Integrand size = 30, antiderivative size = 61 \[ \int \frac {e^{a+i b x}}{(g \cos (d+b x)+f \sin (d+b x))^3} \, dx=\frac {2 e^{a-i d+4 i (d+b x)}}{b (f-i g) \left (f-e^{2 i (d+b x)} (f+i g)-i g\right )^2} \] Output:
2*exp(a-I*d+4*I*(b*x+d))/b/(f-I*g)/(f-exp(2*I*(b*x+d))*(f+I*g)-I*g)^2
Time = 0.46 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.59 \[ \int \frac {e^{a+i b x}}{(g \cos (d+b x)+f \sin (d+b x))^3} \, dx=\frac {2 e^{a-i d} \left (\left (-1+2 e^{2 i (d+b x)}\right ) f+i \left (1+2 e^{2 i (d+b x)}\right ) g\right )}{b (f+i g)^2 \left (\left (-1+e^{2 i (d+b x)}\right ) f+i \left (1+e^{2 i (d+b x)}\right ) g\right )^2} \] Input:
Integrate[E^(a + I*b*x)/(g*Cos[d + b*x] + f*Sin[d + b*x])^3,x]
Output:
(2*E^(a - I*d)*((-1 + 2*E^((2*I)*(d + b*x)))*f + I*(1 + 2*E^((2*I)*(d + b* x)))*g))/(b*(f + I*g)^2*((-1 + E^((2*I)*(d + b*x)))*f + I*(1 + E^((2*I)*(d + b*x)))*g)^2)
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.
\(\displaystyle \int \frac {e^{a+i b x}}{(f \sin (b x+d)+g \cos (b x+d))^3} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {e^{a+i b x}}{(f \sin (b x+d)+g \cos (b x+d))^3}dx\) |
Input:
Int[E^(a + I*b*x)/(g*Cos[d + b*x] + f*Sin[d + b*x])^3,x]
Output:
$Aborted
Time = 2.71 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.98
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{4 i b x +3 i d +a}}{b \left (f \,{\mathrm e}^{2 i \left (b x +d \right )}+i g \,{\mathrm e}^{2 i \left (b x +d \right )}-f +i g \right )^{2} \left (-i g +f \right )}\) | \(60\) |
parallelrisch | \(\frac {{\mathrm e}^{i b x +a} \left (i \sin \left (b x +d \right )+\cos \left (b x +d \right )\right )}{b \left (i g -f \right ) \left (g^{2} \cos \left (2 b x +2 d \right )+g^{2}+2 f g \sin \left (2 b x +2 d \right )+f^{2}-f^{2} \cos \left (2 b x +2 d \right )\right )}\) | \(88\) |
norman | \(\frac {-\frac {{\mathrm e}^{i b x +a}}{2 \left (-i b g +f b \right )}+\frac {{\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{4}}{2 b \left (-i g +f \right )}-\frac {i {\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )}{b \left (-i g +f \right )}-\frac {i {\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{3}}{b \left (-i g +f \right )}}{\left (-\tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2} g +2 f \tan \left (\frac {b x}{2}+\frac {d}{2}\right )+g \right )^{2}}\) | \(150\) |
Input:
int(exp(a+I*b*x)/(g*cos(b*x+d)+f*sin(b*x+d))^3,x,method=_RETURNVERBOSE)
Output:
2/b/(f*exp(2*I*(b*x+d))+I*g*exp(2*I*(b*x+d))-f+I*g)^2/(f-I*g)*exp(4*I*b*x+ 3*I*d+a)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (46) = 92\).
Time = 0.07 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.23 \[ \int \frac {e^{a+i b x}}{(g \cos (d+b x)+f \sin (d+b x))^3} \, dx=\frac {2 \, {\left (2 \, {\left (f + i \, g\right )} e^{\left (2 i \, b x + a + i \, d\right )} - {\left (f - i \, g\right )} e^{\left (a - i \, d\right )}\right )}}{b f^{4} + 2 \, b f^{2} g^{2} + b g^{4} + {\left (b f^{4} + 4 i \, b f^{3} g - 6 \, b f^{2} g^{2} - 4 i \, b f g^{3} + b g^{4}\right )} e^{\left (4 i \, b x + 4 i \, d\right )} - 2 \, {\left (b f^{4} + 2 i \, b f^{3} g + 2 i \, b f g^{3} - b g^{4}\right )} e^{\left (2 i \, b x + 2 i \, d\right )}} \] Input:
integrate(exp(a+I*b*x)/(g*cos(b*x+d)+f*sin(b*x+d))^3,x, algorithm="fricas" )
Output:
2*(2*(f + I*g)*e^(2*I*b*x + a + I*d) - (f - I*g)*e^(a - I*d))/(b*f^4 + 2*b *f^2*g^2 + b*g^4 + (b*f^4 + 4*I*b*f^3*g - 6*b*f^2*g^2 - 4*I*b*f*g^3 + b*g^ 4)*e^(4*I*b*x + 4*I*d) - 2*(b*f^4 + 2*I*b*f^3*g + 2*I*b*f*g^3 - b*g^4)*e^( 2*I*b*x + 2*I*d))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (44) = 88\).
Time = 1.61 (sec) , antiderivative size = 240, normalized size of antiderivative = 3.93 \[ \int \frac {e^{a+i b x}}{(g \cos (d+b x)+f \sin (d+b x))^3} \, dx=\frac {- 2 f e^{a} + 2 i g e^{a} + \left (4 f e^{a} e^{2 i d} + 4 i g e^{a} e^{2 i d}\right ) e^{2 i b x}}{b f^{4} e^{i d} + 2 b f^{2} g^{2} e^{i d} + b g^{4} e^{i d} + \left (- 2 b f^{4} e^{3 i d} - 4 i b f^{3} g e^{3 i d} - 4 i b f g^{3} e^{3 i d} + 2 b g^{4} e^{3 i d}\right ) e^{2 i b x} + \left (b f^{4} e^{5 i d} + 4 i b f^{3} g e^{5 i d} - 6 b f^{2} g^{2} e^{5 i d} - 4 i b f g^{3} e^{5 i d} + b g^{4} e^{5 i d}\right ) e^{4 i b x}} \] Input:
integrate(exp(a+I*b*x)/(g*cos(b*x+d)+f*sin(b*x+d))**3,x)
Output:
(-2*f*exp(a) + 2*I*g*exp(a) + (4*f*exp(a)*exp(2*I*d) + 4*I*g*exp(a)*exp(2* I*d))*exp(2*I*b*x))/(b*f**4*exp(I*d) + 2*b*f**2*g**2*exp(I*d) + b*g**4*exp (I*d) + (-2*b*f**4*exp(3*I*d) - 4*I*b*f**3*g*exp(3*I*d) - 4*I*b*f*g**3*exp (3*I*d) + 2*b*g**4*exp(3*I*d))*exp(2*I*b*x) + (b*f**4*exp(5*I*d) + 4*I*b*f **3*g*exp(5*I*d) - 6*b*f**2*g**2*exp(5*I*d) - 4*I*b*f*g**3*exp(5*I*d) + b* g**4*exp(5*I*d))*exp(4*I*b*x))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (46) = 92\).
Time = 0.07 (sec) , antiderivative size = 271, normalized size of antiderivative = 4.44 \[ \int \frac {e^{a+i b x}}{(g \cos (d+b x)+f \sin (d+b x))^3} \, dx=-\frac {2 \, {\left (2 \, {\left (i \, f e^{a} - g e^{a}\right )} \cos \left (2 \, b x + 2 \, d\right ) - i \, f e^{a} - g e^{a} - 2 \, {\left (f e^{a} + i \, g e^{a}\right )} \sin \left (2 \, b x + 2 \, d\right )\right )}}{{\left (-i \, b \cos \left (d\right ) + b \sin \left (d\right )\right )} f^{4} - 2 \, {\left (i \, b \cos \left (d\right ) - b \sin \left (d\right )\right )} f^{2} g^{2} + {\left (-i \, b \cos \left (d\right ) + b \sin \left (d\right )\right )} g^{4} + {\left (-i \, b f^{4} + 4 \, b f^{3} g + 6 i \, b f^{2} g^{2} - 4 \, b f g^{3} - i \, b g^{4}\right )} \cos \left (4 \, b x + 5 \, d\right ) - 2 \, {\left (-i \, b f^{4} + 2 \, b f^{3} g + 2 \, b f g^{3} + i \, b g^{4}\right )} \cos \left (2 \, b x + 3 \, d\right ) + {\left (b f^{4} + 4 i \, b f^{3} g - 6 \, b f^{2} g^{2} - 4 i \, b f g^{3} + b g^{4}\right )} \sin \left (4 \, b x + 5 \, d\right ) - 2 \, {\left (b f^{4} + 2 i \, b f^{3} g + 2 i \, b f g^{3} - b g^{4}\right )} \sin \left (2 \, b x + 3 \, d\right )} \] Input:
integrate(exp(a+I*b*x)/(g*cos(b*x+d)+f*sin(b*x+d))^3,x, algorithm="maxima" )
Output:
-2*(2*(I*f*e^a - g*e^a)*cos(2*b*x + 2*d) - I*f*e^a - g*e^a - 2*(f*e^a + I* g*e^a)*sin(2*b*x + 2*d))/((-I*b*cos(d) + b*sin(d))*f^4 - 2*(I*b*cos(d) - b *sin(d))*f^2*g^2 + (-I*b*cos(d) + b*sin(d))*g^4 + (-I*b*f^4 + 4*b*f^3*g + 6*I*b*f^2*g^2 - 4*b*f*g^3 - I*b*g^4)*cos(4*b*x + 5*d) - 2*(-I*b*f^4 + 2*b* f^3*g + 2*b*f*g^3 + I*b*g^4)*cos(2*b*x + 3*d) + (b*f^4 + 4*I*b*f^3*g - 6*b *f^2*g^2 - 4*I*b*f*g^3 + b*g^4)*sin(4*b*x + 5*d) - 2*(b*f^4 + 2*I*b*f^3*g + 2*I*b*f*g^3 - b*g^4)*sin(2*b*x + 3*d))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (46) = 92\).
Time = 0.22 (sec) , antiderivative size = 204, normalized size of antiderivative = 3.34 \[ \int \frac {e^{a+i b x}}{(g \cos (d+b x)+f \sin (d+b x))^3} \, dx=\frac {2 \, {\left (2 \, f e^{\left (2 i \, b x + a + 2 i \, d\right )} + 2 i \, g e^{\left (2 i \, b x + a + 2 i \, d\right )} - f e^{a} + i \, g e^{a}\right )}}{{\left (f^{4} e^{\left (4 i \, b x + 5 i \, d\right )} + 4 i \, f^{3} g e^{\left (4 i \, b x + 5 i \, d\right )} - 6 \, f^{2} g^{2} e^{\left (4 i \, b x + 5 i \, d\right )} - 4 i \, f g^{3} e^{\left (4 i \, b x + 5 i \, d\right )} + g^{4} e^{\left (4 i \, b x + 5 i \, d\right )} - 2 \, f^{4} e^{\left (2 i \, b x + 3 i \, d\right )} - 4 i \, f^{3} g e^{\left (2 i \, b x + 3 i \, d\right )} - 4 i \, f g^{3} e^{\left (2 i \, b x + 3 i \, d\right )} + 2 \, g^{4} e^{\left (2 i \, b x + 3 i \, d\right )} + f^{4} e^{\left (i \, d\right )} + 2 \, f^{2} g^{2} e^{\left (i \, d\right )} + g^{4} e^{\left (i \, d\right )}\right )} b} \] Input:
integrate(exp(a+I*b*x)/(g*cos(b*x+d)+f*sin(b*x+d))^3,x, algorithm="giac")
Output:
2*(2*f*e^(2*I*b*x + a + 2*I*d) + 2*I*g*e^(2*I*b*x + a + 2*I*d) - f*e^a + I *g*e^a)/((f^4*e^(4*I*b*x + 5*I*d) + 4*I*f^3*g*e^(4*I*b*x + 5*I*d) - 6*f^2* g^2*e^(4*I*b*x + 5*I*d) - 4*I*f*g^3*e^(4*I*b*x + 5*I*d) + g^4*e^(4*I*b*x + 5*I*d) - 2*f^4*e^(2*I*b*x + 3*I*d) - 4*I*f^3*g*e^(2*I*b*x + 3*I*d) - 4*I* f*g^3*e^(2*I*b*x + 3*I*d) + 2*g^4*e^(2*I*b*x + 3*I*d) + f^4*e^(I*d) + 2*f^ 2*g^2*e^(I*d) + g^4*e^(I*d))*b)
Timed out. \[ \int \frac {e^{a+i b x}}{(g \cos (d+b x)+f \sin (d+b x))^3} \, dx=\int \frac {{\mathrm {e}}^{a+b\,x\,1{}\mathrm {i}}}{{\left (g\,\cos \left (d+b\,x\right )+f\,\sin \left (d+b\,x\right )\right )}^3} \,d x \] Input:
int(exp(a + b*x*1i)/(g*cos(d + b*x) + f*sin(d + b*x))^3,x)
Output:
int(exp(a + b*x*1i)/(g*cos(d + b*x) + f*sin(d + b*x))^3, x)
Time = 0.20 (sec) , antiderivative size = 874, normalized size of antiderivative = 14.33 \[ \int \frac {e^{a+i b x}}{(g \cos (d+b x)+f \sin (d+b x))^3} \, dx =\text {Too large to display} \] Input:
int(exp(a+I*b*x)/(g*cos(b*x+d)+f*sin(b*x+d))^3,x)
Output:
(e**(a + b*i*x)*( - 2*cos(b*x + d)*f**10*i - 19*cos(b*x + d)*f**9*g + 81*c os(b*x + d)*f**8*g**2*i + 204*cos(b*x + d)*f**7*g**3 - 336*cos(b*x + d)*f* *6*g**4*i - 378*cos(b*x + d)*f**5*g**5 + 294*cos(b*x + d)*f**4*g**6*i + 15 6*cos(b*x + d)*f**3*g**7 - 54*cos(b*x + d)*f**2*g**8*i - 11*cos(b*x + d)*f *g**9 + cos(b*x + d)*g**10*i + 2*sin(b*x + d)*f**10 - 19*sin(b*x + d)*f**9 *g*i - 81*sin(b*x + d)*f**8*g**2 + 204*sin(b*x + d)*f**7*g**3*i + 336*sin( b*x + d)*f**6*g**4 - 378*sin(b*x + d)*f**5*g**5*i - 294*sin(b*x + d)*f**4* g**6 + 156*sin(b*x + d)*f**3*g**7*i + 54*sin(b*x + d)*f**2*g**8 - 11*sin(b *x + d)*f*g**9*i - sin(b*x + d)*g**10))/(2*b*(4*cos(b*x + d)*sin(b*x + d)* f**12*g*i + 42*cos(b*x + d)*sin(b*x + d)*f**11*g**2 - 200*cos(b*x + d)*sin (b*x + d)*f**10*g**3*i - 570*cos(b*x + d)*sin(b*x + d)*f**9*g**4 + 1080*co s(b*x + d)*sin(b*x + d)*f**8*g**5*i + 1428*cos(b*x + d)*sin(b*x + d)*f**7* g**6 - 1344*cos(b*x + d)*sin(b*x + d)*f**6*g**7*i - 900*cos(b*x + d)*sin(b *x + d)*f**5*g**8 + 420*cos(b*x + d)*sin(b*x + d)*f**4*g**9*i + 130*cos(b* x + d)*sin(b*x + d)*f**3*g**10 - 24*cos(b*x + d)*sin(b*x + d)*f**2*g**11*i - 2*cos(b*x + d)*sin(b*x + d)*f*g**12 + 2*sin(b*x + d)**2*f**13*i + 21*si n(b*x + d)**2*f**12*g - 102*sin(b*x + d)**2*f**11*g**2*i - 306*sin(b*x + d )**2*f**10*g**3 + 640*sin(b*x + d)**2*f**9*g**4*i + 999*sin(b*x + d)**2*f* *8*g**5 - 1212*sin(b*x + d)**2*f**7*g**6*i - 1164*sin(b*x + d)**2*f**6*g** 7 + 882*sin(b*x + d)**2*f**5*g**8*i + 515*sin(b*x + d)**2*f**4*g**9 - 2...