Integrand size = 25, antiderivative size = 106 \[ \int e^{a+i b x} \cot (d+b x) \csc ^2(d+b x) \, dx=\frac {2 e^{a-i d+i (d+b x)}}{b \left (1-e^{2 i (d+b x)}\right )^2}-\frac {3 e^{a-i d+i (d+b x)}}{b \left (1-e^{2 i (d+b x)}\right )}+\frac {e^{a-i d} \text {arctanh}\left (e^{i (d+b x)}\right )}{b} \] Output:
2*exp(a-I*d+I*(b*x+d))/b/(1-exp(2*I*(b*x+d)))^2-3*exp(a-I*d+I*(b*x+d))/b/( 1-exp(2*I*(b*x+d)))+exp(a-I*d)*arctanh(exp(I*(b*x+d)))/b
Time = 0.28 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.14 \[ \int e^{a+i b x} \cot (d+b x) \csc ^2(d+b x) \, dx=\frac {e^{a-i d} \left (-2 e^{i (d+b x)}+6 e^{3 i (d+b x)}-\left (-1+e^{2 i (d+b x)}\right )^2 \log \left (1-e^{i (d+b x)}\right )+\left (-1+e^{2 i (d+b x)}\right )^2 \log \left (1+e^{i (d+b x)}\right )\right )}{2 b \left (-1+e^{2 i (d+b x)}\right )^2} \] Input:
Integrate[E^(a + I*b*x)*Cot[d + b*x]*Csc[d + b*x]^2,x]
Output:
(E^(a - I*d)*(-2*E^(I*(d + b*x)) + 6*E^((3*I)*(d + b*x)) - (-1 + E^((2*I)* (d + b*x)))^2*Log[1 - E^(I*(d + b*x))] + (-1 + E^((2*I)*(d + b*x)))^2*Log[ 1 + E^(I*(d + b*x))]))/(2*b*(-1 + E^((2*I)*(d + b*x)))^2)
Time = 0.39 (sec) , antiderivative size = 106, 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.
\(\displaystyle \int e^{a+i b x} \cot (b x+d) \csc ^2(b x+d) \, dx\) |
\(\Big \downarrow \) 4974 |
\(\displaystyle \int \left (-\frac {4 i e^{a+3 i b x+2 i d}}{\left (-1+e^{2 i (b x+d)}\right )^2}-\frac {8 i e^{a+3 i b x+2 i d}}{\left (-1+e^{2 i (b x+d)}\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^{a-i d} \text {arctanh}\left (e^{i (b x+d)}\right )}{b}-\frac {3 e^{a+i (b x+d)-i d}}{b \left (1-e^{2 i (b x+d)}\right )}+\frac {2 e^{a+i (b x+d)-i d}}{b \left (1-e^{2 i (b x+d)}\right )^2}\) |
Input:
Int[E^(a + I*b*x)*Cot[d + b*x]*Csc[d + b*x]^2,x]
Output:
(2*E^(a - I*d + I*(d + b*x)))/(b*(1 - E^((2*I)*(d + b*x)))^2) - (3*E^(a - I*d + I*(d + b*x)))/(b*(1 - E^((2*I)*(d + b*x)))) + (E^(a - 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 = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.62
method | result | size |
risch | \(\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{3 i b x} {\mathrm e}^{2 i d}-{\mathrm e}^{a} {\mathrm e}^{i b x}}{b \left (-1+{\mathrm e}^{2 i \left (b x +d \right )}\right )^{2}}+\frac {{\mathrm e}^{a} {\mathrm e}^{-i d} \operatorname {arctanh}\left ({\mathrm e}^{i \left (b x +d \right )}\right )}{b}\) | \(66\) |
Input:
int(exp(a+I*b*x)*cot(b*x+d)*csc(b*x+d)^2,x,method=_RETURNVERBOSE)
Output:
1/b/(-1+exp(2*I*(b*x+d)))^2*(3*exp(a)*exp(3*I*b*x)*exp(2*I*d)-exp(a)*exp(I *b*x))+exp(a)/b*exp(-I*d)*arctanh(exp(I*(b*x+d)))
Time = 0.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.28 \[ \int e^{a+i b x} \cot (d+b x) \csc ^2(d+b x) \, dx=\frac {{\left (e^{\left (4 i \, b x + a + 3 i \, d\right )} - 2 \, e^{\left (2 i \, b x + a + i \, d\right )} + e^{\left (a - i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + 1\right ) - {\left (e^{\left (4 i \, b x + a + 3 i \, d\right )} - 2 \, e^{\left (2 i \, b x + a + i \, d\right )} + e^{\left (a - i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - 1\right ) + 6 \, e^{\left (3 i \, b x + a + 2 i \, d\right )} - 2 \, e^{\left (i \, b x + a\right )}}{2 \, {\left (b e^{\left (4 i \, b x + 4 i \, d\right )} - 2 \, b e^{\left (2 i \, b x + 2 i \, d\right )} + b\right )}} \] Input:
integrate(exp(a+I*b*x)*cot(b*x+d)*csc(b*x+d)^2,x, algorithm="fricas")
Output:
1/2*((e^(4*I*b*x + a + 3*I*d) - 2*e^(2*I*b*x + a + I*d) + e^(a - I*d))*log (e^(I*b*x + I*d) + 1) - (e^(4*I*b*x + a + 3*I*d) - 2*e^(2*I*b*x + a + I*d) + e^(a - I*d))*log(e^(I*b*x + I*d) - 1) + 6*e^(3*I*b*x + a + 2*I*d) - 2*e ^(I*b*x + a))/(b*e^(4*I*b*x + 4*I*d) - 2*b*e^(2*I*b*x + 2*I*d) + b)
\[ \int e^{a+i b x} \cot (d+b x) \csc ^2(d+b x) \, dx=e^{a} \int e^{i b x} \cot {\left (b x + d \right )} \csc ^{2}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(a+I*b*x)*cot(b*x+d)*csc(b*x+d)**2,x)
Output:
exp(a)*Integral(exp(I*b*x)*cot(b*x + d)*csc(b*x + d)**2, x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 632 vs. \(2 (71) = 142\).
Time = 0.06 (sec) , antiderivative size = 632, normalized size of antiderivative = 5.96 \[ \int e^{a+i b x} \cot (d+b x) \csc ^2(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(a+I*b*x)*cot(b*x+d)*csc(b*x+d)^2,x, algorithm="maxima")
Output:
-(2*((cos(d)*e^a - I*e^a*sin(d))*cos(4*b*x + 4*d) - 2*(cos(d)*e^a - I*e^a* sin(d))*cos(2*b*x + 2*d) + cos(d)*e^a + (I*cos(d)*e^a + e^a*sin(d))*sin(4* b*x + 4*d) + 2*(-I*cos(d)*e^a - e^a*sin(d))*sin(2*b*x + 2*d) - I*e^a*sin(d ))*arctan2(sin(b*x) + sin(d), cos(b*x) - cos(d)) - 2*((cos(d)*e^a - I*e^a* sin(d))*cos(4*b*x + 4*d) - 2*(cos(d)*e^a - I*e^a*sin(d))*cos(2*b*x + 2*d) + cos(d)*e^a - (-I*cos(d)*e^a - e^a*sin(d))*sin(4*b*x + 4*d) - 2*(I*cos(d) *e^a + e^a*sin(d))*sin(2*b*x + 2*d) - I*e^a*sin(d))*arctan2(sin(b*x) - sin (d), cos(b*x) + cos(d)) + 12*I*cos(3*b*x + 2*d)*e^a - 4*I*cos(b*x)*e^a - ( (-I*cos(d)*e^a - e^a*sin(d))*cos(4*b*x + 4*d) - 2*(-I*cos(d)*e^a - e^a*sin (d))*cos(2*b*x + 2*d) - I*cos(d)*e^a + (cos(d)*e^a - I*e^a*sin(d))*sin(4*b *x + 4*d) - 2*(cos(d)*e^a - I*e^a*sin(d))*sin(2*b*x + 2*d) - e^a*sin(d))*l og(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) - ((I*cos(d)*e^a + e^a*sin(d))*cos(4*b*x + 4*d) - 2*(I*cos (d)*e^a + e^a*sin(d))*cos(2*b*x + 2*d) + I*cos(d)*e^a - (cos(d)*e^a - I*e^ a*sin(d))*sin(4*b*x + 4*d) + 2*(cos(d)*e^a - I*e^a*sin(d))*sin(2*b*x + 2*d ) + e^a*sin(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) - 12*e^a*sin(3*b*x + 2*d) + 4*e^a*sin(b*x ))/(-4*I*b*cos(4*b*x + 4*d) + 8*I*b*cos(2*b*x + 2*d) + 4*b*sin(4*b*x + 4*d ) - 8*b*sin(2*b*x + 2*d) - 4*I*b)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (71) = 142\).
Time = 0.19 (sec) , antiderivative size = 345, normalized size of antiderivative = 3.25 \[ \int e^{a+i b x} \cot (d+b x) \csc ^2(d+b x) \, dx=\frac {3 \, e^{\left (4 i \, b x + a + 3 i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) - 6 \, e^{\left (2 i \, b x + a + i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + 3 \, e^{\left (a - i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + e^{\left (4 i \, b x + a + 3 i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} - i\right ) - 2 \, e^{\left (2 i \, b x + a + i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} - i\right ) + e^{\left (a - i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} - i\right ) - 3 \, e^{\left (4 i \, b x + a + 3 i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + 6 \, e^{\left (2 i \, b x + a + i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right ) - 3 \, e^{\left (a - i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right ) - e^{\left (4 i \, b x + a + 3 i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} - i\right ) + 2 \, e^{\left (2 i \, b x + a + i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} - i\right ) - e^{\left (a - i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} - i\right ) + 12 \, e^{\left (3 i \, b x + a + 2 i \, d\right )} - 4 \, e^{\left (i \, b x + a\right )}}{4 \, b {\left (e^{\left (4 i \, b x + 4 i \, d\right )} - 2 \, e^{\left (2 i \, b x + 2 i \, d\right )} + 1\right )}} \] Input:
integrate(exp(a+I*b*x)*cot(b*x+d)*csc(b*x+d)^2,x, algorithm="giac")
Output:
1/4*(3*e^(4*I*b*x + a + 3*I*d)*log(I*e^(I*b*x + I*d) + I) - 6*e^(2*I*b*x + a + I*d)*log(I*e^(I*b*x + I*d) + I) + 3*e^(a - I*d)*log(I*e^(I*b*x + I*d) + I) + e^(4*I*b*x + a + 3*I*d)*log(I*e^(I*b*x + I*d) - I) - 2*e^(2*I*b*x + a + I*d)*log(I*e^(I*b*x + I*d) - I) + e^(a - I*d)*log(I*e^(I*b*x + I*d) - I) - 3*e^(4*I*b*x + a + 3*I*d)*log(-I*e^(I*b*x + I*d) + I) + 6*e^(2*I*b* x + a + I*d)*log(-I*e^(I*b*x + I*d) + I) - 3*e^(a - I*d)*log(-I*e^(I*b*x + I*d) + I) - e^(4*I*b*x + a + 3*I*d)*log(-I*e^(I*b*x + I*d) - I) + 2*e^(2* I*b*x + a + I*d)*log(-I*e^(I*b*x + I*d) - I) - e^(a - I*d)*log(-I*e^(I*b*x + I*d) - I) + 12*e^(3*I*b*x + a + 2*I*d) - 4*e^(I*b*x + a))/(b*(e^(4*I*b* x + 4*I*d) - 2*e^(2*I*b*x + 2*I*d) + 1))
Timed out. \[ \int e^{a+i b x} \cot (d+b x) \csc ^2(d+b x) \, dx=\int \frac {\mathrm {cot}\left (d+b\,x\right )\,{\mathrm {e}}^{a+b\,x\,1{}\mathrm {i}}}{{\sin \left (d+b\,x\right )}^2} \,d x \] Input:
int((cot(d + b*x)*exp(a + b*x*1i))/sin(d + b*x)^2,x)
Output:
int((cot(d + b*x)*exp(a + b*x*1i))/sin(d + b*x)^2, x)
\[ \int e^{a+i b x} \cot (d+b x) \csc ^2(d+b x) \, dx=e^{a} \left (\int e^{b i x} \cot \left (b x +d \right ) \csc \left (b x +d \right )^{2}d x \right ) \] Input:
int(exp(a+I*b*x)*cot(b*x+d)*csc(b*x+d)^2,x)
Output:
e**a*int(e**(b*i*x)*cot(b*x + d)*csc(b*x + d)**2,x)