sqrt[2]])]/Sqrt[2] - ((2 + Sqrt[2])*Log[Sqrt[2] - 2*Sqrt[1 + Sqrt[2]]*Coth[x] + 2*Coth[x]^2] )/4)/(2*Sqrt[2*(1 + Sqrt[2])]) + (ArcTan[(2*Sqrt[1 + Sqrt[2]] + 4*Coth[x]) /(2*Sqrt[-1 + Sqrt[2]])]/Sqrt[2] + ((2 + Sqrt[2])*Log[1 + Sqrt[2*(1 + Sqrt [2])]*Coth[x] + Sqrt[2]*Coth[x]^2])/4)/(2*Sqrt[2*(1 + Sqrt[2])])
3.1.62.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + 2*a*ff^2*x^2 + ( a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x]] / ; FreeQ[{a, b, e, f}, x] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.20
| method | result | size |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (512 \textit {\_Z}^{4}-32 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (-256 \textit {\_R}^{3}+64 \textit {\_R}^{2}+{\mathrm e}^{2 x}-1\right )\) | \(36\) |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{4}-2 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (2 \tanh \left (\frac {x}{2}\right ) \textit {\_R} +\tanh \left (\frac {x}{2}\right )^{2}+1\right )\right )}{4}\) | \(37\) |
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.76 \[ \int \frac {1}{1+\cosh ^4(x)} \, dx=-\frac {1}{8} \, \sqrt {2} \sqrt {i + 1} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \left (i + 1\right ) \, \sqrt {2} \sqrt {i + 1} + 2 i + 1\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {i + 1} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \left (i + 1\right ) \, \sqrt {2} \sqrt {i + 1} + 2 i + 1\right ) - \frac {1}{8} \, \sqrt {2} \sqrt {-i + 1} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \left (i - 1\right ) \, \sqrt {2} \sqrt {-i + 1} - 2 i + 1\right ) + \frac {1}{8} \, \sqrt {2} \sqrt {-i + 1} \log \left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + \left (i - 1\right ) \, \sqrt {2} \sqrt {-i + 1} - 2 i + 1\right ) \]
-1/8*sqrt(2)*sqrt(I + 1)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + ( I + 1)*sqrt(2)*sqrt(I + 1) + 2*I + 1) + 1/8*sqrt(2)*sqrt(I + 1)*log(cosh(x )^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - (I + 1)*sqrt(2)*sqrt(I + 1) + 2*I + 1) - 1/8*sqrt(2)*sqrt(-I + 1)*log(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^ 2 - (I - 1)*sqrt(2)*sqrt(-I + 1) - 2*I + 1) + 1/8*sqrt(2)*sqrt(-I + 1)*log (cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + (I - 1)*sqrt(2)*sqrt(-I + 1) - 2*I + 1)
Timed out. \[ \int \frac {1}{1+\cosh ^4(x)} \, dx=\text {Timed out} \]
\[ \int \frac {1}{1+\cosh ^4(x)} \, dx=\int { \frac {1}{\cosh \left (x\right )^{4} + 1} \,d x } \]
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.60 \[ \int \frac {1}{1+\cosh ^4(x)} \, dx=-\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2 \, \sqrt {2} - 2} {\left (-\frac {i}{\sqrt {2} - 1} + 1\right )} \log \left (\left (20 i + 10\right ) \, \sqrt {2} e^{\left (2 \, x\right )} + 10 \, \sqrt {2} \sqrt {10 \, \sqrt {2} + 14} + 50 \, \sqrt {2} - \left (2 i - 14\right ) \, \sqrt {10 \, \sqrt {2} + 14} + \left (28 i + 14\right ) \, e^{\left (2 \, x\right )} + 70\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2 \, \sqrt {2} - 2} {\left (-\frac {i}{\sqrt {2} - 1} + 1\right )} \log \left (\left (20 i + 10\right ) \, \sqrt {2} e^{\left (2 \, x\right )} - 10 \, \sqrt {2} \sqrt {10 \, \sqrt {2} + 14} + 50 \, \sqrt {2} + \left (2 i - 14\right ) \, \sqrt {10 \, \sqrt {2} + 14} + \left (28 i + 14\right ) \, e^{\left (2 \, x\right )} + 70\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2 \, \sqrt {2} + 2} {\left (-\frac {i}{\sqrt {2} + 1} + 1\right )} \log \left (2 \, \sqrt {2} e^{\left (2 \, x\right )} + 2 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} + \left (4 i + 2\right ) \, \sqrt {2} + \left (2 i - 2\right ) \, \sqrt {2 \, \sqrt {2} - 2} - 2 \, e^{\left (2 \, x\right )} - 4 i - 2\right ) + \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2 \, \sqrt {2} + 2} {\left (-\frac {i}{\sqrt {2} + 1} + 1\right )} \log \left (2 \, \sqrt {2} e^{\left (2 \, x\right )} - 2 \, \sqrt {2} \sqrt {2 \, \sqrt {2} - 2} + \left (4 i + 2\right ) \, \sqrt {2} - \left (2 i - 2\right ) \, \sqrt {2 \, \sqrt {2} - 2} - 2 \, e^{\left (2 \, x\right )} - 4 i - 2\right ) \]
-(1/16*I + 1/16)*sqrt(2*sqrt(2) - 2)*(-I/(sqrt(2) - 1) + 1)*log((20*I + 10 )*sqrt(2)*e^(2*x) + 10*sqrt(2)*sqrt(10*sqrt(2) + 14) + 50*sqrt(2) - (2*I - 14)*sqrt(10*sqrt(2) + 14) + (28*I + 14)*e^(2*x) + 70) + (1/16*I + 1/16)*s qrt(2*sqrt(2) - 2)*(-I/(sqrt(2) - 1) + 1)*log((20*I + 10)*sqrt(2)*e^(2*x) - 10*sqrt(2)*sqrt(10*sqrt(2) + 14) + 50*sqrt(2) + (2*I - 14)*sqrt(10*sqrt( 2) + 14) + (28*I + 14)*e^(2*x) + 70) - (1/16*I + 1/16)*sqrt(2*sqrt(2) + 2) *(-I/(sqrt(2) + 1) + 1)*log(2*sqrt(2)*e^(2*x) + 2*sqrt(2)*sqrt(2*sqrt(2) - 2) + (4*I + 2)*sqrt(2) + (2*I - 2)*sqrt(2*sqrt(2) - 2) - 2*e^(2*x) - 4*I - 2) + (1/16*I + 1/16)*sqrt(2*sqrt(2) + 2)*(-I/(sqrt(2) + 1) + 1)*log(2*sq rt(2)*e^(2*x) - 2*sqrt(2)*sqrt(2*sqrt(2) - 2) + (4*I + 2)*sqrt(2) - (2*I - 2)*sqrt(2*sqrt(2) - 2) - 2*e^(2*x) - 4*I - 2)
Time = 1.11 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.16 \[ \int \frac {1}{1+\cosh ^4(x)} \, dx=\frac {\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (436273152+91291648{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\left (-9830400+56623104{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (218890240+149422080{}\mathrm {i}\right )+21168128+94306304{}\mathrm {i}\right )}{8}-\frac {\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (436273152+91291648{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,\left (9830400-56623104{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1-\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (-218890240-149422080{}\mathrm {i}\right )+21168128+94306304{}\mathrm {i}\right )}{8}+\frac {\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (436273152-91291648{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\left (-9830400-56623104{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (218890240-149422080{}\mathrm {i}\right )+21168128-94306304{}\mathrm {i}\right )}{8}-\frac {\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\ln \left ({\mathrm {e}}^{2\,x}\,\left (436273152-91291648{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,\left (9830400+56623104{}\mathrm {i}\right )+\sqrt {2}\,\sqrt {1+1{}\mathrm {i}}\,{\mathrm {e}}^{2\,x}\,\left (-218890240+149422080{}\mathrm {i}\right )+21168128-94306304{}\mathrm {i}\right )}{8} \]
(2^(1/2)*(1 - 1i)^(1/2)*log(exp(2*x)*(436273152 + 91291648i) - 2^(1/2)*(1 - 1i)^(1/2)*(9830400 - 56623104i) + 2^(1/2)*(1 - 1i)^(1/2)*exp(2*x)*(21889 0240 + 149422080i) + (21168128 + 94306304i)))/8 - (2^(1/2)*(1 - 1i)^(1/2)* log(exp(2*x)*(436273152 + 91291648i) + 2^(1/2)*(1 - 1i)^(1/2)*(9830400 - 5 6623104i) - 2^(1/2)*(1 - 1i)^(1/2)*exp(2*x)*(218890240 + 149422080i) + (21 168128 + 94306304i)))/8 + (2^(1/2)*(1 + 1i)^(1/2)*log(exp(2*x)*(436273152 - 91291648i) - 2^(1/2)*(1 + 1i)^(1/2)*(9830400 + 56623104i) + 2^(1/2)*(1 + 1i)^(1/2)*exp(2*x)*(218890240 - 149422080i) + (21168128 - 94306304i)))/8 - (2^(1/2)*(1 + 1i)^(1/2)*log(exp(2*x)*(436273152 - 91291648i) + 2^(1/2)*( 1 + 1i)^(1/2)*(9830400 + 56623104i) - 2^(1/2)*(1 + 1i)^(1/2)*exp(2*x)*(218 890240 - 149422080i) + (21168128 - 94306304i)))/8