Integrand size = 30, antiderivative size = 107 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^2 \, dx=-\frac {i e^{a-i d-i (d+b x)} (f-i g)^2}{4 b}+\frac {i e^{a-i d+3 i (d+b x)} (f+i g)^2}{12 b}-\frac {i e^{a-i d+i (d+b x)} \left (f^2+g^2\right )}{2 b} \] Output:
-1/4*I*exp(a-I*d-I*(b*x+d))*(f-I*g)^2/b+1/12*I*exp(a-I*d+3*I*(b*x+d))*(f+I *g)^2/b-1/2*I*exp(a-I*d+I*(b*x+d))*(f^2+g^2)/b
Time = 0.81 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^2 \, dx=-\frac {i e^{a-i d+3 i (d+b x)} (-i f+g)^2}{12 b}+\frac {i e^{a-i d-i (d+b x)} (i f+g)^2}{4 b}-\frac {i e^{a-i d+i (d+b x)} \left (f^2+g^2\right )}{2 b} \] Input:
Integrate[E^(a + I*b*x)*(g*Cos[d + b*x] + f*Sin[d + b*x])^2,x]
Output:
((-1/12*I)*E^(a - I*d + (3*I)*(d + b*x))*((-I)*f + g)^2)/b + ((I/4)*E^(a - I*d - I*(d + b*x))*(I*f + g)^2)/b - ((I/2)*E^(a - I*d + I*(d + b*x))*(f^2 + g^2))/b
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(227\) vs. \(2(107)=214\).
Time = 0.49 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {7293, 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.
\(\displaystyle \int e^{a+i b x} (f \sin (b x+d)+g \cos (b x+d))^2 \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (f^2 e^{a+i b x} \sin ^2(b x+d)+f g e^{a+i b x} \sin (2 b x+2 d)+g^2 e^{a+i b x} \cos ^2(b x+d)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i f^2 e^{a+i b x} \sin ^2(b x+d)}{3 b}-\frac {2 f^2 e^{a+i b x} \sin (b x+d) \cos (b x+d)}{3 b}+\frac {i f g e^{a+i b x} \sin (2 b x+2 d)}{3 b}-\frac {2 f g e^{a+i b x} \cos (2 b x+2 d)}{3 b}+\frac {i g^2 e^{a+i b x} \cos ^2(b x+d)}{3 b}+\frac {2 g^2 e^{a+i b x} \sin (b x+d) \cos (b x+d)}{3 b}-\frac {2 i f^2 e^{a+i b x}}{3 b}-\frac {2 i g^2 e^{a+i b x}}{3 b}\) |
Input:
Int[E^(a + I*b*x)*(g*Cos[d + b*x] + f*Sin[d + b*x])^2,x]
Output:
(((-2*I)/3)*E^(a + I*b*x)*f^2)/b - (((2*I)/3)*E^(a + I*b*x)*g^2)/b + ((I/3 )*E^(a + I*b*x)*g^2*Cos[d + b*x]^2)/b - (2*E^(a + I*b*x)*f*g*Cos[2*d + 2*b *x])/(3*b) - (2*E^(a + I*b*x)*f^2*Cos[d + b*x]*Sin[d + b*x])/(3*b) + (2*E^ (a + I*b*x)*g^2*Cos[d + b*x]*Sin[d + b*x])/(3*b) + ((I/3)*E^(a + I*b*x)*f^ 2*Sin[d + b*x]^2)/b + ((I/3)*E^(a + I*b*x)*f*g*Sin[2*d + 2*b*x])/b
Time = 0.90 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.38
method | result | size |
norman | \(\frac {-\frac {\left (4 i f^{2}+2 i g^{2}+4 f g \right ) {\mathrm e}^{i b x +a}}{3 b}-\frac {2 g^{2} {\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{3}}{b}+\frac {2 \left (-i g^{2}+2 f g \right ) {\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}}{b}-\frac {2 \left (-4 i f g +4 f^{2}-g^{2}\right ) {\mathrm e}^{i b x +a} \tan \left (\frac {b x}{2}+\frac {d}{2}\right )}{3 b}}{\left (1+\tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}\right )^{2}}\) | \(148\) |
parallelrisch | \(-\frac {2 \,{\mathrm e}^{i b x +a} \left (3 i \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2} g^{2}+3 \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{3} g^{2}-4 i \tan \left (\frac {b x}{2}+\frac {d}{2}\right ) f g -6 \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2} f g +2 i f^{2}+i g^{2}+4 \tan \left (\frac {b x}{2}+\frac {d}{2}\right ) f^{2}-\tan \left (\frac {b x}{2}+\frac {d}{2}\right ) g^{2}+2 f g \right )}{3 b \left (\tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{4}+2 \tan \left (\frac {b x}{2}+\frac {d}{2}\right )^{2}+1\right )}\) | \(149\) |
parts | \(g^{2} \left (-\frac {i {\mathrm e}^{i b x +a}}{2 b}+\frac {i {\mathrm e}^{i b x +a} \cos \left (2 b x +2 d \right )}{6 b}+\frac {\sin \left (2 b x +2 d \right ) {\mathrm e}^{i b x +a}}{3 b}\right )+f^{2} \left (-\frac {i {\mathrm e}^{i b x +a}}{2 b}-\frac {i {\mathrm e}^{i b x +a} \cos \left (2 b x +2 d \right )}{6 b}-\frac {\sin \left (2 b x +2 d \right ) {\mathrm e}^{i b x +a}}{3 b}\right )+2 f g \left (-\frac {{\mathrm e}^{i b x +a} \cos \left (2 b x +2 d \right )}{3 b}+\frac {i {\mathrm e}^{i b x +a} \sin \left (2 b x +2 d \right )}{6 b}\right )\) | \(180\) |
default | \(-\frac {i g^{2} {\mathrm e}^{i b x +a}}{2 b}-\frac {i f^{2} {\mathrm e}^{i b x +a}}{2 b}-\frac {f^{2} \left (\frac {i {\mathrm e}^{i b x +a} \cos \left (2 b x +2 d \right )}{3 b}+\frac {2 \sin \left (2 b x +2 d \right ) {\mathrm e}^{i b x +a}}{3 b}\right )}{2}+\frac {g^{2} \left (\frac {i {\mathrm e}^{i b x +a} \cos \left (2 b x +2 d \right )}{3 b}+\frac {2 \sin \left (2 b x +2 d \right ) {\mathrm e}^{i b x +a}}{3 b}\right )}{2}+f g \left (-\frac {2 \,{\mathrm e}^{i b x +a} \cos \left (2 b x +2 d \right )}{3 b}+\frac {i {\mathrm e}^{i b x +a} \sin \left (2 b x +2 d \right )}{3 b}\right )\) | \(187\) |
orering | \(-\frac {i {\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{2}}{3 b}-\frac {i b \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{2}+2 \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right ) \left (-g b \sin \left (b x +d \right )+f b \cos \left (b x +d \right )\right )}{b^{2}}-\frac {i \left (-b^{2} {\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right )^{2}+4 i b \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right ) \left (-g b \sin \left (b x +d \right )+f b \cos \left (b x +d \right )\right )+2 \,{\mathrm e}^{i b x +a} \left (-g b \sin \left (b x +d \right )+f b \cos \left (b x +d \right )\right )^{2}+2 \,{\mathrm e}^{i b x +a} \left (g \cos \left (b x +d \right )+f \sin \left (b x +d \right )\right ) \left (-g \,b^{2} \cos \left (b x +d \right )-f \,b^{2} \sin \left (b x +d \right )\right )\right )}{3 b^{3}}\) | \(291\) |
Input:
int(exp(a+I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^2,x,method=_RETURNVERBOSE)
Output:
(-1/3*(4*I*f^2+2*I*g^2+4*f*g)/b*exp(a+I*b*x)-2*g^2/b*exp(a+I*b*x)*tan(1/2* b*x+1/2*d)^3+2*(-I*g^2+2*f*g)/b*exp(a+I*b*x)*tan(1/2*b*x+1/2*d)^2-2/3*(-4* I*f*g+4*f^2-g^2)/b*exp(a+I*b*x)*tan(1/2*b*x+1/2*d))/(1+tan(1/2*b*x+1/2*d)^ 2)^2
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.81 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^2 \, dx=\frac {{\left ({\left (i \, f^{2} - 2 \, f g - i \, g^{2}\right )} e^{\left (4 i \, b x + a + 3 i \, d\right )} - 6 \, {\left (i \, f^{2} + i \, g^{2}\right )} e^{\left (2 i \, b x + a + i \, d\right )} - 3 \, {\left (i \, f^{2} + 2 \, f g - i \, g^{2}\right )} e^{\left (a - i \, d\right )}\right )} e^{\left (-i \, b x - i \, d\right )}}{12 \, b} \] Input:
integrate(exp(a+I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^2,x, algorithm="fricas" )
Output:
1/12*((I*f^2 - 2*f*g - I*g^2)*e^(4*I*b*x + a + 3*I*d) - 6*(I*f^2 + I*g^2)* e^(2*I*b*x + a + I*d) - 3*(I*f^2 + 2*f*g - I*g^2)*e^(a - I*d))*e^(-I*b*x - I*d)/b
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (76) = 152\).
Time = 0.27 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.72 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^2 \, dx=\begin {cases} \frac {\left (\left (- 48 i b^{2} f^{2} e^{a} e^{2 i d} - 48 i b^{2} g^{2} e^{a} e^{2 i d}\right ) e^{i b x} + \left (- 24 i b^{2} f^{2} e^{a} - 48 b^{2} f g e^{a} + 24 i b^{2} g^{2} e^{a}\right ) e^{- i b x} + \left (8 i b^{2} f^{2} e^{a} e^{4 i d} - 16 b^{2} f g e^{a} e^{4 i d} - 8 i b^{2} g^{2} e^{a} e^{4 i d}\right ) e^{3 i b x}\right ) e^{- 2 i d}}{96 b^{3}} & \text {for}\: b^{3} e^{2 i d} \neq 0 \\\frac {x \left (- f^{2} e^{a} e^{4 i d} + 2 f^{2} e^{a} e^{2 i d} - f^{2} e^{a} - 2 i f g e^{a} e^{4 i d} + 2 i f g e^{a} + g^{2} e^{a} e^{4 i d} + 2 g^{2} e^{a} e^{2 i d} + g^{2} e^{a}\right ) e^{- 2 i d}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(exp(a+I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))**2,x)
Output:
Piecewise((((-48*I*b**2*f**2*exp(a)*exp(2*I*d) - 48*I*b**2*g**2*exp(a)*exp (2*I*d))*exp(I*b*x) + (-24*I*b**2*f**2*exp(a) - 48*b**2*f*g*exp(a) + 24*I* b**2*g**2*exp(a))*exp(-I*b*x) + (8*I*b**2*f**2*exp(a)*exp(4*I*d) - 16*b**2 *f*g*exp(a)*exp(4*I*d) - 8*I*b**2*g**2*exp(a)*exp(4*I*d))*exp(3*I*b*x))*ex p(-2*I*d)/(96*b**3), Ne(b**3*exp(2*I*d), 0)), (x*(-f**2*exp(a)*exp(4*I*d) + 2*f**2*exp(a)*exp(2*I*d) - f**2*exp(a) - 2*I*f*g*exp(a)*exp(4*I*d) + 2*I *f*g*exp(a) + g**2*exp(a)*exp(4*I*d) + 2*g**2*exp(a)*exp(2*I*d) + g**2*exp (a))*exp(-2*I*d)/4, True))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (64) = 128\).
Time = 0.04 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.76 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^2 \, dx=-\frac {{\left (-i \, \cos \left (3 \, b x + 2 \, d\right ) e^{a} + 3 i \, \cos \left (b x + 2 \, d\right ) e^{a} + 6 i \, \cos \left (b x\right ) e^{a} + e^{a} \sin \left (3 \, b x + 2 \, d\right ) + 3 \, e^{a} \sin \left (b x + 2 \, d\right ) - 6 \, e^{a} \sin \left (b x\right )\right )} f^{2}}{12 \, b} + \frac {{\left (-i \, \cos \left (3 \, b x + 2 \, d\right ) e^{a} + 3 i \, \cos \left (b x + 2 \, d\right ) e^{a} - 6 i \, \cos \left (b x\right ) e^{a} + e^{a} \sin \left (3 \, b x + 2 \, d\right ) + 3 \, e^{a} \sin \left (b x + 2 \, d\right ) + 6 \, e^{a} \sin \left (b x\right )\right )} g^{2}}{12 \, b} - \frac {{\left (2 \, b \cos \left (2 \, b x + 2 \, d\right ) - i \, b \sin \left (2 \, b x + 2 \, d\right )\right )} f g e^{\left (i \, b x + a\right )}}{3 \, b^{2}} \] Input:
integrate(exp(a+I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^2,x, algorithm="maxima" )
Output:
-1/12*(-I*cos(3*b*x + 2*d)*e^a + 3*I*cos(b*x + 2*d)*e^a + 6*I*cos(b*x)*e^a + e^a*sin(3*b*x + 2*d) + 3*e^a*sin(b*x + 2*d) - 6*e^a*sin(b*x))*f^2/b + 1 /12*(-I*cos(3*b*x + 2*d)*e^a + 3*I*cos(b*x + 2*d)*e^a - 6*I*cos(b*x)*e^a + e^a*sin(3*b*x + 2*d) + 3*e^a*sin(b*x + 2*d) + 6*e^a*sin(b*x))*g^2/b - 1/3 *(2*b*cos(2*b*x + 2*d) - I*b*sin(2*b*x + 2*d))*f*g*e^(I*b*x + a)/b^2
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (64) = 128\).
Time = 0.13 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.47 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^2 \, dx=\frac {12 \, {\left (f^{2} + g^{2}\right )} e^{a} \sin \left (b x\right ) + i \, {\left ({\left (f^{2} + 2 i \, f g - g^{2}\right )} e^{\left (3 i \, b x + 2 i \, d\right )} + {\left (f^{2} + 2 i \, f g - g^{2}\right )} e^{\left (-3 i \, b x - 2 i \, d\right )}\right )} e^{a} - {\left ({\left (-i \, f^{2} + 2 \, f g + i \, g^{2}\right )} e^{\left (3 i \, b x + 2 i \, d\right )} + {\left (i \, f^{2} - 2 \, f g - i \, g^{2}\right )} e^{\left (-3 i \, b x - 2 i \, d\right )}\right )} e^{a} - 3 i \, {\left ({\left (f^{2} - 2 i \, f g - g^{2}\right )} e^{\left (i \, b x + 2 i \, d\right )} + {\left (f^{2} - 2 i \, f g - g^{2}\right )} e^{\left (-i \, b x - 2 i \, d\right )}\right )} e^{a} - 3 \, {\left ({\left (-i \, f^{2} - 2 \, f g + i \, g^{2}\right )} e^{\left (i \, b x + 2 i \, d\right )} + {\left (i \, f^{2} + 2 \, f g - i \, g^{2}\right )} e^{\left (-i \, b x - 2 i \, d\right )}\right )} e^{a} - 6 i \, {\left ({\left (f^{2} + g^{2}\right )} e^{\left (i \, b x\right )} + {\left (f^{2} + g^{2}\right )} e^{\left (-i \, b x\right )}\right )} e^{a}}{24 \, b} \] Input:
integrate(exp(a+I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^2,x, algorithm="giac")
Output:
1/24*(12*(f^2 + g^2)*e^a*sin(b*x) + I*((f^2 + 2*I*f*g - g^2)*e^(3*I*b*x + 2*I*d) + (f^2 + 2*I*f*g - g^2)*e^(-3*I*b*x - 2*I*d))*e^a - ((-I*f^2 + 2*f* g + I*g^2)*e^(3*I*b*x + 2*I*d) + (I*f^2 - 2*f*g - I*g^2)*e^(-3*I*b*x - 2*I *d))*e^a - 3*I*((f^2 - 2*I*f*g - g^2)*e^(I*b*x + 2*I*d) + (f^2 - 2*I*f*g - g^2)*e^(-I*b*x - 2*I*d))*e^a - 3*((-I*f^2 - 2*f*g + I*g^2)*e^(I*b*x + 2*I *d) + (I*f^2 + 2*f*g - I*g^2)*e^(-I*b*x - 2*I*d))*e^a - 6*I*((f^2 + g^2)*e ^(I*b*x) + (f^2 + g^2)*e^(-I*b*x))*e^a)/b
Time = 16.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.05 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^2 \, dx=-\frac {{\cos \left (d+b\,x\right )}^2\,{\mathrm {e}}^{a+b\,x\,1{}\mathrm {i}}\,\left (2\,f^2-f\,g\,2{}\mathrm {i}+g^2\right )\,1{}\mathrm {i}}{3\,b}-\frac {{\mathrm {e}}^{a+b\,x\,1{}\mathrm {i}}\,{\sin \left (d+b\,x\right )}^2\,\left (f^2+f\,g\,2{}\mathrm {i}+2\,g^2\right )\,1{}\mathrm {i}}{3\,b}+\frac {2\,\cos \left (d+b\,x\right )\,{\mathrm {e}}^{a+b\,x\,1{}\mathrm {i}}\,\sin \left (d+b\,x\right )\,\left (-f^2+f\,g\,1{}\mathrm {i}+g^2\right )}{3\,b} \] Input:
int(exp(a + b*x*1i)*(g*cos(d + b*x) + f*sin(d + b*x))^2,x)
Output:
(2*cos(d + b*x)*exp(a + b*x*1i)*sin(d + b*x)*(f*g*1i - f^2 + g^2))/(3*b) - (exp(a + b*x*1i)*sin(d + b*x)^2*(f*g*2i + f^2 + 2*g^2)*1i)/(3*b) - (cos(d + b*x)^2*exp(a + b*x*1i)*(2*f^2 - f*g*2i + g^2)*1i)/(3*b)
Time = 0.16 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.36 \[ \int e^{a+i b x} (g \cos (d+b x)+f \sin (d+b x))^2 \, dx=\frac {e^{b i x +a} \left (-2 \cos \left (b x +d \right )^{2} f^{2} i -2 \cos \left (b x +d \right )^{2} f g -\cos \left (b x +d \right )^{2} g^{2} i -2 \cos \left (b x +d \right ) \sin \left (b x +d \right ) f^{2}+2 \cos \left (b x +d \right ) \sin \left (b x +d \right ) f g i +2 \cos \left (b x +d \right ) \sin \left (b x +d \right ) g^{2}-\sin \left (b x +d \right )^{2} f^{2} i +2 \sin \left (b x +d \right )^{2} f g -2 \sin \left (b x +d \right )^{2} g^{2} i \right )}{3 b} \] Input:
int(exp(a+I*b*x)*(g*cos(b*x+d)+f*sin(b*x+d))^2,x)
Output:
(e**(a + b*i*x)*( - 2*cos(b*x + d)**2*f**2*i - 2*cos(b*x + d)**2*f*g - cos (b*x + d)**2*g**2*i - 2*cos(b*x + d)*sin(b*x + d)*f**2 + 2*cos(b*x + d)*si n(b*x + d)*f*g*i + 2*cos(b*x + d)*sin(b*x + d)*g**2 - sin(b*x + d)**2*f**2 *i + 2*sin(b*x + d)**2*f*g - 2*sin(b*x + d)**2*g**2*i))/(3*b)