Integrand size = 24, antiderivative size = 129 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=-\frac {3 d e^{a+b x} \cos (2 c+2 d x)}{16 \left (b^2+4 d^2\right )}+\frac {3 d e^{a+b x} \cos (6 c+6 d x)}{16 \left (b^2+36 d^2\right )}+\frac {3 b e^{a+b x} \sin (2 c+2 d x)}{32 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (6 c+6 d x)}{32 \left (b^2+36 d^2\right )} \]
-3/16*d*exp(b*x+a)*cos(2*d*x+2*c)/(b^2+4*d^2)+3/16*d*exp(b*x+a)*cos(6*d*x+ 6*c)/(b^2+36*d^2)+3/32*b*exp(b*x+a)*sin(2*d*x+2*c)/(b^2+4*d^2)-1/32*b*exp( b*x+a)*sin(6*d*x+6*c)/(b^2+36*d^2)
Time = 0.79 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.86 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=\frac {e^{a+b x} \left (-6 d \left (b^2+36 d^2\right ) \cos (2 (c+d x))+6 d \left (b^2+4 d^2\right ) \cos (6 (c+d x))-2 b \left (-b^2-52 d^2+\left (b^2+4 d^2\right ) \cos (4 (c+d x))\right ) \sin (2 (c+d x))\right )}{32 \left (b^4+40 b^2 d^2+144 d^4\right )} \]
(E^(a + b*x)*(-6*d*(b^2 + 36*d^2)*Cos[2*(c + d*x)] + 6*d*(b^2 + 4*d^2)*Cos [6*(c + d*x)] - 2*b*(-b^2 - 52*d^2 + (b^2 + 4*d^2)*Cos[4*(c + d*x)])*Sin[2 *(c + d*x)]))/(32*(b^4 + 40*b^2*d^2 + 144*d^4))
Time = 0.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4972, 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.
| \(\displaystyle \int e^{a+b x} \sin ^3(c+d x) \cos ^3(c+d x) \, dx\) |
| \(\Big \downarrow \) 4972 |
| \(\displaystyle \int \left (\frac {3}{32} e^{a+b x} \sin (2 c+2 d x)-\frac {1}{32} e^{a+b x} \sin (6 c+6 d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b e^{a+b x} \sin (2 c+2 d x)}{32 \left (b^2+4 d^2\right )}-\frac {b e^{a+b x} \sin (6 c+6 d x)}{32 \left (b^2+36 d^2\right )}-\frac {3 d e^{a+b x} \cos (2 c+2 d x)}{16 \left (b^2+4 d^2\right )}+\frac {3 d e^{a+b x} \cos (6 c+6 d x)}{16 \left (b^2+36 d^2\right )}\) |
(-3*d*E^(a + b*x)*Cos[2*c + 2*d*x])/(16*(b^2 + 4*d^2)) + (3*d*E^(a + b*x)* Cos[6*c + 6*d*x])/(16*(b^2 + 36*d^2)) + (3*b*E^(a + b*x)*Sin[2*c + 2*d*x]) /(32*(b^2 + 4*d^2)) - (b*E^(a + b*x)*Sin[6*c + 6*d*x])/(32*(b^2 + 36*d^2))
3.1.46.3.1 Defintions of rubi rules used
Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_ .) + (e_.)*(x_)]^(m_.), x_Symbol] :> Int[ExpandTrigReduce[F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Time = 1.68 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84
| method | result | size |
| parallelrisch | \(-\frac {\left (\left (b^{3}+4 b \,d^{2}\right ) \sin \left (6 d x +6 c \right )+\left (-6 b^{2} d -24 d^{3}\right ) \cos \left (6 d x +6 c \right )-3 \left (b^{2}+36 d^{2}\right ) \left (b \sin \left (2 d x +2 c \right )-2 d \cos \left (2 d x +2 c \right )\right )\right ) {\mathrm e}^{x b +a}}{32 b^{4}+1280 b^{2} d^{2}+4608 d^{4}}\) | \(108\) |
| default | \(-\frac {3 d \,{\mathrm e}^{x b +a} \cos \left (2 d x +2 c \right )}{16 \left (b^{2}+4 d^{2}\right )}+\frac {3 d \,{\mathrm e}^{x b +a} \cos \left (6 d x +6 c \right )}{16 \left (b^{2}+36 d^{2}\right )}+\frac {3 b \,{\mathrm e}^{x b +a} \sin \left (2 d x +2 c \right )}{32 \left (b^{2}+4 d^{2}\right )}-\frac {b \,{\mathrm e}^{x b +a} \sin \left (6 d x +6 c \right )}{32 \left (b^{2}+36 d^{2}\right )}\) | \(118\) |
| risch | \(\frac {i {\mathrm e}^{x b +a} \left (-12 i d \left (b^{2}+4 d^{2}\right ) \cos \left (6 d x +6 c \right )-i \left (-2 b^{3}-8 b \,d^{2}\right ) \sin \left (6 d x +6 c \right )+12 i d \left (b^{2}+36 d^{2}\right ) \cos \left (2 d x +2 c \right )-i \left (6 b^{3}+216 b \,d^{2}\right ) \sin \left (2 d x +2 c \right )\right )}{64 \left (6 i d +b \right ) \left (2 i d +b \right ) \left (2 i d -b \right ) \left (6 i d -b \right )}\) | \(139\) |
-((b^3+4*b*d^2)*sin(6*d*x+6*c)+(-6*b^2*d-24*d^3)*cos(6*d*x+6*c)-3*(b^2+36* d^2)*(b*sin(2*d*x+2*c)-2*d*cos(2*d*x+2*c)))*exp(b*x+a)/(32*b^4+1280*b^2*d^ 2+4608*d^4)
Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.21 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=-\frac {{\left ({\left (b^{3} + 4 \, b d^{2}\right )} \cos \left (d x + c\right )^{5} - 6 \, b d^{2} \cos \left (d x + c\right ) - {\left (b^{3} + 4 \, b d^{2}\right )} \cos \left (d x + c\right )^{3}\right )} e^{\left (b x + a\right )} \sin \left (d x + c\right ) - 3 \, {\left (2 \, {\left (b^{2} d + 4 \, d^{3}\right )} \cos \left (d x + c\right )^{6} + b^{2} d \cos \left (d x + c\right )^{2} - 3 \, {\left (b^{2} d + 4 \, d^{3}\right )} \cos \left (d x + c\right )^{4} + 2 \, d^{3}\right )} e^{\left (b x + a\right )}}{b^{4} + 40 \, b^{2} d^{2} + 144 \, d^{4}} \]
-(((b^3 + 4*b*d^2)*cos(d*x + c)^5 - 6*b*d^2*cos(d*x + c) - (b^3 + 4*b*d^2) *cos(d*x + c)^3)*e^(b*x + a)*sin(d*x + c) - 3*(2*(b^2*d + 4*d^3)*cos(d*x + c)^6 + b^2*d*cos(d*x + c)^2 - 3*(b^2*d + 4*d^3)*cos(d*x + c)^4 + 2*d^3)*e ^(b*x + a))/(b^4 + 40*b^2*d^2 + 144*d^4)
Result contains complex when optimal does not.
Time = 71.62 (sec) , antiderivative size = 1991, normalized size of antiderivative = 15.43 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=\text {Too large to display} \]
Piecewise((x*exp(a)*sin(c)**3*cos(c)**3, Eq(b, 0) & Eq(d, 0)), (-I*x*exp(a )*exp(-6*I*d*x)*sin(c + d*x)**6/64 - 3*x*exp(a)*exp(-6*I*d*x)*sin(c + d*x) **5*cos(c + d*x)/32 + 15*I*x*exp(a)*exp(-6*I*d*x)*sin(c + d*x)**4*cos(c + d*x)**2/64 + 5*x*exp(a)*exp(-6*I*d*x)*sin(c + d*x)**3*cos(c + d*x)**3/16 - 15*I*x*exp(a)*exp(-6*I*d*x)*sin(c + d*x)**2*cos(c + d*x)**4/64 - 3*x*exp( a)*exp(-6*I*d*x)*sin(c + d*x)*cos(c + d*x)**5/32 + I*x*exp(a)*exp(-6*I*d*x )*cos(c + d*x)**6/64 - exp(a)*exp(-6*I*d*x)*sin(c + d*x)**6/(160*d) + 7*I* exp(a)*exp(-6*I*d*x)*sin(c + d*x)**5*cos(c + d*x)/(320*d) + 11*I*exp(a)*ex p(-6*I*d*x)*sin(c + d*x)**3*cos(c + d*x)**3/(96*d) + 7*I*exp(a)*exp(-6*I*d *x)*sin(c + d*x)*cos(c + d*x)**5/(320*d) + exp(a)*exp(-6*I*d*x)*cos(c + d* x)**6/(160*d), Eq(b, -6*I*d)), (3*I*x*exp(a)*exp(-2*I*d*x)*sin(c + d*x)**6 /64 + 3*x*exp(a)*exp(-2*I*d*x)*sin(c + d*x)**5*cos(c + d*x)/32 + 3*I*x*exp (a)*exp(-2*I*d*x)*sin(c + d*x)**4*cos(c + d*x)**2/64 + 3*x*exp(a)*exp(-2*I *d*x)*sin(c + d*x)**3*cos(c + d*x)**3/16 - 3*I*x*exp(a)*exp(-2*I*d*x)*sin( c + d*x)**2*cos(c + d*x)**4/64 + 3*x*exp(a)*exp(-2*I*d*x)*sin(c + d*x)*cos (c + d*x)**5/32 - 3*I*x*exp(a)*exp(-2*I*d*x)*cos(c + d*x)**6/64 - 3*exp(a) *exp(-2*I*d*x)*sin(c + d*x)**6/(32*d) + 15*I*exp(a)*exp(-2*I*d*x)*sin(c + d*x)**5*cos(c + d*x)/(64*d) + 13*I*exp(a)*exp(-2*I*d*x)*sin(c + d*x)**3*co s(c + d*x)**3/(32*d) + 15*I*exp(a)*exp(-2*I*d*x)*sin(c + d*x)*cos(c + d*x) **5/(64*d) + 3*exp(a)*exp(-2*I*d*x)*cos(c + d*x)**6/(32*d), Eq(b, -2*I*...
Leaf count of result is larger than twice the leaf count of optimal. 550 vs. \(2 (117) = 234\).
Time = 0.22 (sec) , antiderivative size = 550, normalized size of antiderivative = 4.26 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=\frac {{\left (6 \, b^{2} d \cos \left (6 \, c\right ) e^{a} + 24 \, d^{3} \cos \left (6 \, c\right ) e^{a} - b^{3} e^{a} \sin \left (6 \, c\right ) - 4 \, b d^{2} e^{a} \sin \left (6 \, c\right )\right )} \cos \left (6 \, d x\right ) e^{\left (b x\right )} + {\left (6 \, b^{2} d \cos \left (6 \, c\right ) e^{a} + 24 \, d^{3} \cos \left (6 \, c\right ) e^{a} + b^{3} e^{a} \sin \left (6 \, c\right ) + 4 \, b d^{2} e^{a} \sin \left (6 \, c\right )\right )} \cos \left (6 \, d x + 12 \, c\right ) e^{\left (b x\right )} - 3 \, {\left (2 \, b^{2} d \cos \left (6 \, c\right ) e^{a} + 72 \, d^{3} \cos \left (6 \, c\right ) e^{a} + b^{3} e^{a} \sin \left (6 \, c\right ) + 36 \, b d^{2} e^{a} \sin \left (6 \, c\right )\right )} \cos \left (2 \, d x + 8 \, c\right ) e^{\left (b x\right )} - 3 \, {\left (2 \, b^{2} d \cos \left (6 \, c\right ) e^{a} + 72 \, d^{3} \cos \left (6 \, c\right ) e^{a} - b^{3} e^{a} \sin \left (6 \, c\right ) - 36 \, b d^{2} e^{a} \sin \left (6 \, c\right )\right )} \cos \left (2 \, d x - 4 \, c\right ) e^{\left (b x\right )} - {\left (b^{3} \cos \left (6 \, c\right ) e^{a} + 4 \, b d^{2} \cos \left (6 \, c\right ) e^{a} + 6 \, b^{2} d e^{a} \sin \left (6 \, c\right ) + 24 \, d^{3} e^{a} \sin \left (6 \, c\right )\right )} e^{\left (b x\right )} \sin \left (6 \, d x\right ) - {\left (b^{3} \cos \left (6 \, c\right ) e^{a} + 4 \, b d^{2} \cos \left (6 \, c\right ) e^{a} - 6 \, b^{2} d e^{a} \sin \left (6 \, c\right ) - 24 \, d^{3} e^{a} \sin \left (6 \, c\right )\right )} e^{\left (b x\right )} \sin \left (6 \, d x + 12 \, c\right ) + 3 \, {\left (b^{3} \cos \left (6 \, c\right ) e^{a} + 36 \, b d^{2} \cos \left (6 \, c\right ) e^{a} - 2 \, b^{2} d e^{a} \sin \left (6 \, c\right ) - 72 \, d^{3} e^{a} \sin \left (6 \, c\right )\right )} e^{\left (b x\right )} \sin \left (2 \, d x + 8 \, c\right ) + 3 \, {\left (b^{3} \cos \left (6 \, c\right ) e^{a} + 36 \, b d^{2} \cos \left (6 \, c\right ) e^{a} + 2 \, b^{2} d e^{a} \sin \left (6 \, c\right ) + 72 \, d^{3} e^{a} \sin \left (6 \, c\right )\right )} e^{\left (b x\right )} \sin \left (2 \, d x - 4 \, c\right )}{64 \, {\left (b^{4} \cos \left (6 \, c\right )^{2} + b^{4} \sin \left (6 \, c\right )^{2} + 144 \, {\left (\cos \left (6 \, c\right )^{2} + \sin \left (6 \, c\right )^{2}\right )} d^{4} + 40 \, {\left (b^{2} \cos \left (6 \, c\right )^{2} + b^{2} \sin \left (6 \, c\right )^{2}\right )} d^{2}\right )}} \]
1/64*((6*b^2*d*cos(6*c)*e^a + 24*d^3*cos(6*c)*e^a - b^3*e^a*sin(6*c) - 4*b *d^2*e^a*sin(6*c))*cos(6*d*x)*e^(b*x) + (6*b^2*d*cos(6*c)*e^a + 24*d^3*cos (6*c)*e^a + b^3*e^a*sin(6*c) + 4*b*d^2*e^a*sin(6*c))*cos(6*d*x + 12*c)*e^( b*x) - 3*(2*b^2*d*cos(6*c)*e^a + 72*d^3*cos(6*c)*e^a + b^3*e^a*sin(6*c) + 36*b*d^2*e^a*sin(6*c))*cos(2*d*x + 8*c)*e^(b*x) - 3*(2*b^2*d*cos(6*c)*e^a + 72*d^3*cos(6*c)*e^a - b^3*e^a*sin(6*c) - 36*b*d^2*e^a*sin(6*c))*cos(2*d* x - 4*c)*e^(b*x) - (b^3*cos(6*c)*e^a + 4*b*d^2*cos(6*c)*e^a + 6*b^2*d*e^a* sin(6*c) + 24*d^3*e^a*sin(6*c))*e^(b*x)*sin(6*d*x) - (b^3*cos(6*c)*e^a + 4 *b*d^2*cos(6*c)*e^a - 6*b^2*d*e^a*sin(6*c) - 24*d^3*e^a*sin(6*c))*e^(b*x)* sin(6*d*x + 12*c) + 3*(b^3*cos(6*c)*e^a + 36*b*d^2*cos(6*c)*e^a - 2*b^2*d* e^a*sin(6*c) - 72*d^3*e^a*sin(6*c))*e^(b*x)*sin(2*d*x + 8*c) + 3*(b^3*cos( 6*c)*e^a + 36*b*d^2*cos(6*c)*e^a + 2*b^2*d*e^a*sin(6*c) + 72*d^3*e^a*sin(6 *c))*e^(b*x)*sin(2*d*x - 4*c))/(b^4*cos(6*c)^2 + b^4*sin(6*c)^2 + 144*(cos (6*c)^2 + sin(6*c)^2)*d^4 + 40*(b^2*cos(6*c)^2 + b^2*sin(6*c)^2)*d^2)
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.86 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=\frac {1}{32} \, {\left (\frac {6 \, d \cos \left (6 \, d x + 6 \, c\right )}{b^{2} + 36 \, d^{2}} - \frac {b \sin \left (6 \, d x + 6 \, c\right )}{b^{2} + 36 \, d^{2}}\right )} e^{\left (b x + a\right )} - \frac {3}{32} \, {\left (\frac {2 \, d \cos \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}} - \frac {b \sin \left (2 \, d x + 2 \, c\right )}{b^{2} + 4 \, d^{2}}\right )} e^{\left (b x + a\right )} \]
1/32*(6*d*cos(6*d*x + 6*c)/(b^2 + 36*d^2) - b*sin(6*d*x + 6*c)/(b^2 + 36*d ^2))*e^(b*x + a) - 3/32*(2*d*cos(2*d*x + 2*c)/(b^2 + 4*d^2) - b*sin(2*d*x + 2*c)/(b^2 + 4*d^2))*e^(b*x + a)
Time = 1.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.38 \[ \int e^{a+b x} \cos ^3(c+d x) \sin ^3(c+d x) \, dx=-\frac {3\,{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )-\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )-\sin \left (2\,c\right )\,1{}\mathrm {i}\right )}{64\,\left (2\,d+b\,1{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (6\,d\,x\right )-\sin \left (6\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (6\,c\right )-\sin \left (6\,c\right )\,1{}\mathrm {i}\right )}{64\,\left (6\,d+b\,1{}\mathrm {i}\right )}-\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (2\,d\,x\right )+\sin \left (2\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (2\,c\right )+\sin \left (2\,c\right )\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{64\,\left (b+d\,2{}\mathrm {i}\right )}+\frac {{\mathrm {e}}^{a+b\,x}\,\left (\cos \left (6\,d\,x\right )+\sin \left (6\,d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (6\,c\right )+\sin \left (6\,c\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{64\,\left (b+d\,6{}\mathrm {i}\right )} \]
(exp(a + b*x)*(cos(6*d*x) - sin(6*d*x)*1i)*(cos(6*c) - sin(6*c)*1i))/(64*( b*1i + 6*d)) - (3*exp(a + b*x)*(cos(2*d*x) - sin(2*d*x)*1i)*(cos(2*c) - si n(2*c)*1i))/(64*(b*1i + 2*d)) - (exp(a + b*x)*(cos(2*d*x) + sin(2*d*x)*1i) *(cos(2*c) + sin(2*c)*1i)*3i)/(64*(b + d*2i)) + (exp(a + b*x)*(cos(6*d*x) + sin(6*d*x)*1i)*(cos(6*c) + sin(6*c)*1i)*1i)/(64*(b + d*6i))