Integrand size = 27, antiderivative size = 133 \[ \int e^{a+i b x} \csc ^3(d+b x) \sec ^3(d+b x) \, dx=\frac {8 e^{a-i d+3 i (d+b x)}}{b \left (1-e^{4 i (d+b x)}\right )^2}-\frac {6 e^{a-i d+3 i (d+b x)}}{b \left (1-e^{4 i (d+b x)}\right )}+\frac {3 e^{a-i d} \arctan \left (e^{i (d+b x)}\right )}{b}-\frac {3 e^{a-i d} \text {arctanh}\left (e^{i (d+b x)}\right )}{b} \] Output:
8*exp(a-I*d+3*I*(b*x+d))/b/(1-exp(4*I*(b*x+d)))^2-6*exp(a-I*d+3*I*(b*x+d)) /b/(1-exp(4*I*(b*x+d)))+3*exp(a-I*d)*arctan(exp(I*(b*x+d)))/b-3*exp(a-I*d) *arctanh(exp(I*(b*x+d)))/b
Time = 0.34 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.91 \[ \int e^{a+i b x} \csc ^3(d+b x) \sec ^3(d+b x) \, dx=\frac {e^{a-i d} \left (\frac {16 e^{3 i (d+b x)}}{\left (-1+e^{4 i (d+b x)}\right )^2}+\frac {12 e^{3 i (d+b x)}}{-1+e^{4 i (d+b x)}}+6 \arctan \left (e^{i (d+b x)}\right )+3 \log \left (1-e^{i (d+b x)}\right )-3 \log \left (1+e^{i (d+b x)}\right )\right )}{2 b} \] Input:
Integrate[E^(a + I*b*x)*Csc[d + b*x]^3*Sec[d + b*x]^3,x]
Output:
(E^(a - I*d)*((16*E^((3*I)*(d + b*x)))/(-1 + E^((4*I)*(d + b*x)))^2 + (12* E^((3*I)*(d + b*x)))/(-1 + E^((4*I)*(d + b*x))) + 6*ArcTan[E^(I*(d + b*x)) ] + 3*Log[1 - E^(I*(d + b*x))] - 3*Log[1 + E^(I*(d + b*x))]))/(2*b)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(436\) vs. \(2(133)=266\).
Time = 1.24 (sec) , antiderivative size = 436, normalized size of antiderivative = 3.28, 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.
\(\displaystyle \int e^{a+i b x} \csc ^3(b x+d) \sec ^3(b x+d) \, dx\) |
\(\Big \downarrow \) 4974 |
\(\displaystyle \int \left (\frac {21 i e^{a+7 i b x+6 i d}}{2 \left (1+e^{i b x+i d}\right )}+\frac {12 i e^{a+7 i b x+6 i d}}{1+e^{2 i b x+2 i d}}+\frac {9 i e^{a+7 i b x+6 i d}}{2 \left (1+e^{i b x+i d}\right )^2}+\frac {12 i e^{a+7 i b x+6 i d}}{\left (1+e^{2 i b x+2 i d}\right )^2}+\frac {i e^{a+7 i b x+6 i d}}{\left (1+e^{i b x+i d}\right )^3}+\frac {8 i e^{a+7 i b x+6 i d}}{\left (1+e^{2 i b x+2 i d}\right )^3}-\frac {21 i e^{a+7 i b x+6 i d}}{2 \left (-1+e^{i b x+i d}\right )}+\frac {9 i e^{a+7 i b x+6 i d}}{2 \left (-1+e^{i b x+i d}\right )^2}-\frac {i e^{a+7 i b x+6 i d}}{\left (-1+e^{i b x+i d}\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 e^{a-i d} \arctan \left (e^{i b x+i d}\right )}{b}-\frac {12 e^{a+5 i b x+4 i d}}{5 b}+\frac {6 e^{a+3 i (b x+d)-i d}}{b}+\frac {12 e^{a+5 i (b x+d)-i d}}{5 b}-\frac {3 e^{a-i d}}{2 b \left (1-e^{i b x+i d}\right )}+\frac {3 e^{a-i d}}{2 b \left (1+e^{i b x+i d}\right )}-\frac {5 e^{a+3 i (b x+d)-i d}}{b \left (1+e^{2 i (b x+d)}\right )}-\frac {6 e^{a+5 i (b x+d)-i d}}{b \left (1+e^{2 i (b x+d)}\right )}+\frac {e^{a-i d}}{2 b \left (1-e^{i b x+i d}\right )^2}-\frac {e^{a-i d}}{2 b \left (1+e^{i b x+i d}\right )^2}-\frac {2 e^{a+5 i (b x+d)-i d}}{b \left (1+e^{2 i (b x+d)}\right )^2}+\frac {3 e^{a-i d} \log \left (1-e^{i b x+i d}\right )}{2 b}-\frac {3 e^{a-i d} \log \left (1+e^{i b x+i d}\right )}{2 b}+\frac {e^{a+i b x}}{b}\) |
Input:
Int[E^(a + I*b*x)*Csc[d + b*x]^3*Sec[d + b*x]^3,x]
Output:
E^(a + I*b*x)/b - (12*E^(a + (4*I)*d + (5*I)*b*x))/(5*b) + (6*E^(a - I*d + (3*I)*(d + b*x)))/b + (12*E^(a - I*d + (5*I)*(d + b*x)))/(5*b) + E^(a - I *d)/(2*b*(1 - E^(I*d + I*b*x))^2) - (3*E^(a - I*d))/(2*b*(1 - E^(I*d + I*b *x))) - E^(a - I*d)/(2*b*(1 + E^(I*d + I*b*x))^2) + (3*E^(a - I*d))/(2*b*( 1 + E^(I*d + I*b*x))) - (2*E^(a - I*d + (5*I)*(d + b*x)))/(b*(1 + E^((2*I) *(d + b*x)))^2) - (5*E^(a - I*d + (3*I)*(d + b*x)))/(b*(1 + E^((2*I)*(d + b*x)))) - (6*E^(a - I*d + (5*I)*(d + b*x)))/(b*(1 + E^((2*I)*(d + b*x)))) + (3*E^(a - I*d)*ArcTan[E^(I*d + I*b*x)])/b + (3*E^(a - I*d)*Log[1 - E^(I* d + I*b*x)])/(2*b) - (3*E^(a - I*d)*Log[1 + E^(I*d + I*b*x)])/(2*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 = 12.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {6 \,{\mathrm e}^{a} {\mathrm e}^{7 i b x} {\mathrm e}^{6 i d}+2 \,{\mathrm e}^{a} {\mathrm e}^{3 i b x} {\mathrm e}^{2 i d}}{\left (-1+{\mathrm e}^{2 i \left (b x +d \right )}\right )^{2} \left (1+{\mathrm e}^{2 i \left (b x +d \right )}\right )^{2} b}+\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{-i d} \arctan \left ({\mathrm e}^{i \left (b x +d \right )}\right )}{b}-\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{-i d} \operatorname {arctanh}\left ({\mathrm e}^{i \left (b x +d \right )}\right )}{b}\) | \(107\) |
Input:
int(exp(a+I*b*x)*csc(b*x+d)^3*sec(b*x+d)^3,x,method=_RETURNVERBOSE)
Output:
2/(-1+exp(2*I*(b*x+d)))^2/(1+exp(2*I*(b*x+d)))^2/b*(3*exp(a)*exp(7*I*b*x)* exp(6*I*d)+exp(a)*exp(3*I*b*x)*exp(2*I*d))+3*exp(a)/b*exp(-I*d)*arctan(exp (I*(b*x+d)))-3/b*exp(a)*exp(-I*d)*arctanh(exp(I*(b*x+d)))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (99) = 198\).
Time = 0.08 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.76 \[ \int e^{a+i b x} \csc ^3(d+b x) \sec ^3(d+b x) \, dx=-\frac {3 \, {\left (e^{\left (8 i \, b x + a + 7 i \, d\right )} - 2 \, e^{\left (4 i \, b x + a + 3 i \, d\right )} + e^{\left (a - i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + 1\right ) + 3 \, {\left (-i \, e^{\left (8 i \, b x + a + 7 i \, d\right )} + 2 i \, e^{\left (4 i \, b x + a + 3 i \, d\right )} - i \, e^{\left (a - i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + i\right ) + 3 \, {\left (i \, e^{\left (8 i \, b x + a + 7 i \, d\right )} - 2 i \, e^{\left (4 i \, b x + a + 3 i \, d\right )} + i \, e^{\left (a - i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - i\right ) - 3 \, {\left (e^{\left (8 i \, b x + a + 7 i \, d\right )} - 2 \, e^{\left (4 i \, b x + a + 3 i \, d\right )} + e^{\left (a - i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - 1\right ) - 12 \, e^{\left (7 i \, b x + a + 6 i \, d\right )} - 4 \, e^{\left (3 i \, b x + a + 2 i \, d\right )}}{2 \, {\left (b e^{\left (8 i \, b x + 8 i \, d\right )} - 2 \, b e^{\left (4 i \, b x + 4 i \, d\right )} + b\right )}} \] Input:
integrate(exp(a+I*b*x)*csc(b*x+d)^3*sec(b*x+d)^3,x, algorithm="fricas")
Output:
-1/2*(3*(e^(8*I*b*x + a + 7*I*d) - 2*e^(4*I*b*x + a + 3*I*d) + e^(a - I*d) )*log(e^(I*b*x + I*d) + 1) + 3*(-I*e^(8*I*b*x + a + 7*I*d) + 2*I*e^(4*I*b* x + a + 3*I*d) - I*e^(a - I*d))*log(e^(I*b*x + I*d) + I) + 3*(I*e^(8*I*b*x + a + 7*I*d) - 2*I*e^(4*I*b*x + a + 3*I*d) + I*e^(a - I*d))*log(e^(I*b*x + I*d) - I) - 3*(e^(8*I*b*x + a + 7*I*d) - 2*e^(4*I*b*x + a + 3*I*d) + e^( a - I*d))*log(e^(I*b*x + I*d) - 1) - 12*e^(7*I*b*x + a + 6*I*d) - 4*e^(3*I *b*x + a + 2*I*d))/(b*e^(8*I*b*x + 8*I*d) - 2*b*e^(4*I*b*x + 4*I*d) + b)
Timed out. \[ \int e^{a+i b x} \csc ^3(d+b x) \sec ^3(d+b x) \, dx=\text {Timed out} \] Input:
integrate(exp(a+I*b*x)*csc(b*x+d)**3*sec(b*x+d)**3,x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1133 vs. \(2 (99) = 198\).
Time = 0.18 (sec) , antiderivative size = 1133, normalized size of antiderivative = 8.52 \[ \int e^{a+i b x} \csc ^3(d+b x) \sec ^3(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(a+I*b*x)*csc(b*x+d)^3*sec(b*x+d)^3,x, algorithm="maxima")
Output:
(6*((I*cos(d)*e^a + e^a*sin(d))*cos(8*b*x + 8*d) + 2*(-I*cos(d)*e^a - e^a* sin(d))*cos(4*b*x + 4*d) + I*cos(d)*e^a - (cos(d)*e^a - I*e^a*sin(d))*sin( 8*b*x + 8*d) + 2*(cos(d)*e^a - I*e^a*sin(d))*sin(4*b*x + 4*d) + e^a*sin(d) )*arctan2(2*(cos(b*x + 2*d)*cos(d) + sin(b*x + 2*d)*sin(d))/(cos(b*x + 2*d )^2 + cos(d)^2 + 2*cos(d)*sin(b*x + 2*d) + sin(b*x + 2*d)^2 - 2*cos(b*x + 2*d)*sin(d) + sin(d)^2), (cos(b*x + 2*d)^2 - cos(d)^2 + sin(b*x + 2*d)^2 - sin(d)^2)/(cos(b*x + 2*d)^2 + cos(d)^2 + 2*cos(d)*sin(b*x + 2*d) + sin(b* x + 2*d)^2 - 2*cos(b*x + 2*d)*sin(d) + sin(d)^2)) + 6*((cos(d)*e^a - I*e^a *sin(d))*cos(8*b*x + 8*d) - 2*(cos(d)*e^a - I*e^a*sin(d))*cos(4*b*x + 4*d) + cos(d)*e^a + (I*cos(d)*e^a + e^a*sin(d))*sin(8*b*x + 8*d) + 2*(-I*cos(d )*e^a - e^a*sin(d))*sin(4*b*x + 4*d) - I*e^a*sin(d))*arctan2(sin(b*x) + si n(d), cos(b*x) - cos(d)) - 6*((cos(d)*e^a - I*e^a*sin(d))*cos(8*b*x + 8*d) - 2*(cos(d)*e^a - I*e^a*sin(d))*cos(4*b*x + 4*d) + cos(d)*e^a - (-I*cos(d )*e^a - e^a*sin(d))*sin(8*b*x + 8*d) - 2*(I*cos(d)*e^a + e^a*sin(d))*sin(4 *b*x + 4*d) - I*e^a*sin(d))*arctan2(sin(b*x) - sin(d), cos(b*x) + cos(d)) - 24*I*cos(7*b*x + 6*d)*e^a - 8*I*cos(3*b*x + 2*d)*e^a + 3*((I*cos(d)*e^a + e^a*sin(d))*cos(8*b*x + 8*d) + 2*(-I*cos(d)*e^a - e^a*sin(d))*cos(4*b*x + 4*d) + I*cos(d)*e^a - (cos(d)*e^a - I*e^a*sin(d))*sin(8*b*x + 8*d) + 2*( cos(d)*e^a - I*e^a*sin(d))*sin(4*b*x + 4*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)...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (99) = 198\).
Time = 0.17 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.44 \[ \int e^{a+i b x} \csc ^3(d+b x) \sec ^3(d+b x) \, dx=\frac {3 i \, e^{\left (8 i \, b x + a + 8 i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + i\right ) - 6 i \, e^{\left (4 i \, b x + a + 4 i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + i\right ) + 3 i \, e^{a} \log \left (e^{\left (i \, b x + i \, d\right )} + i\right ) - 3 i \, e^{\left (8 i \, b x + a + 8 i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - i\right ) + 6 i \, e^{\left (4 i \, b x + a + 4 i \, d\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - i\right ) - 3 i \, e^{a} \log \left (e^{\left (i \, b x + i \, d\right )} - i\right ) - 3 \, e^{\left (8 i \, b x + a + 8 i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + 6 \, e^{\left (4 i \, b x + a + 4 i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) - 3 \, e^{a} \log \left (i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + 3 \, e^{\left (8 i \, b x + a + 8 i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right ) - 6 \, e^{\left (4 i \, b x + a + 4 i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + 3 \, e^{a} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} + i\right ) + 12 \, e^{\left (7 i \, b x + a + 7 i \, d\right )} + 4 \, e^{\left (3 i \, b x + a + 3 i \, d\right )}}{2 \, b {\left (e^{\left (8 i \, b x + 9 i \, d\right )} - 2 \, e^{\left (4 i \, b x + 5 i \, d\right )} + e^{\left (i \, d\right )}\right )}} \] Input:
integrate(exp(a+I*b*x)*csc(b*x+d)^3*sec(b*x+d)^3,x, algorithm="giac")
Output:
1/2*(3*I*e^(8*I*b*x + a + 8*I*d)*log(e^(I*b*x + I*d) + I) - 6*I*e^(4*I*b*x + a + 4*I*d)*log(e^(I*b*x + I*d) + I) + 3*I*e^a*log(e^(I*b*x + I*d) + I) - 3*I*e^(8*I*b*x + a + 8*I*d)*log(e^(I*b*x + I*d) - I) + 6*I*e^(4*I*b*x + a + 4*I*d)*log(e^(I*b*x + I*d) - I) - 3*I*e^a*log(e^(I*b*x + I*d) - I) - 3 *e^(8*I*b*x + a + 8*I*d)*log(I*e^(I*b*x + I*d) + I) + 6*e^(4*I*b*x + a + 4 *I*d)*log(I*e^(I*b*x + I*d) + I) - 3*e^a*log(I*e^(I*b*x + I*d) + I) + 3*e^ (8*I*b*x + a + 8*I*d)*log(-I*e^(I*b*x + I*d) + I) - 6*e^(4*I*b*x + a + 4*I *d)*log(-I*e^(I*b*x + I*d) + I) + 3*e^a*log(-I*e^(I*b*x + I*d) + I) + 12*e ^(7*I*b*x + a + 7*I*d) + 4*e^(3*I*b*x + a + 3*I*d))/(b*(e^(8*I*b*x + 9*I*d ) - 2*e^(4*I*b*x + 5*I*d) + e^(I*d)))
Timed out. \[ \int e^{a+i b x} \csc ^3(d+b x) \sec ^3(d+b x) \, dx=\int \frac {{\mathrm {e}}^{a+b\,x\,1{}\mathrm {i}}}{{\cos \left (d+b\,x\right )}^3\,{\sin \left (d+b\,x\right )}^3} \,d x \] Input:
int(exp(a + b*x*1i)/(cos(d + b*x)^3*sin(d + b*x)^3),x)
Output:
int(exp(a + b*x*1i)/(cos(d + b*x)^3*sin(d + b*x)^3), x)
\[ \int e^{a+i b x} \csc ^3(d+b x) \sec ^3(d+b x) \, dx=e^{a} \left (\int e^{b i x} \csc \left (b x +d \right )^{3} \sec \left (b x +d \right )^{3}d x \right ) \] Input:
int(exp(a+I*b*x)*csc(b*x+d)^3*sec(b*x+d)^3,x)
Output:
e**a*int(e**(b*i*x)*csc(b*x + d)**3*sec(b*x + d)**3,x)