Integrand size = 21, antiderivative size = 117 \[ \int e^{2 (a+i b x)} \sec ^3(d+b x) \, dx=\frac {2 i e^{2 (a-i d)+3 i (d+b x)}}{b \left (1+e^{2 i (d+b x)}\right )^2}+\frac {3 i e^{2 (a-i d)+i (d+b x)}}{b \left (1+e^{2 i (d+b x)}\right )}-\frac {3 i e^{2 a-2 i d} \arctan \left (e^{i (d+b x)}\right )}{b} \] Output:
2*I*exp(2*a-2*I*d+3*I*(b*x+d))/b/(1+exp(2*I*(b*x+d)))^2+3*I*exp(2*a-2*I*d+ I*(b*x+d))/b/(1+exp(2*I*(b*x+d)))-3*I*exp(2*a-2*I*d)*arctan(exp(I*(b*x+d)) )/b
Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.68 \[ \int e^{2 (a+i b x)} \sec ^3(d+b x) \, dx=\frac {e^{2 a-2 i d} \left (\frac {2 i e^{i (d+b x)} \left (3+5 e^{2 i (d+b x)}\right )}{\left (1+e^{2 i (d+b x)}\right )^2}-6 i \arctan \left (e^{i (d+b x)}\right )\right )}{2 b} \] Input:
Integrate[E^(2*(a + I*b*x))*Sec[d + b*x]^3,x]
Output:
(E^(2*a - (2*I)*d)*(((2*I)*E^(I*(d + b*x))*(3 + 5*E^((2*I)*(d + b*x))))/(1 + E^((2*I)*(d + b*x)))^2 - (6*I)*ArcTan[E^(I*(d + b*x))]))/(2*b)
Time = 0.30 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4948, 4951}
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.
\(\displaystyle \int e^{2 (a+i b x)} \sec ^3(b x+d) \, dx\) |
\(\Big \downarrow \) 4948 |
\(\displaystyle -\frac {3}{2} \int e^{2 (a+i b x)} \sec (d+b x)dx-\frac {i e^{2 (a+i b x)} \sec (b x+d)}{b}+\frac {e^{2 (a+i b x)} \tan (b x+d) \sec (b x+d)}{2 b}\) |
\(\Big \downarrow \) 4951 |
\(\displaystyle \frac {3 i e^{2 (a+i b x)-3 i (b x+d)} \left (e^{2 i (b x+d)}-e^{i (b x+d)} \arctan \left (e^{i (b x+d)}\right )\right )}{b}-\frac {i e^{2 (a+i b x)} \sec (b x+d)}{b}+\frac {e^{2 (a+i b x)} \tan (b x+d) \sec (b x+d)}{2 b}\) |
Input:
Int[E^(2*(a + I*b*x))*Sec[d + b*x]^3,x]
Output:
((3*I)*E^(2*(a + I*b*x) - (3*I)*(d + b*x))*(E^((2*I)*(d + b*x)) - E^(I*(d + b*x))*ArcTan[E^(I*(d + b*x))]))/b - (I*E^(2*(a + I*b*x))*Sec[d + b*x])/b + (E^(2*(a + I*b*x))*Sec[d + b*x]*Tan[d + b*x])/(2*b)
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sec[(d_.) + (e_.)*(x_)]^(n_), x_Symbo l] :> Simp[(-b)*c*Log[F]*F^(c*(a + b*x))*(Sec[d + e*x]^(n - 2)/(e^2*(n - 1) *(n - 2))), x] + (Simp[F^(c*(a + b*x))*Sec[d + e*x]^(n - 1)*(Sin[d + e*x]/( e*(n - 1))), x] + Simp[(e^2*(n - 2)^2 + b^2*c^2*Log[F]^2)/(e^2*(n - 1)*(n - 2)) Int[F^(c*(a + b*x))*Sec[d + e*x]^(n - 2), x], x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b^2*c^2*Log[F]^2 + e^2*(n - 2)^2, 0] && GtQ[n, 1] && N eQ[n, 2]
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sec[(d_.) + (e_.)*(x_)]^(n_.), x_Symb ol] :> Simp[2^n*E^(I*n*(d + e*x))*(F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))*Hy pergeometric2F1[n, n/2 - I*b*c*(Log[F]/(2*e)), 1 + n/2 - I*b*c*(Log[F]/(2*e )), -E^(2*I*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
Time = 0.81 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.86
method | result | size |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{5 i b x} {\mathrm e}^{3 i d} {\mathrm e}^{2 a}+{\mathrm e}^{3 i b x} {\mathrm e}^{i d} {\mathrm e}^{2 a}\right )}{\left (1+{\mathrm e}^{2 i \left (b x +d \right )}\right )^{2} b}+\frac {3 i {\mathrm e}^{2 a} {\mathrm e}^{i b x} {\mathrm e}^{-i d}}{b}-\frac {3 i {\mathrm e}^{2 a} {\mathrm e}^{-2 i d} \arctan \left ({\mathrm e}^{i \left (b x +d \right )}\right )}{b}\) | \(101\) |
Input:
int(exp(2*a+2*I*b*x)*sec(b*x+d)^3,x,method=_RETURNVERBOSE)
Output:
-I/(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))+3*I/b*exp(2*a)*exp(I*b*x)*exp(-I*d)-3*I/b*exp(2*a)*e xp(-2*I*d)*arctan(exp(I*(b*x+d)))
Time = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.28 \[ \int e^{2 (a+i b x)} \sec ^3(d+b x) \, dx=\frac {3 \, {\left (e^{\left (4 i \, b x + 2 \, a + 2 i \, d\right )} + 2 \, e^{\left (2 i \, b x + 2 \, a\right )} + e^{\left (2 \, a - 2 i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} + i\right ) - 3 \, {\left (e^{\left (4 i \, b x + 2 \, a + 2 i \, d\right )} + 2 \, e^{\left (2 i \, b x + 2 \, a\right )} + e^{\left (2 \, a - 2 i \, d\right )}\right )} \log \left (e^{\left (i \, b x + i \, d\right )} - i\right ) + 10 i \, e^{\left (3 i \, b x + 2 \, a + i \, d\right )} + 6 i \, e^{\left (i \, b x + 2 \, a - i \, d\right )}}{2 \, {\left (b e^{\left (4 i \, b x + 4 i \, d\right )} + 2 \, b e^{\left (2 i \, b x + 2 i \, d\right )} + b\right )}} \] Input:
integrate(exp(2*a+2*I*b*x)*sec(b*x+d)^3,x, algorithm="fricas")
Output:
1/2*(3*(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) + I) - 3*(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) - I) + 10*I*e^(3*I*b*x + 2*a + I*d) + 6*I*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)
\[ \int e^{2 (a+i b x)} \sec ^3(d+b x) \, dx=e^{2 a} \int e^{2 i b x} \sec ^{3}{\left (b x + d \right )}\, dx \] Input:
integrate(exp(2*a+2*I*b*x)*sec(b*x+d)**3,x)
Output:
exp(2*a)*Integral(exp(2*I*b*x)*sec(b*x + d)**3, x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 735 vs. \(2 (84) = 168\).
Time = 0.17 (sec) , antiderivative size = 735, normalized size of antiderivative = 6.28 \[ \int e^{2 (a+i b x)} \sec ^3(d+b x) \, dx=\text {Too large to display} \] Input:
integrate(exp(2*a+2*I*b*x)*sec(b*x+d)^3,x, algorithm="maxima")
Output:
(6*((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(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 + co s(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)) + 20*cos(3*b*x + 2*d)*e^(2*a) + 12*cos(b*x)*e^(2*a) - 3*( (-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^(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*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) + si n(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)) + 20*I*e^(2*a)*sin(3*b*x + 2*d) + 12*I*e^(2*a)*sin(b*x))/(-4*I*b*cos(4*b*x + 5*d) - 8*I*b*cos(2*b*x + 3*d) - 4*I*b*cos(d) + 4*b*sin(4*b*x + 5*d) + 8*b*sin(2*b*x + 3*d) + 4*...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (84) = 168\).
Time = 0.16 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.81 \[ \int e^{2 (a+i b x)} \sec ^3(d+b x) \, dx=\frac {3 \, e^{\left (4 i \, b x + 2 \, a + 2 i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} - 1\right ) + 6 \, e^{\left (2 i \, b x + 2 \, a\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} - 1\right ) + 3 \, e^{\left (2 \, a - 2 i \, d\right )} \log \left (i \, e^{\left (i \, b x + i \, d\right )} - 1\right ) - 3 \, e^{\left (4 i \, b x + 2 \, a + 2 i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} - 1\right ) - 6 \, e^{\left (2 i \, b x + 2 \, a\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} - 1\right ) - 3 \, e^{\left (2 \, a - 2 i \, d\right )} \log \left (-i \, e^{\left (i \, b x + i \, d\right )} - 1\right ) + 10 i \, e^{\left (3 i \, b x + 2 \, a + i \, d\right )} + 6 i \, e^{\left (i \, b x + 2 \, a - i \, d\right )}}{2 \, b {\left (e^{\left (4 i \, b x + 4 i \, d\right )} + 2 \, e^{\left (2 i \, b x + 2 i \, d\right )} + 1\right )}} \] Input:
integrate(exp(2*a+2*I*b*x)*sec(b*x+d)^3,x, algorithm="giac")
Output:
1/2*(3*e^(4*I*b*x + 2*a + 2*I*d)*log(I*e^(I*b*x + I*d) - 1) + 6*e^(2*I*b*x + 2*a)*log(I*e^(I*b*x + I*d) - 1) + 3*e^(2*a - 2*I*d)*log(I*e^(I*b*x + I* d) - 1) - 3*e^(4*I*b*x + 2*a + 2*I*d)*log(-I*e^(I*b*x + I*d) - 1) - 6*e^(2 *I*b*x + 2*a)*log(-I*e^(I*b*x + I*d) - 1) - 3*e^(2*a - 2*I*d)*log(-I*e^(I* b*x + I*d) - 1) + 10*I*e^(3*I*b*x + 2*a + I*d) + 6*I*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. \[ \int e^{2 (a+i b x)} \sec ^3(d+b x) \, dx=\int \frac {{\mathrm {e}}^{2\,a+b\,x\,2{}\mathrm {i}}}{{\cos \left (d+b\,x\right )}^3} \,d x \] Input:
int(exp(2*a + b*x*2i)/cos(d + b*x)^3,x)
Output:
int(exp(2*a + b*x*2i)/cos(d + b*x)^3, x)
\[ \int e^{2 (a+i b x)} \sec ^3(d+b x) \, dx=\text {too large to display} \] Input:
int(exp(2*a+2*I*b*x)*sec(b*x+d)^3,x)
Output:
(e**(2*a)*( - 70*e**(2*b*i*x)*cos(b*x + d)*sec(b*x + d)**3*sin(b*x + d)**2 *tan((b*x + d)/2)**6*i + 210*e**(2*b*i*x)*cos(b*x + d)*sec(b*x + d)**3*sin (b*x + d)**2*tan((b*x + d)/2)**4*i - 210*e**(2*b*i*x)*cos(b*x + d)*sec(b*x + d)**3*sin(b*x + d)**2*tan((b*x + d)/2)**2*i + 70*e**(2*b*i*x)*cos(b*x + d)*sec(b*x + d)**3*sin(b*x + d)**2*i + 70*e**(2*b*i*x)*cos(b*x + d)*sec(b *x + d)**3*tan((b*x + d)/2)**6*i - 210*e**(2*b*i*x)*cos(b*x + d)*sec(b*x + d)**3*tan((b*x + d)/2)**4*i + 210*e**(2*b*i*x)*cos(b*x + d)*sec(b*x + d)* *3*tan((b*x + d)/2)**2*i - 70*e**(2*b*i*x)*cos(b*x + d)*sec(b*x + d)**3*i + 6*e**(2*b*i*x)*cos(b*x + d)*sin(b*x + d)**2*tan((b*x + d)/2)**6*i + 6*e* *(2*b*i*x)*cos(b*x + d)*sin(b*x + d)**2*tan((b*x + d)/2)**5 - 6*e**(2*b*i* x)*cos(b*x + d)*sin(b*x + d)**2*tan((b*x + d)/2)**4*i - 36*e**(2*b*i*x)*co s(b*x + d)*sin(b*x + d)**2*tan((b*x + d)/2)**3 - 42*e**(2*b*i*x)*cos(b*x + d)*sin(b*x + d)**2*tan((b*x + d)/2)**2*i + 102*e**(2*b*i*x)*cos(b*x + d)* sin(b*x + d)**2*tan((b*x + d)/2) + 138*e**(2*b*i*x)*cos(b*x + d)*sin(b*x + d)**2*i - 52*e**(2*b*i*x)*cos(b*x + d)*sin(b*x + d)*tan((b*x + d)/2)**6 + 156*e**(2*b*i*x)*cos(b*x + d)*sin(b*x + d)*tan((b*x + d)/2)**4 - 156*e**( 2*b*i*x)*cos(b*x + d)*sin(b*x + d)*tan((b*x + d)/2)**2 + 52*e**(2*b*i*x)*c os(b*x + d)*sin(b*x + d) - 65*e**(2*b*i*x)*cos(b*x + d)*tan((b*x + d)/2)** 6*i - 6*e**(2*b*i*x)*cos(b*x + d)*tan((b*x + d)/2)**5 + 183*e**(2*b*i*x)*c os(b*x + d)*tan((b*x + d)/2)**4*i + 36*e**(2*b*i*x)*cos(b*x + d)*tan((b...