Integrand size = 15, antiderivative size = 46 \[ \int \cos (a+b x) \cot ^2(c+b x) \, dx=-\frac {\cos (a-c) \csc (c+b x)}{b}+\frac {\text {arctanh}(\cos (c+b x)) \sin (a-c)}{b}-\frac {\sin (a+b x)}{b} \]
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.43 \[ \int \cos (a+b x) \cot ^2(c+b x) \, dx=-\frac {\cos (a-c) \csc (c+b x)}{b}-\frac {\cos (b x) \sin (a)}{b}+\frac {2 i \arctan \left (\frac {(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac {b x}{2}\right )-\sin (c) \sin \left (\frac {b x}{2}\right )\right )}{i \cos (c) \cos \left (\frac {b x}{2}\right )+\cos \left (\frac {b x}{2}\right ) \sin (c)}\right ) \sin (a-c)}{b}-\frac {\cos (a) \sin (b x)}{b} \]
-((Cos[a - c]*Csc[c + b*x])/b) - (Cos[b*x]*Sin[a])/b + ((2*I)*ArcTan[((Cos [c] - I*Sin[c])*(Cos[c]*Cos[(b*x)/2] - Sin[c]*Sin[(b*x)/2]))/(I*Cos[c]*Cos [(b*x)/2] + Cos[(b*x)/2]*Sin[c])]*Sin[a - c])/b - (Cos[a]*Sin[b*x])/b
Time = 0.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5088, 3042, 25, 3086, 24, 5089, 3042, 3117, 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.
| \(\displaystyle \int \cos (a+b x) \cot ^2(b x+c) \, dx\) |
| \(\Big \downarrow \) 5088 |
| \(\displaystyle \cos (a-c) \int \cot (c+b x) \csc (c+b x)dx-\int \cot (c+b x) \sin (a+b x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \cos (a-c) \int -\sec \left (c+b x-\frac {\pi }{2}\right ) \tan \left (c+b x-\frac {\pi }{2}\right )dx-\int \cot (c+b x) \sin (a+b x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \cot (c+b x) \sin (a+b x)dx-\cos (a-c) \int \sec \left (\frac {1}{2} (2 c-\pi )+b x\right ) \tan \left (\frac {1}{2} (2 c-\pi )+b x\right )dx\) |
| \(\Big \downarrow \) 3086 |
| \(\displaystyle -\frac {\cos (a-c) \int 1d\csc (c+b x)}{b}-\int \cot (c+b x) \sin (a+b x)dx\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\int \cot (c+b x) \sin (a+b x)dx-\frac {\cos (a-c) \csc (b x+c)}{b}\) |
| \(\Big \downarrow \) 5089 |
| \(\displaystyle -\sin (a-c) \int \csc (c+b x)dx-\int \cos (a+b x)dx-\frac {\cos (a-c) \csc (b x+c)}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\sin (a-c) \int \csc (c+b x)dx-\int \sin \left (a+b x+\frac {\pi }{2}\right )dx-\frac {\cos (a-c) \csc (b x+c)}{b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -\sin (a-c) \int \csc (c+b x)dx-\frac {\cos (a-c) \csc (b x+c)}{b}-\frac {\sin (a+b x)}{b}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\sin (a-c) \text {arctanh}(\cos (b x+c))}{b}-\frac {\cos (a-c) \csc (b x+c)}{b}-\frac {\sin (a+b x)}{b}\) |
3.3.51.3.1 Defintions of rubi rules used
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[Cos[v_]*Cot[w_]^(n_.), x_Symbol] :> -Int[Sin[v]*Cot[w]^(n - 1), x] + Si mp[Cos[v - w] Int[Csc[w]*Cot[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
Int[Cot[w_]^(n_.)*Sin[v_], x_Symbol] :> Int[Cos[v]*Cot[w]^(n - 1), x] + Sim p[Sin[v - w] Int[Csc[w]*Cot[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 3.15
| method | result | size |
| risch | \(\frac {i {\mathrm e}^{i \left (x b +a \right )}}{2 b}-\frac {i {\mathrm e}^{-i \left (x b +a \right )}}{2 b}+\frac {i \left ({\mathrm e}^{i \left (x b +3 a \right )}+{\mathrm e}^{i \left (x b +a +2 c \right )}\right )}{b \left (-{\mathrm e}^{2 i \left (x b +a +c \right )}+{\mathrm e}^{2 i a}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \sin \left (a -c \right )}{b}\) | \(145\) |
1/2*I*exp(I*(b*x+a))/b-1/2*I/b*exp(-I*(b*x+a))+I/b/(-exp(2*I*(b*x+a+c))+ex p(2*I*a))*(exp(I*(b*x+3*a))+exp(I*(b*x+a+2*c)))-ln(exp(I*(b*x+a))-exp(I*(a -c)))/b*sin(a-c)+ln(exp(I*(b*x+a))+exp(I*(a-c)))/b*sin(a-c)
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (46) = 92\).
Time = 0.28 (sec) , antiderivative size = 316, normalized size of antiderivative = 6.87 \[ \int \cos (a+b x) \cot ^2(c+b x) \, dx=\frac {4 \, {\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right )^{2} - 4 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + \frac {\sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - {\left (\cos \left (-2 \, a + 2 \, c\right )^{2} - 1\right )} \cos \left (b x + a\right )\right )} \log \left (-\frac {2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) + 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) - 1}\right )}{\sqrt {\cos \left (-2 \, a + 2 \, c\right ) + 1}} - 8 \, \cos \left (-2 \, a + 2 \, c\right ) - 8}{4 \, {\left (b \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) + {\left (b \cos \left (-2 \, a + 2 \, c\right ) + b\right )} \sin \left (b x + a\right )\right )}} \]
1/4*(4*(cos(-2*a + 2*c) + 1)*cos(b*x + a)^2 - 4*cos(b*x + a)*sin(b*x + a)* sin(-2*a + 2*c) + sqrt(2)*((cos(-2*a + 2*c) + 1)*sin(b*x + a)*sin(-2*a + 2 *c) - (cos(-2*a + 2*c)^2 - 1)*cos(b*x + a))*log(-(2*cos(b*x + a)^2*cos(-2* a + 2*c) - 2*cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - 2*sqrt(2)*((cos(- 2*a + 2*c) + 1)*cos(b*x + a) - sin(b*x + a)*sin(-2*a + 2*c))/sqrt(cos(-2*a + 2*c) + 1) - cos(-2*a + 2*c) + 3)/(2*cos(b*x + a)^2*cos(-2*a + 2*c) - 2* cos(b*x + a)*sin(b*x + a)*sin(-2*a + 2*c) - cos(-2*a + 2*c) - 1))/sqrt(cos (-2*a + 2*c) + 1) - 8*cos(-2*a + 2*c) - 8)/(b*cos(b*x + a)*sin(-2*a + 2*c) + (b*cos(-2*a + 2*c) + b)*sin(b*x + a))
\[ \int \cos (a+b x) \cot ^2(c+b x) \, dx=\int \cos {\left (a + b x \right )} \cot ^{2}{\left (b x + c \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (46) = 92\).
Time = 0.25 (sec) , antiderivative size = 613, normalized size of antiderivative = 13.33 \[ \int \cos (a+b x) \cot ^2(c+b x) \, dx=\frac {{\left (\sin \left (3 \, b x + a + 2 \, c\right ) - \sin \left (b x + a\right )\right )} \cos \left (4 \, b x + 2 \, a + 2 \, c\right ) + 3 \, {\left (\sin \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, c\right )\right )} \cos \left (3 \, b x + a + 2 \, c\right ) - {\left (\cos \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) \sin \left (-a + c\right ) + \cos \left (b x + a\right )^{2} \sin \left (-a + c\right ) + \sin \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) \sin \left (-a + c\right ) + \sin \left (b x + a\right )^{2} \sin \left (-a + c\right )\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) + {\left (\cos \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) \sin \left (-a + c\right ) + \cos \left (b x + a\right )^{2} \sin \left (-a + c\right ) + \sin \left (3 \, b x + a + 2 \, c\right )^{2} \sin \left (-a + c\right ) - 2 \, \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) \sin \left (-a + c\right ) + \sin \left (b x + a\right )^{2} \sin \left (-a + c\right )\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (c\right ) + \cos \left (c\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right ) - {\left (\cos \left (3 \, b x + a + 2 \, c\right ) - \cos \left (b x + a\right )\right )} \sin \left (4 \, b x + 2 \, a + 2 \, c\right ) - {\left (3 \, \cos \left (2 \, b x + 2 \, a\right ) + 3 \, \cos \left (2 \, b x + 2 \, c\right ) - 1\right )} \sin \left (3 \, b x + a + 2 \, c\right ) - 3 \, \cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 3 \, \cos \left (b x + a\right ) \sin \left (2 \, b x + 2 \, c\right ) + 3 \, \cos \left (2 \, b x + 2 \, a\right ) \sin \left (b x + a\right ) + 3 \, \cos \left (2 \, b x + 2 \, c\right ) \sin \left (b x + a\right ) - \sin \left (b x + a\right )}{2 \, {\left (b \cos \left (3 \, b x + a + 2 \, c\right )^{2} - 2 \, b \cos \left (3 \, b x + a + 2 \, c\right ) \cos \left (b x + a\right ) + b \cos \left (b x + a\right )^{2} + b \sin \left (3 \, b x + a + 2 \, c\right )^{2} - 2 \, b \sin \left (3 \, b x + a + 2 \, c\right ) \sin \left (b x + a\right ) + b \sin \left (b x + a\right )^{2}\right )}} \]
1/2*((sin(3*b*x + a + 2*c) - sin(b*x + a))*cos(4*b*x + 2*a + 2*c) + 3*(sin (2*b*x + 2*a) + sin(2*b*x + 2*c))*cos(3*b*x + a + 2*c) - (cos(3*b*x + a + 2*c)^2*sin(-a + c) - 2*cos(3*b*x + a + 2*c)*cos(b*x + a)*sin(-a + c) + cos (b*x + a)^2*sin(-a + c) + sin(3*b*x + a + 2*c)^2*sin(-a + c) - 2*sin(3*b*x + a + 2*c)*sin(b*x + a)*sin(-a + c) + sin(b*x + a)^2*sin(-a + c))*log(cos (b*x)^2 + 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(c) + sin(c)^2) + (cos(3*b*x + a + 2*c)^2*sin(-a + c) - 2*cos(3*b*x + a + 2*c)*c os(b*x + a)*sin(-a + c) + cos(b*x + a)^2*sin(-a + c) + sin(3*b*x + a + 2*c )^2*sin(-a + c) - 2*sin(3*b*x + a + 2*c)*sin(b*x + a)*sin(-a + c) + sin(b* x + a)^2*sin(-a + c))*log(cos(b*x)^2 - 2*cos(b*x)*cos(c) + cos(c)^2 + sin( b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2) - (cos(3*b*x + a + 2*c) - cos(b*x + a))*sin(4*b*x + 2*a + 2*c) - (3*cos(2*b*x + 2*a) + 3*cos(2*b*x + 2*c) - 1 )*sin(3*b*x + a + 2*c) - 3*cos(b*x + a)*sin(2*b*x + 2*a) - 3*cos(b*x + a)* sin(2*b*x + 2*c) + 3*cos(2*b*x + 2*a)*sin(b*x + a) + 3*cos(2*b*x + 2*c)*si n(b*x + a) - sin(b*x + a))/(b*cos(3*b*x + a + 2*c)^2 - 2*b*cos(3*b*x + a + 2*c)*cos(b*x + a) + b*cos(b*x + a)^2 + b*sin(3*b*x + a + 2*c)^2 - 2*b*sin (3*b*x + a + 2*c)*sin(b*x + a) + b*sin(b*x + a)^2)
Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (46) = 92\).
Time = 0.33 (sec) , antiderivative size = 627, normalized size of antiderivative = 13.63 \[ \int \cos (a+b x) \cot ^2(c+b x) \, dx=\text {Too large to display} \]
1/2*(4*(tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*a)*t an(1/2*c) - tan(1/2*c)^2)*log(abs(tan(1/2*b*x)*tan(1/2*c) - 1))/(tan(1/2*a )^2*tan(1/2*c)^3 + tan(1/2*a)^2*tan(1/2*c) + tan(1/2*c)^3 + tan(1/2*c)) - 4*(tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*a) - tan(1/ 2*c))*log(abs(tan(1/2*b*x) + tan(1/2*c)))/(tan(1/2*a)^2*tan(1/2*c)^2 + tan (1/2*a)^2 + tan(1/2*c)^2 + 1) - (tan(1/2*b*x)^3*tan(1/2*a)^2*tan(1/2*c)^4 - 6*tan(1/2*b*x)^3*tan(1/2*a)^2*tan(1/2*c)^2 + 4*tan(1/2*b*x)^3*tan(1/2*a) *tan(1/2*c)^3 - 6*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*c)^3 - tan(1/2*b*x)^ 3*tan(1/2*c)^4 + tan(1/2*b*x)*tan(1/2*a)^2*tan(1/2*c)^4 + tan(1/2*b*x)^3*t an(1/2*a)^2 - 4*tan(1/2*b*x)^3*tan(1/2*a)*tan(1/2*c) + 6*tan(1/2*b*x)^2*ta n(1/2*a)^2*tan(1/2*c) + 6*tan(1/2*b*x)^3*tan(1/2*c)^2 + 2*tan(1/2*b*x)*tan (1/2*a)^2*tan(1/2*c)^2 + 6*tan(1/2*b*x)^2*tan(1/2*c)^3 + 12*tan(1/2*b*x)*t an(1/2*a)*tan(1/2*c)^3 - 2*tan(1/2*a)^2*tan(1/2*c)^3 - tan(1/2*b*x)*tan(1/ 2*c)^4 - tan(1/2*b*x)^3 + tan(1/2*b*x)*tan(1/2*a)^2 - 6*tan(1/2*b*x)^2*tan (1/2*c) - 12*tan(1/2*b*x)*tan(1/2*a)*tan(1/2*c) + 2*tan(1/2*a)^2*tan(1/2*c ) - 2*tan(1/2*b*x)*tan(1/2*c)^2 - 16*tan(1/2*a)*tan(1/2*c)^2 + 2*tan(1/2*c )^3 - tan(1/2*b*x) - 2*tan(1/2*c))/((tan(1/2*b*x)^4*tan(1/2*c) + tan(1/2*b *x)^3*tan(1/2*c)^2 - tan(1/2*b*x)^3 + tan(1/2*b*x)*tan(1/2*c)^2 - tan(1/2* b*x) - tan(1/2*c))*(tan(1/2*a)^2*tan(1/2*c) + tan(1/2*c))))/b
Time = 27.02 (sec) , antiderivative size = 289, normalized size of antiderivative = 6.28 \[ \int \cos (a+b x) \cot ^2(c+b x) \, dx=-\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,b}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2\,b}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}+1\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}\,1{}\mathrm {i}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}\right )}-\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}}+\frac {\ln \left ({\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )+\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}-1\right )\,1{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{-c\,2{}\mathrm {i}}}}\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}-1\right )}{2\,b\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}-c\,2{}\mathrm {i}}}} \]
(exp(a*1i + b*x*1i)*1i)/(2*b) - (exp(- a*1i - b*x*1i)*1i)/(2*b) - (exp(a*1 i + b*x*1i)*(exp(a*2i - c*2i) + 1))/(b*(exp(a*2i - c*2i)*1i - exp(a*2i + b *x*2i)*1i)) - (log(exp(a*1i)*exp(b*x*1i)*(exp(a*2i)*exp(-c*2i) - 1) - (exp (a*2i)*exp(-c*2i)*(exp(a*2i)*exp(-c*2i) - 1)*1i)/(-exp(a*2i)*exp(-c*2i))^( 1/2))*(exp(a*2i - c*2i) - 1))/(2*b*(-exp(a*2i - c*2i))^(1/2)) + (log(exp(a *1i)*exp(b*x*1i)*(exp(a*2i)*exp(-c*2i) - 1) + (exp(a*2i)*exp(-c*2i)*(exp(a *2i)*exp(-c*2i) - 1)*1i)/(-exp(a*2i)*exp(-c*2i))^(1/2))*(exp(a*2i - c*2i) - 1))/(2*b*(-exp(a*2i - c*2i))^(1/2))