Integrand size = 25, antiderivative size = 104 \[ \int e^{2 (a+i b x)} \cot (d+b x) \csc (d+b x) \, dx=\frac {2 i e^{2 (a-i d)+i (d+b x)}}{b}+\frac {2 i e^{2 (a-i d)+i (d+b x)}}{b \left (1-e^{2 i (d+b x)}\right )}-\frac {4 i e^{2 a-2 i d} \text {arctanh}\left (e^{i (d+b x)}\right )}{b} \] Output:
2*I*exp(2*a-2*I*d+I*(b*x+d))/b+2*I*exp(2*a-2*I*d+I*(b*x+d))/b/(1-exp(2*I*( b*x+d)))-4*I*exp(2*a-2*I*d)*arctanh(exp(I*(b*x+d)))/b
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(386\) vs. \(2(104)=208\).
Time = 0.55 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.71 \[ \int e^{2 (a+i b x)} \cot (d+b x) \csc (d+b x) \, dx=e^{2 a} \left (\frac {2 i e^{i b x} \cos (d)}{b}+\frac {2 \arctan \left (\frac {\left (-1+e^{i b x}\right ) \tan \left (\frac {d}{2}\right )}{1+e^{i b x}}\right ) \cos (2 d)}{b}-\frac {2 \arctan \left (\frac {\left (1+e^{i b x}\right ) \tan \left (\frac {d}{2}\right )}{-1+e^{i b x}}\right ) \cos (2 d)}{b}+\frac {i \cos (2 d) \log \left (1+e^{2 i b x}-2 e^{i b x} \cos (d)\right )}{b}-\frac {i \cos (2 d) \log \left (1+e^{2 i b x}+2 e^{i b x} \cos (d)\right )}{b}+\frac {2 e^{i b x} \sin (d)}{b}+\frac {2 e^{i b x} (\cos (d)-i \sin (d))^2}{i b \left (-1+e^{2 i b x}\right ) \cos (d)-b \left (1+e^{2 i b x}\right ) \sin (d)}-\frac {2 i \arctan \left (\frac {\left (-1+e^{i b x}\right ) \tan \left (\frac {d}{2}\right )}{1+e^{i b x}}\right ) \sin (2 d)}{b}+\frac {2 i \arctan \left (\frac {\left (1+e^{i b x}\right ) \tan \left (\frac {d}{2}\right )}{-1+e^{i b x}}\right ) \sin (2 d)}{b}+\frac {\log \left (1+e^{2 i b x}-2 e^{i b x} \cos (d)\right ) \sin (2 d)}{b}-\frac {\log \left (1+e^{2 i b x}+2 e^{i b x} \cos (d)\right ) \sin (2 d)}{b}\right ) \] Input:
Integrate[E^(2*(a + I*b*x))*Cot[d + b*x]*Csc[d + b*x],x]
Output:
E^(2*a)*(((2*I)*E^(I*b*x)*Cos[d])/b + (2*ArcTan[((-1 + E^(I*b*x))*Tan[d/2] )/(1 + E^(I*b*x))]*Cos[2*d])/b - (2*ArcTan[((1 + E^(I*b*x))*Tan[d/2])/(-1 + E^(I*b*x))]*Cos[2*d])/b + (I*Cos[2*d]*Log[1 + E^((2*I)*b*x) - 2*E^(I*b*x )*Cos[d]])/b - (I*Cos[2*d]*Log[1 + E^((2*I)*b*x) + 2*E^(I*b*x)*Cos[d]])/b + (2*E^(I*b*x)*Sin[d])/b + (2*E^(I*b*x)*(Cos[d] - I*Sin[d])^2)/(I*b*(-1 + E^((2*I)*b*x))*Cos[d] - b*(1 + E^((2*I)*b*x))*Sin[d]) - ((2*I)*ArcTan[((-1 + E^(I*b*x))*Tan[d/2])/(1 + E^(I*b*x))]*Sin[2*d])/b + ((2*I)*ArcTan[((1 + E^(I*b*x))*Tan[d/2])/(-1 + E^(I*b*x))]*Sin[2*d])/b + (Log[1 + E^((2*I)*b* x) - 2*E^(I*b*x)*Cos[d]]*Sin[2*d])/b - (Log[1 + E^((2*I)*b*x) + 2*E^(I*b*x )*Cos[d]]*Sin[2*d])/b)
Time = 0.36 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {4974, 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.
\(\displaystyle \int e^{2 (a+i b x)} \cot (b x+d) \csc (b x+d) \, dx\) |
\(\Big \downarrow \) 4974 |
\(\displaystyle \int \left (\frac {2 e^{2 a+3 i b x+i d}}{1-e^{2 i (b x+d)}}-\frac {4 e^{2 a+3 i b x+i d}}{\left (-1+e^{2 i (b x+d)}\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 i e^{2 a-2 i d} \text {arctanh}\left (e^{i (b x+d)}\right )}{b}+\frac {2 i e^{2 (a-i d)+i (b x+d)}}{b}+\frac {2 i e^{2 (a-i d)+i (b x+d)}}{b \left (1-e^{2 i (b x+d)}\right )}\) |
Input:
Int[E^(2*(a + I*b*x))*Cot[d + b*x]*Csc[d + b*x],x]
Output:
((2*I)*E^(2*(a - I*d) + I*(d + b*x)))/b + ((2*I)*E^(2*(a - I*d) + I*(d + b *x)))/(b*(1 - E^((2*I)*(d + b*x)))) - ((4*I)*E^(2*a - (2*I)*d)*ArcTanh[E^( I*(d + b*x))])/b
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(G_)[(d_.) + (e_.)*(x_)]^(m_.)*(H_)[( d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^(c*(a + b*x)), G[d + e*x]^m*H[d + e*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IGtQ[ m, 0] && IGtQ[n, 0] && TrigQ[G] && TrigQ[H]
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.79
method | result | size |
risch | \(-\frac {2 i {\mathrm e}^{2 a} {\mathrm e}^{3 i b x} {\mathrm e}^{i d}}{\left (-1+{\mathrm e}^{2 i \left (b x +d \right )}\right ) b}+\frac {4 i {\mathrm e}^{2 a} {\mathrm e}^{i b x} {\mathrm e}^{-i d}}{b}-\frac {4 i {\mathrm e}^{2 a} {\mathrm e}^{-2 i d} \operatorname {arctanh}\left ({\mathrm e}^{i \left (b x +d \right )}\right )}{b}\) | \(82\) |
Input:
int(exp(2*a+2*I*b*x)*cot(b*x+d)*csc(b*x+d),x,method=_RETURNVERBOSE)
Output:
-2*I/(-1+exp(2*I*(b*x+d)))/b*exp(2*a)*exp(3*I*b*x)*exp(I*d)+4*I/b*exp(2*a) *exp(I*b*x)*exp(-I*d)-4*I/b*exp(2*a)*exp(-2*I*d)*arctanh(exp(I*(b*x+d)))
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.13 \[ \int e^{2 (a+i b x)} \cot (d+b x) \csc (d+b x) \, dx=-\frac {2 \, {\left ({\left (i \, e^{\left (2 i \, b x + 2 \, a\right )} - i \, e^{\left (2 \, a - 2 i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + 1\right ) + {\left (-i \, e^{\left (2 i \, b x + 2 \, a\right )} + i \, e^{\left (2 \, a - 2 i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - 1\right ) - i \, e^{\left (3 i \, b x + 2 \, a + i \, d\right )} + 2 i \, e^{\left (i \, b x + 2 \, a - i \, d\right )}\right )}}{b e^{\left (2 i \, b x + 2 i \, d\right )} - b} \] Input:
integrate(exp(2*a+2*I*b*x)*cot(b*x+d)*csc(b*x+d),x, algorithm="fricas")
Output:
-2*((I*e^(2*I*b*x + 2*a) - I*e^(2*a - 2*I*d))*log(e^(I*b*x + I*d) + 1) + ( -I*e^(2*I*b*x + 2*a) + I*e^(2*a - 2*I*d))*log(e^(I*b*x + I*d) - 1) - I*e^( 3*I*b*x + 2*a + I*d) + 2*I*e^(I*b*x + 2*a - I*d))/(b*e^(2*I*b*x + 2*I*d) - b)
\[ \int e^{2 (a+i b x)} \cot (d+b x) \csc (d+b x) \, dx=e^{2 a} \int e^{2 i b x} \cot {\left (b x + d \right )} \csc {\left (b x + d \right )}\, dx \] Input:
integrate(exp(2*a+2*I*b*x)*cot(b*x+d)*csc(b*x+d),x)
Output:
exp(2*a)*Integral(exp(2*I*b*x)*cot(b*x + d)*csc(b*x + d), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 628 vs. \(2 (71) = 142\).
Time = 0.06 (sec) , antiderivative size = 628, normalized size of antiderivative = 6.04 \[ \int e^{2 (a+i b x)} \cot (d+b x) \csc (d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(2*a+2*I*b*x)*cot(b*x+d)*csc(b*x+d),x, algorithm="maxima")
Output:
-(2*((-I*cos(2*d)*e^(2*a) - e^(2*a)*sin(2*d))*cos(2*b*x + 3*d) + (I*cos(d) *e^(2*a) - e^(2*a)*sin(d))*cos(2*d) + (cos(2*d)*e^(2*a) - I*e^(2*a)*sin(2* d))*sin(2*b*x + 3*d) + (cos(d)*e^(2*a) + I*e^(2*a)*sin(d))*sin(2*d))*arcta n2(sin(b*x) + sin(d), cos(b*x) - cos(d)) + 2*((I*cos(2*d)*e^(2*a) + e^(2*a )*sin(2*d))*cos(2*b*x + 3*d) + (-I*cos(d)*e^(2*a) + e^(2*a)*sin(d))*cos(2* d) - (cos(2*d)*e^(2*a) - I*e^(2*a)*sin(2*d))*sin(2*b*x + 3*d) - (cos(d)*e^ (2*a) + I*e^(2*a)*sin(d))*sin(2*d))*arctan2(sin(b*x) - sin(d), cos(b*x) + cos(d)) - 2*cos(3*b*x + 2*d)*e^(2*a) + 4*cos(b*x)*e^(2*a) + ((cos(2*d)*e^( 2*a) - I*e^(2*a)*sin(2*d))*cos(2*b*x + 3*d) - (cos(d)*e^(2*a) + I*e^(2*a)* sin(d))*cos(2*d) - (-I*cos(2*d)*e^(2*a) - e^(2*a)*sin(2*d))*sin(2*b*x + 3* d) - (-I*cos(d)*e^(2*a) + e^(2*a)*sin(d))*sin(2*d))*log(cos(b*x)^2 + 2*cos (b*x)*cos(d) + cos(d)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(d) + sin(d)^2) - ((c os(2*d)*e^(2*a) - I*e^(2*a)*sin(2*d))*cos(2*b*x + 3*d) - (cos(d)*e^(2*a) + I*e^(2*a)*sin(d))*cos(2*d) + (I*cos(2*d)*e^(2*a) + e^(2*a)*sin(2*d))*sin( 2*b*x + 3*d) + (I*cos(d)*e^(2*a) - e^(2*a)*sin(d))*sin(2*d))*log(cos(b*x)^ 2 - 2*cos(b*x)*cos(d) + cos(d)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(d) + sin(d) ^2) - 2*I*e^(2*a)*sin(3*b*x + 2*d) + 4*I*e^(2*a)*sin(b*x))/(-I*b*cos(2*b*x + 3*d) + I*b*cos(d) + b*sin(2*b*x + 3*d) - b*sin(d))
Time = 0.12 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.32 \[ \int e^{2 (a+i b x)} \cot (d+b x) \csc (d+b x) \, dx=-\frac {2 \, {\left (i \, e^{\left (2 i \, b x + 2 \, a\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + 1\right ) - i \, e^{\left (2 \, a - 2 i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + 1\right ) - i \, e^{\left (2 i \, b x + 2 \, a\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - 1\right ) + i \, e^{\left (2 \, a - 2 i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - 1\right ) - i \, e^{\left (3 i \, b x + 2 \, a + i \, d\right )} + 2 i \, e^{\left (i \, b x + 2 \, a - i \, d\right )}\right )}}{b {\left (e^{\left (2 i \, b x + 2 i \, d\right )} - 1\right )}} \] Input:
integrate(exp(2*a+2*I*b*x)*cot(b*x+d)*csc(b*x+d),x, algorithm="giac")
Output:
-2*(I*e^(2*I*b*x + 2*a)*log(e^(I*b*x + I*d) + 1) - I*e^(2*a - 2*I*d)*log(e ^(I*b*x + I*d) + 1) - I*e^(2*I*b*x + 2*a)*log(e^(I*b*x + I*d) - 1) + I*e^( 2*a - 2*I*d)*log(e^(I*b*x + I*d) - 1) - I*e^(3*I*b*x + 2*a + I*d) + 2*I*e^ (I*b*x + 2*a - I*d))/(b*(e^(2*I*b*x + 2*I*d) - 1))
Time = 19.46 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.82 \[ \int e^{2 (a+i b x)} \cot (d+b x) \csc (d+b x) \, dx=\frac {{\mathrm {e}}^{2\,a-d\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,2{}\mathrm {i}}{b}+\frac {{\mathrm {e}}^{4\,a-d\,3{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,2{}\mathrm {i}}{b\,\left ({\mathrm {e}}^{2\,a-d\,2{}\mathrm {i}}-{\mathrm {e}}^{2\,a+b\,x\,2{}\mathrm {i}}\right )}-\frac {2\,\sqrt {-{\mathrm {e}}^{4\,a-d\,4{}\mathrm {i}}}\,\ln \left (4\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-d\,3{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}-{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-d\,2{}\mathrm {i}}\,\sqrt {-{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-d\,4{}\mathrm {i}}}\,4{}\mathrm {i}\right )}{b}+\frac {2\,\sqrt {-{\mathrm {e}}^{4\,a-d\,4{}\mathrm {i}}}\,\ln \left (4\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-d\,3{}\mathrm {i}}\,{\mathrm {e}}^{b\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-d\,2{}\mathrm {i}}\,\sqrt {-{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{-d\,4{}\mathrm {i}}}\,4{}\mathrm {i}\right )}{b} \] Input:
int((cot(d + b*x)*exp(2*a + b*x*2i))/sin(d + b*x),x)
Output:
(exp(2*a - d*1i + b*x*1i)*2i)/b + (exp(4*a - d*3i + b*x*1i)*2i)/(b*(exp(2* a - d*2i) - exp(2*a + b*x*2i))) - (2*(-exp(4*a - d*4i))^(1/2)*log(4*exp(4* a)*exp(-d*3i)*exp(b*x*1i) - exp(2*a)*exp(-d*2i)*(-exp(4*a)*exp(-d*4i))^(1/ 2)*4i))/b + (2*(-exp(4*a - d*4i))^(1/2)*log(4*exp(4*a)*exp(-d*3i)*exp(b*x* 1i) + exp(2*a)*exp(-d*2i)*(-exp(4*a)*exp(-d*4i))^(1/2)*4i))/b
\[ \int e^{2 (a+i b x)} \cot (d+b x) \csc (d+b x) \, dx=e^{2 a} \left (\int e^{2 b i x} \cot \left (b x +d \right ) \csc \left (b x +d \right )d x \right ) \] Input:
int(exp(2*a+2*I*b*x)*cot(b*x+d)*csc(b*x+d),x)
Output:
e**(2*a)*int(e**(2*b*i*x)*cot(b*x + d)*csc(b*x + d),x)