Integrand size = 14, antiderivative size = 33 \[ \int \csc (c-b x) \csc (a+b x) \, dx=-\frac {\csc (a+c) \log (\sin (c-b x))}{b}+\frac {\csc (a+c) \log (\sin (a+b x))}{b} \]
Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \csc (c-b x) \csc (a+b x) \, dx=-\frac {\csc (a+c) (\log (\sin (c-b x))-\log (-\sin (a+b x)))}{b} \]
Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5122, 3042, 25, 3956}
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.
| \(\displaystyle \int \csc (a+b x) \csc (c-b x) \, dx\) |
| \(\Big \downarrow \) 5122 |
| \(\displaystyle \csc (a+c) \int \cot (c-b x)dx+\csc (a+c) \int \cot (a+b x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \csc (a+c) \int -\tan \left (c-b x+\frac {\pi }{2}\right )dx+\csc (a+c) \int -\tan \left (a+b x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\csc (a+c) \int \tan \left (\frac {1}{2} (2 c+\pi )-b x\right )dx-\csc (a+c) \int \tan \left (\frac {1}{2} (2 a+\pi )+b x\right )dx\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\csc (a+c) \log (-\sin (a+b x))}{b}-\frac {\csc (a+c) \log (-\sin (c-b x))}{b}\) |
3.2.46.3.1 Defintions of rubi rules used
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[Csc[(a_.) + (b_.)*(x_)]*Csc[(c_) + (d_.)*(x_)], x_Symbol] :> Simp[Csc[( b*c - a*d)/b] Int[Cot[a + b*x], x], x] - Simp[Csc[(b*c - a*d)/d] Int[Co t[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b* c - a*d, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(36)=72\).
Time = 1.48 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42
| method | result | size |
| default | \(-\frac {-\frac {\ln \left (\tan \left (x b +a \right )\right )}{\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )}+\frac {\ln \left (\tan \left (x b +a \right ) \cos \left (a \right ) \cos \left (c \right )-\tan \left (x b +a \right ) \sin \left (a \right ) \sin \left (c \right )-\sin \left (a \right ) \cos \left (c \right )-\cos \left (a \right ) \sin \left (c \right )\right )}{\sin \left (a \right ) \cos \left (c \right )+\cos \left (a \right ) \sin \left (c \right )}}{b}\) | \(80\) |
| risch | \(\frac {2 i \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right ) {\mathrm e}^{i \left (a +c \right )}}{\left ({\mathrm e}^{2 i \left (a +c \right )}-1\right ) b}-\frac {2 i \ln \left (-{\mathrm e}^{2 i \left (a +c \right )}+{\mathrm e}^{2 i \left (x b +a \right )}\right ) {\mathrm e}^{i \left (a +c \right )}}{\left ({\mathrm e}^{2 i \left (a +c \right )}-1\right ) b}\) | \(82\) |
-1/b*(-1/(sin(a)*cos(c)+cos(a)*sin(c))*ln(tan(b*x+a))+1/(sin(a)*cos(c)+cos (a)*sin(c))*ln(tan(b*x+a)*cos(a)*cos(c)-tan(b*x+a)*sin(a)*sin(c)-sin(a)*co s(c)-cos(a)*sin(c)))
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (36) = 72\).
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.91 \[ \int \csc (c-b x) \csc (a+b x) \, dx=\frac {\log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) - \log \left (-\frac {2 \, \cos \left (b x + a\right ) \cos \left (a + c\right ) \sin \left (b x + a\right ) \sin \left (a + c\right ) + {\left (2 \, \cos \left (a + c\right )^{2} - 1\right )} \cos \left (b x + a\right )^{2} - \cos \left (a + c\right )^{2}}{\cos \left (a + c\right )^{2} + 2 \, \cos \left (a + c\right ) + 1}\right )}{2 \, b \sin \left (a + c\right )} \]
1/2*(log(-1/4*cos(b*x + a)^2 + 1/4) - log(-(2*cos(b*x + a)*cos(a + c)*sin( b*x + a)*sin(a + c) + (2*cos(a + c)^2 - 1)*cos(b*x + a)^2 - cos(a + c)^2)/ (cos(a + c)^2 + 2*cos(a + c) + 1)))/(b*sin(a + c))
Leaf count of result is larger than twice the leaf count of optimal. 1824 vs. \(2 (31) = 62\).
Time = 68.55 (sec) , antiderivative size = 1824, normalized size of antiderivative = 55.27 \[ \int \csc (c-b x) \csc (a+b x) \, dx=\text {Too large to display} \]
Piecewise((-log(-tan(c/2) + tan(b*x/2))*tan(c/2)/(2*b) - log(-tan(c/2) + t an(b*x/2))/(2*b*tan(c/2)) - log(tan(b*x/2) + 1/tan(c/2))*tan(c/2)/(2*b) - log(tan(b*x/2) + 1/tan(c/2))/(2*b*tan(c/2)) + log(tan(b*x/2))*tan(c/2)/(2* b) + log(tan(b*x/2))/(2*b*tan(c/2)), Eq(a, 0)), (log(tan(a/2) + tan(b*x/2) )*tan(a/2)/(2*b) + log(tan(a/2) + tan(b*x/2))/(2*b*tan(a/2)) + log(tan(b*x /2) - 1/tan(a/2))*tan(a/2)/(2*b) + log(tan(b*x/2) - 1/tan(a/2))/(2*b*tan(a /2)) - log(tan(b*x/2))*tan(a/2)/(2*b) - log(tan(b*x/2))/(2*b*tan(a/2)), Eq (c, 0)), (-tan(c/2)**4*tan(b*x/2)/(-2*b*tan(c/2)**3*tan(b*x/2) + 2*b*tan(c /2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 + 2*b*tan(c/2)*tan(b*x/2)) - 2*tan( c/2)**2*tan(b*x/2)/(-2*b*tan(c/2)**3*tan(b*x/2) + 2*b*tan(c/2)**2*tan(b*x/ 2)**2 - 2*b*tan(c/2)**2 + 2*b*tan(c/2)*tan(b*x/2)) - tan(b*x/2)/(-2*b*tan( c/2)**3*tan(b*x/2) + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 + 2*b *tan(c/2)*tan(b*x/2)), Eq(a, 2*atan(1/tan(c/2)))), (tan(c/2)**4*tan(b*x/2) /(-2*b*tan(c/2)**3*tan(b*x/2) + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/ 2)**2 + 2*b*tan(c/2)*tan(b*x/2)) + 2*tan(c/2)**2*tan(b*x/2)/(-2*b*tan(c/2) **3*tan(b*x/2) + 2*b*tan(c/2)**2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 + 2*b*tan (c/2)*tan(b*x/2)) + tan(b*x/2)/(-2*b*tan(c/2)**3*tan(b*x/2) + 2*b*tan(c/2) **2*tan(b*x/2)**2 - 2*b*tan(c/2)**2 + 2*b*tan(c/2)*tan(b*x/2)), Eq(a, -2*a tan(tan(c/2)) - 2*pi*floor((c/2 - pi/2)/pi))), (x/(sin(a)*sin(c)), Eq(b, 0 )), (-log(tan(a/2) + tan(b*x/2))*tan(a/2)**2*tan(c/2)**2/(2*b*tan(a/2)*...
Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (36) = 72\).
Time = 0.22 (sec) , antiderivative size = 536, normalized size of antiderivative = 16.24 \[ \int \csc (c-b x) \csc (a+b x) \, dx=-\frac {2 \, {\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\right )} \arctan \left (\sin \left (b x\right ) + \sin \left (a\right ), \cos \left (b x\right ) - \cos \left (a\right )\right ) + 2 \, {\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\right )} \arctan \left (\sin \left (b x\right ) - \sin \left (a\right ), \cos \left (b x\right ) + \cos \left (a\right )\right ) - 2 \, {\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\right )} \arctan \left (\sin \left (b x\right ) + \sin \left (c\right ), \cos \left (b x\right ) + \cos \left (c\right )\right ) - 2 \, {\left (\cos \left (2 \, a + 2 \, c\right ) \cos \left (a + c\right ) + \sin \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) - \cos \left (a + c\right )\right )} \arctan \left (\sin \left (b x\right ) - \sin \left (c\right ), \cos \left (b x\right ) - \cos \left (c\right )\right ) - {\left (\cos \left (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \sin \left (a + c\right )\right )} \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) - {\left (\cos \left (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \sin \left (a + c\right )\right )} \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + {\left (\cos \left (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \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 (a + c\right ) \sin \left (2 \, a + 2 \, c\right ) - \cos \left (2 \, a + 2 \, c\right ) \sin \left (a + c\right ) + \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 )}{b \cos \left (2 \, a + 2 \, c\right )^{2} + b \sin \left (2 \, a + 2 \, c\right )^{2} - 2 \, b \cos \left (2 \, a + 2 \, c\right ) + b} \]
-(2*(cos(2*a + 2*c)*cos(a + c) + sin(2*a + 2*c)*sin(a + c) - cos(a + c))*a rctan2(sin(b*x) + sin(a), cos(b*x) - cos(a)) + 2*(cos(2*a + 2*c)*cos(a + c ) + sin(2*a + 2*c)*sin(a + c) - cos(a + c))*arctan2(sin(b*x) - sin(a), cos (b*x) + cos(a)) - 2*(cos(2*a + 2*c)*cos(a + c) + sin(2*a + 2*c)*sin(a + c) - cos(a + c))*arctan2(sin(b*x) + sin(c), cos(b*x) + cos(c)) - 2*(cos(2*a + 2*c)*cos(a + c) + sin(2*a + 2*c)*sin(a + c) - cos(a + c))*arctan2(sin(b* x) - sin(c), cos(b*x) - cos(c)) - (cos(a + c)*sin(2*a + 2*c) - cos(2*a + 2 *c)*sin(a + c) + sin(a + c))*log(cos(b*x)^2 + 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(a) + sin(a)^2) - (cos(a + c)*sin(2*a + 2*c) - cos(2*a + 2*c)*sin(a + c) + sin(a + c))*log(cos(b*x)^2 - 2*cos(b*x)*cos (a) + cos(a)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a) + sin(a)^2) + (cos(a + c)* sin(2*a + 2*c) - cos(2*a + 2*c)*sin(a + c) + 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(a + c)*sin(2*a + 2*c) - cos(2*a + 2*c)*sin(a + c) + 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))/(b*cos(2*a + 2*c)^2 + b*sin(2*a + 2*c)^2 - 2*b*cos(2*a + 2* c) + b)
Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (36) = 72\).
Time = 0.31 (sec) , antiderivative size = 397, normalized size of antiderivative = 12.03 \[ \int \csc (c-b x) \csc (a+b x) \, dx=\frac {\frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{4} - 4 \, \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, a\right )^{4} - 4 \, \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right ) - 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, c\right )^{4} - 4 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 1\right )} \log \left ({\left | \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - 4 \, \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right ) + 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - \tan \left (b x + a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (b x + a\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right ) - 2 \, \tan \left (\frac {1}{2} \, c\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{4} - \tan \left (\frac {1}{2} \, a\right )^{4} \tan \left (\frac {1}{2} \, c\right ) - 6 \, \tan \left (\frac {1}{2} \, a\right )^{3} \tan \left (\frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{4} + \tan \left (\frac {1}{2} \, a\right )^{3} + 6 \, \tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 6 \, \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )} - \frac {{\left (\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, a\right )^{2} + \tan \left (\frac {1}{2} \, c\right )^{2} + 1\right )} \log \left ({\left | \tan \left (b x + a\right ) \right |}\right )}{\tan \left (\frac {1}{2} \, a\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + \tan \left (\frac {1}{2} \, a\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, a\right ) - \tan \left (\frac {1}{2} \, c\right )}}{2 \, b} \]
1/2*((tan(1/2*a)^4*tan(1/2*c)^4 - 4*tan(1/2*a)^3*tan(1/2*c)^3 - tan(1/2*a) ^4 - 4*tan(1/2*a)^3*tan(1/2*c) - 4*tan(1/2*a)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*a)*tan(1/2*c) + 1)*log(abs(tan(b*x + a)*tan(1/2*a)^2*tan(1/2*c )^2 - tan(b*x + a)*tan(1/2*a)^2 - 4*tan(b*x + a)*tan(1/2*a)*tan(1/2*c) + 2 *tan(1/2*a)^2*tan(1/2*c) - tan(b*x + a)*tan(1/2*c)^2 + 2*tan(1/2*a)*tan(1/ 2*c)^2 + tan(b*x + a) - 2*tan(1/2*a) - 2*tan(1/2*c)))/(tan(1/2*a)^4*tan(1/ 2*c)^3 + tan(1/2*a)^3*tan(1/2*c)^4 - tan(1/2*a)^4*tan(1/2*c) - 6*tan(1/2*a )^3*tan(1/2*c)^2 - 6*tan(1/2*a)^2*tan(1/2*c)^3 - tan(1/2*a)*tan(1/2*c)^4 + tan(1/2*a)^3 + 6*tan(1/2*a)^2*tan(1/2*c) + 6*tan(1/2*a)*tan(1/2*c)^2 + ta n(1/2*c)^3 - tan(1/2*a) - 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)*log(abs(tan(b*x + a)))/(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)))/b
Time = 32.91 (sec) , antiderivative size = 249, normalized size of antiderivative = 7.55 \[ \int \csc (c-b x) \csc (a+b x) \, dx=\frac {2\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}}\,\left (\ln \left (\frac {2\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}\,\left (-4\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}-1\right )}+{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\,4{}\mathrm {i}\right )-\ln \left (\frac {2\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}\,\left (-4\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\right )}{b-b\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,2{}\mathrm {i}}}+{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,2{}\mathrm {i}}\,4{}\mathrm {i}\right )\right )}{b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+c\,2{}\mathrm {i}}-1\right )} \]
(2*(-exp(a*2i + c*2i))^(1/2)*(log((2*(-exp(a*2i)*exp(c*2i))^(1/2)*(2*b*exp (a*2i)*exp(b*x*2i) - 4*b*exp(a*2i)*exp(c*2i) + 2*b*exp(a*4i)*exp(c*2i)*exp (b*x*2i)))/(b*(exp(a*2i)*exp(c*2i) - 1)) + exp(a*1i)*exp(a*2i)*exp(c*1i)*e xp(b*x*2i)*4i) - log((2*(-exp(a*2i)*exp(c*2i))^(1/2)*(2*b*exp(a*2i)*exp(b* x*2i) - 4*b*exp(a*2i)*exp(c*2i) + 2*b*exp(a*4i)*exp(c*2i)*exp(b*x*2i)))/(b - b*exp(a*2i)*exp(c*2i)) + exp(a*1i)*exp(a*2i)*exp(c*1i)*exp(b*x*2i)*4i)) )/(b*(exp(a*2i + c*2i) - 1))