Integrand size = 27, antiderivative size = 141 \[ \int e^{2 (a+i b x)} \cot ^2(d+b x) \csc (d+b x) \, dx=-\frac {2 e^{2 (a-i d)+i (d+b x)}}{b}+\frac {2 e^{2 (a-i d)+i (d+b x)}}{b \left (1-e^{2 i (d+b x)}\right )^2}-\frac {5 e^{2 (a-i d)+i (d+b x)}}{b \left (1-e^{2 i (d+b x)}\right )}+\frac {5 e^{2 a-2 i d} \text {arctanh}\left (e^{i (d+b x)}\right )}{b} \] Output:
-2*exp(2*a-2*I*d+I*(b*x+d))/b+2*exp(2*a-2*I*d+I*(b*x+d))/b/(1-exp(2*I*(b*x +d)))^2-5*exp(2*a-2*I*d+I*(b*x+d))/b/(1-exp(2*I*(b*x+d)))+5*exp(2*a-2*I*d) *arctanh(exp(I*(b*x+d)))/b
Time = 0.37 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.77 \[ \int e^{2 (a+i b x)} \cot ^2(d+b x) \csc (d+b x) \, dx=\frac {e^{2 a-2 i d} \left (-\frac {2 e^{i (d+b x)} \left (5-9 e^{2 i (d+b x)}+2 e^{4 i (d+b x)}\right )}{\left (-1+e^{2 i (d+b x)}\right )^2}-5 \log \left (1-e^{i (d+b x)}\right )+5 \log \left (1+e^{i (d+b x)}\right )\right )}{2 b} \] Input:
Integrate[E^(2*(a + I*b*x))*Cot[d + b*x]^2*Csc[d + b*x],x]
Output:
(E^(2*a - (2*I)*d)*((-2*E^(I*(d + b*x))*(5 - 9*E^((2*I)*(d + b*x)) + 2*E^( (4*I)*(d + b*x))))/(-1 + E^((2*I)*(d + b*x)))^2 - 5*Log[1 - E^(I*(d + b*x) )] + 5*Log[1 + E^(I*(d + b*x))]))/(2*b)
Time = 0.45 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.
\(\displaystyle \int e^{2 (a+i b x)} \cot ^2(b x+d) \csc (b x+d) \, dx\) |
\(\Big \downarrow \) 4974 |
\(\displaystyle \int \left (-\frac {2 i e^{2 a+3 i b x+i d}}{-1+e^{2 i (b x+d)}}-\frac {8 i e^{2 a+3 i b x+i d}}{\left (-1+e^{2 i (b x+d)}\right )^2}-\frac {8 i e^{2 a+3 i b x+i d}}{\left (-1+e^{2 i (b x+d)}\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {5 e^{2 a-2 i d} \text {arctanh}\left (e^{i (b x+d)}\right )}{b}-\frac {2 e^{2 (a-i d)+i (b x+d)}}{b}-\frac {5 e^{2 (a-i d)+i (b x+d)}}{b \left (1-e^{2 i (b x+d)}\right )}+\frac {2 e^{2 (a-i d)+i (b x+d)}}{b \left (1-e^{2 i (b x+d)}\right )^2}\) |
Input:
Int[E^(2*(a + I*b*x))*Cot[d + b*x]^2*Csc[d + b*x],x]
Output:
(-2*E^(2*(a - I*d) + I*(d + b*x)))/b + (2*E^(2*(a - I*d) + I*(d + b*x)))/( b*(1 - E^((2*I)*(d + b*x)))^2) - (5*E^(2*(a - I*d) + I*(d + b*x)))/(b*(1 - E^((2*I)*(d + b*x)))) + (5*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 = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.70
method | result | size |
risch | \(\frac {3 \,{\mathrm e}^{5 i b x} {\mathrm e}^{3 i d} {\mathrm e}^{2 a}-{\mathrm e}^{3 i b x} {\mathrm e}^{i d} {\mathrm e}^{2 a}}{\left (-1+{\mathrm e}^{2 i \left (b x +d \right )}\right )^{2} b}+\frac {5 \,{\mathrm e}^{2 a} {\mathrm e}^{-2 i d} \operatorname {arctanh}\left ({\mathrm e}^{i \left (b x +d \right )}\right )}{b}-\frac {5 \,{\mathrm e}^{2 a} {\mathrm e}^{-i d} {\mathrm e}^{i b x}}{b}\) | \(98\) |
Input:
int(exp(2*a+2*I*b*x)*cot(b*x+d)^2*csc(b*x+d),x,method=_RETURNVERBOSE)
Output:
1/(-1+exp(2*I*(b*x+d)))^2/b*(3*exp(5*I*b*x)*exp(3*I*d)*exp(2*a)-exp(3*I*b* x)*exp(I*d)*exp(2*a))+5/b*exp(2*a)*exp(-2*I*d)*arctanh(exp(I*(b*x+d)))-5/b *exp(2*a)*exp(-I*d)*exp(I*b*x)
Time = 0.08 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.16 \[ \int e^{2 (a+i b x)} \cot ^2(d+b x) \csc (d+b x) \, dx=\frac {5 \, {\left (e^{\left (4 i \, b x + 2 \, a + 2 i \, d\right )} - 2 \, e^{\left (2 i \, b x + 2 \, a\right )} + e^{\left (2 \, a - 2 i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + 1\right ) - 5 \, {\left (e^{\left (4 i \, b x + 2 \, a + 2 i \, d\right )} - 2 \, e^{\left (2 i \, b x + 2 \, a\right )} + e^{\left (2 \, a - 2 i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - 1\right ) - 4 \, e^{\left (5 i \, b x + 2 \, a + 3 i \, d\right )} + 18 \, e^{\left (3 i \, b x + 2 \, a + i \, d\right )} - 10 \, e^{\left (i \, b x + 2 \, a - i \, d\right )}}{2 \, {\left (b e^{\left (4 i \, b x + 4 i \, d\right )} - 2 \, b e^{\left (2 i \, b x + 2 i \, d\right )} + b\right )}} \] Input:
integrate(exp(2*a+2*I*b*x)*cot(b*x+d)^2*csc(b*x+d),x, algorithm="fricas")
Output:
1/2*(5*(e^(4*I*b*x + 2*a + 2*I*d) - 2*e^(2*I*b*x + 2*a) + e^(2*a - 2*I*d)) *log(e^(I*b*x + I*d) + 1) - 5*(e^(4*I*b*x + 2*a + 2*I*d) - 2*e^(2*I*b*x + 2*a) + e^(2*a - 2*I*d))*log(e^(I*b*x + I*d) - 1) - 4*e^(5*I*b*x + 2*a + 3* I*d) + 18*e^(3*I*b*x + 2*a + I*d) - 10*e^(I*b*x + 2*a - I*d))/(b*e^(4*I*b* x + 4*I*d) - 2*b*e^(2*I*b*x + 2*I*d) + b)
\[ \int e^{2 (a+i b x)} \cot ^2(d+b x) \csc (d+b x) \, dx=e^{2 a} \int e^{2 i b x} \cot ^{2}{\left (b x + d \right )} \csc {\left (b x + d \right )}\, dx \] Input:
integrate(exp(2*a+2*I*b*x)*cot(b*x+d)**2*csc(b*x+d),x)
Output:
exp(2*a)*Integral(exp(2*I*b*x)*cot(b*x + d)**2*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. 932 vs. \(2 (101) = 202\).
Time = 0.08 (sec) , antiderivative size = 932, normalized size of antiderivative = 6.61 \[ \int e^{2 (a+i b x)} \cot ^2(d+b x) \csc (d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(2*a+2*I*b*x)*cot(b*x+d)^2*csc(b*x+d),x, algorithm="maxima")
Output:
-(10*((cos(2*d)*e^(2*a) - I*e^(2*a)*sin(2*d))*cos(4*b*x + 5*d) - 2*(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(4*b*x + 5*d) + 2*(-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))*arctan2(sin(b*x) + sin(d), co s(b*x) - cos(d)) - 10*((cos(2*d)*e^(2*a) - I*e^(2*a)*sin(2*d))*cos(4*b*x + 5*d) - 2*(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)*s in(2*d))*sin(4*b*x + 5*d) - 2*(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))*arctan2(sin(b *x) - sin(d), cos(b*x) + cos(d)) - 8*I*cos(5*b*x + 4*d)*e^(2*a) + 36*I*cos (3*b*x + 2*d)*e^(2*a) - 20*I*cos(b*x)*e^(2*a) + 5*((I*cos(2*d)*e^(2*a) + e ^(2*a)*sin(2*d))*cos(4*b*x + 5*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) - (co s(2*d)*e^(2*a) - I*e^(2*a)*sin(2*d))*sin(4*b*x + 5*d) + 2*(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))*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) + 5*((-I*cos(2*d)*e^(2*a) - e^(2*a)*sin(2* d))*cos(4*b*x + 5*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^(...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (101) = 202\).
Time = 0.24 (sec) , antiderivative size = 378, normalized size of antiderivative = 2.68 \[ \int e^{2 (a+i b x)} \cot ^2(d+b x) \csc (d+b x) \, dx=\frac {21 \, e^{\left (4 i \, b x + 2 \, a + 2 i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) - 42 \, e^{\left (2 i \, b x + 2 \, a\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + 21 \, e^{\left (2 \, a - 2 i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + e^{\left (4 i \, b x + 2 \, a + 2 i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} - i\right ) - 2 \, e^{\left (2 i \, b x + 2 \, a\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} - i\right ) + e^{\left (2 \, a - 2 i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} - i\right ) - 21 \, e^{\left (4 i \, b x + 2 \, a + 2 i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + 42 \, e^{\left (2 i \, b x + 2 \, a\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right ) - 21 \, e^{\left (2 \, a - 2 i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right ) - e^{\left (4 i \, b x + 2 \, a + 2 i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} - i\right ) + 2 \, e^{\left (2 i \, b x + 2 \, a\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} - i\right ) - e^{\left (2 \, a - 2 i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} - i\right ) - 16 \, e^{\left (5 i \, b x + 2 \, a + 3 i \, d\right )} + 72 \, e^{\left (3 i \, b x + 2 \, a + i \, d\right )} - 40 \, e^{\left (i \, b x + 2 \, a - i \, d\right )}}{8 \, b {\left (e^{\left (4 i \, b x + 4 i \, d\right )} - 2 \, e^{\left (2 i \, b x + 2 i \, d\right )} + 1\right )}} \] Input:
integrate(exp(2*a+2*I*b*x)*cot(b*x+d)^2*csc(b*x+d),x, algorithm="giac")
Output:
1/8*(21*e^(4*I*b*x + 2*a + 2*I*d)*log(I*e^(I*b*x + I*d) + I) - 42*e^(2*I*b *x + 2*a)*log(I*e^(I*b*x + I*d) + I) + 21*e^(2*a - 2*I*d)*log(I*e^(I*b*x + I*d) + I) + e^(4*I*b*x + 2*a + 2*I*d)*log(I*e^(I*b*x + I*d) - I) - 2*e^(2 *I*b*x + 2*a)*log(I*e^(I*b*x + I*d) - I) + e^(2*a - 2*I*d)*log(I*e^(I*b*x + I*d) - I) - 21*e^(4*I*b*x + 2*a + 2*I*d)*log(-I*e^(I*b*x + I*d) + I) + 4 2*e^(2*I*b*x + 2*a)*log(-I*e^(I*b*x + I*d) + I) - 21*e^(2*a - 2*I*d)*log(- I*e^(I*b*x + I*d) + I) - e^(4*I*b*x + 2*a + 2*I*d)*log(-I*e^(I*b*x + I*d) - I) + 2*e^(2*I*b*x + 2*a)*log(-I*e^(I*b*x + I*d) - I) - e^(2*a - 2*I*d)*l og(-I*e^(I*b*x + I*d) - I) - 16*e^(5*I*b*x + 2*a + 3*I*d) + 72*e^(3*I*b*x + 2*a + I*d) - 40*e^(I*b*x + 2*a - I*d))/(b*(e^(4*I*b*x + 4*I*d) - 2*e^(2* I*b*x + 2*I*d) + 1))
Timed out. \[ \int e^{2 (a+i b x)} \cot ^2(d+b x) \csc (d+b x) \, dx=\int \frac {{\mathrm {cot}\left (d+b\,x\right )}^2\,{\mathrm {e}}^{2\,a+b\,x\,2{}\mathrm {i}}}{\sin \left (d+b\,x\right )} \,d x \] Input:
int((cot(d + b*x)^2*exp(2*a + b*x*2i))/sin(d + b*x),x)
Output:
int((cot(d + b*x)^2*exp(2*a + b*x*2i))/sin(d + b*x), x)
\[ \int e^{2 (a+i b x)} \cot ^2(d+b x) \csc (d+b x) \, dx=e^{2 a} \left (\int e^{2 b i x} \cot \left (b x +d \right )^{2} \csc \left (b x +d \right )d x \right ) \] Input:
int(exp(2*a+2*I*b*x)*cot(b*x+d)^2*csc(b*x+d),x)
Output:
e**(2*a)*int(e**(2*b*i*x)*cot(b*x + d)**2*csc(b*x + d),x)