Integrand size = 24, antiderivative size = 179 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^2} \, dx=\frac {3 b \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^2}-\frac {3 b \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^2}-\frac {3 \sin (2 a+2 b x)}{32 d (c+d x)}+\frac {\sin (6 a+6 b x)}{32 d (c+d x)}-\frac {3 b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^2}+\frac {3 b \sin \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^2} \]
-3/16*b*Ci(6*b*c/d+6*b*x)*cos(6*a-6*b*c/d)/d^2+3/16*b*Ci(2*b*c/d+2*b*x)*co s(2*a-2*b*c/d)/d^2+3/16*b*Si(6*b*c/d+6*b*x)*sin(6*a-6*b*c/d)/d^2-3/16*b*Si (2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d^2-3/32*sin(2*b*x+2*a)/d/(d*x+c)+1/32*si n(6*b*x+6*a)/d/(d*x+c)
Time = 1.00 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^2} \, dx=\frac {6 b (c+d x) \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right )-6 b (c+d x) \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {6 b (c+d x)}{d}\right )-3 d \cos (2 b x) \sin (2 a)+d \cos (6 b x) \sin (6 a)-3 d \cos (2 a) \sin (2 b x)+d \cos (6 a) \sin (6 b x)-6 b (c+d x) \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )+6 b (c+d x) \sin \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b (c+d x)}{d}\right )}{32 d^2 (c+d x)} \]
(6*b*(c + d*x)*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*(c + d*x))/d] - 6*b*( c + d*x)*Cos[6*a - (6*b*c)/d]*CosIntegral[(6*b*(c + d*x))/d] - 3*d*Cos[2*b *x]*Sin[2*a] + d*Cos[6*b*x]*Sin[6*a] - 3*d*Cos[2*a]*Sin[2*b*x] + d*Cos[6*a ]*Sin[6*b*x] - 6*b*(c + d*x)*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d* x))/d] + 6*b*(c + d*x)*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*(c + d*x))/d] )/(32*d^2*(c + d*x))
Time = 0.50 (sec) , antiderivative size = 179, 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 = {4906, 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.
| \(\displaystyle \int \frac {\sin ^3(a+b x) \cos ^3(a+b x)}{(c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {3 \sin (2 a+2 b x)}{32 (c+d x)^2}-\frac {\sin (6 a+6 b x)}{32 (c+d x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^2}-\frac {3 b \cos \left (6 a-\frac {6 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^2}-\frac {3 b \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^2}+\frac {3 b \sin \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^2}-\frac {3 \sin (2 a+2 b x)}{32 d (c+d x)}+\frac {\sin (6 a+6 b x)}{32 d (c+d x)}\) |
(3*b*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x])/(16*d^2) - (3*b* Cos[6*a - (6*b*c)/d]*CosIntegral[(6*b*c)/d + 6*b*x])/(16*d^2) - (3*Sin[2*a + 2*b*x])/(32*d*(c + d*x)) + Sin[6*a + 6*b*x]/(32*d*(c + d*x)) - (3*b*Sin [2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x])/(16*d^2) + (3*b*Sin[6*a - (6*b*c)/d]*SinIntegral[(6*b*c)/d + 6*b*x])/(16*d^2)
3.2.60.3.1 Defintions of rubi rules used
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Time = 1.96 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.43
| method | result | size |
| derivativedivides | \(\frac {-\frac {b^{2} \left (-\frac {6 \sin \left (6 x b +6 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {36 \,\operatorname {Si}\left (-6 x b -6 a -\frac {6 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-6 a d +6 c b}{d}\right )}{d}+\frac {36 \,\operatorname {Ci}\left (6 x b +6 a +\frac {-6 a d +6 c b}{d}\right ) \cos \left (\frac {-6 a d +6 c b}{d}\right )}{d}}{d}\right )}{192}+\frac {3 b^{2} \left (-\frac {2 \sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {4 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}}{d}\right )}{64}}{b}\) | \(256\) |
| default | \(\frac {-\frac {b^{2} \left (-\frac {6 \sin \left (6 x b +6 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {36 \,\operatorname {Si}\left (-6 x b -6 a -\frac {6 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-6 a d +6 c b}{d}\right )}{d}+\frac {36 \,\operatorname {Ci}\left (6 x b +6 a +\frac {-6 a d +6 c b}{d}\right ) \cos \left (\frac {-6 a d +6 c b}{d}\right )}{d}}{d}\right )}{192}+\frac {3 b^{2} \left (-\frac {2 \sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {4 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}}{d}\right )}{64}}{b}\) | \(256\) |
| risch | \(\frac {3 b \,{\mathrm e}^{-\frac {6 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (6 i b x +6 i a -\frac {6 i \left (a d -c b \right )}{d}\right )}{32 d^{2}}-\frac {3 b \,{\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{32 d^{2}}-\frac {3 b \,{\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{32 d^{2}}+\frac {3 b \,{\mathrm e}^{\frac {6 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-6 i b x -6 i a -\frac {6 \left (-i a d +i c b \right )}{d}\right )}{32 d^{2}}+\frac {\left (-2 d x b -2 c b \right ) \sin \left (6 x b +6 a \right )}{64 d \left (-d x b -c b \right ) \left (d x +c \right )}-\frac {3 \left (-2 d x b -2 c b \right ) \sin \left (2 x b +2 a \right )}{64 d \left (-d x b -c b \right ) \left (d x +c \right )}\) | \(280\) |
1/b*(-1/192*b^2*(-6*sin(6*b*x+6*a)/(-a*d+c*b+d*(b*x+a))/d+6*(-6*Si(-6*x*b- 6*a-6*(-a*d+b*c)/d)*sin(6*(-a*d+b*c)/d)/d+6*Ci(6*x*b+6*a+6*(-a*d+b*c)/d)*c os(6*(-a*d+b*c)/d)/d)/d)+3/64*b^2*(-2*sin(2*b*x+2*a)/(-a*d+c*b+d*(b*x+a))/ d+2*(-2*Si(-2*x*b-2*a-2*(-a*d+b*c)/d)*sin(2*(-a*d+b*c)/d)/d+2*Ci(2*x*b+2*a +2*(-a*d+b*c)/d)*cos(2*(-a*d+b*c)/d)/d)/d))
Time = 0.25 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {3 \, {\left (b d x + b c\right )} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {6 \, {\left (b d x + b c\right )}}{d}\right ) - 3 \, {\left (b d x + b c\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - 3 \, {\left (b d x + b c\right )} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {6 \, {\left (b d x + b c\right )}}{d}\right ) + 3 \, {\left (b d x + b c\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - 16 \, {\left (d \cos \left (b x + a\right )^{5} - d \cos \left (b x + a\right )^{3}\right )} \sin \left (b x + a\right )}{16 \, {\left (d^{3} x + c d^{2}\right )}} \]
-1/16*(3*(b*d*x + b*c)*cos(-6*(b*c - a*d)/d)*cos_integral(6*(b*d*x + b*c)/ d) - 3*(b*d*x + b*c)*cos(-2*(b*c - a*d)/d)*cos_integral(2*(b*d*x + b*c)/d) - 3*(b*d*x + b*c)*sin(-6*(b*c - a*d)/d)*sin_integral(6*(b*d*x + b*c)/d) + 3*(b*d*x + b*c)*sin(-2*(b*c - a*d)/d)*sin_integral(2*(b*d*x + b*c)/d) - 1 6*(d*cos(b*x + a)^5 - d*cos(b*x + a)^3)*sin(b*x + a))/(d^3*x + c*d^2)
\[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.72 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {3 \, b^{2} {\left (-i \, E_{2}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{2}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - b^{2} {\left (-i \, E_{2}\left (\frac {6 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{2}\left (-\frac {6 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, b^{2} {\left (E_{2}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{2}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - b^{2} {\left (E_{2}\left (\frac {6 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{2}\left (-\frac {6 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )}{64 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} b} \]
-1/64*(3*b^2*(-I*exp_integral_e(2, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_integral_e(2, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-2*(b*c - a*d)/d) - b^2*(-I*exp_integral_e(2, 6*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_integral_e(2, -6*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-6*(b*c - a*d)/d) + 3*b^2*(exp_integral_e(2, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d ) + exp_integral_e(2, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-2*(b*c - a*d)/d) - b^2*(exp_integral_e(2, 6*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(2, -6*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-6*(b*c - a *d)/d))/((b*c*d + (b*x + a)*d^2 - a*d^2)*b)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.43 (sec) , antiderivative size = 63798, normalized size of antiderivative = 356.41 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^2} \, dx=\text {Too large to display} \]
-1/32*(3*b*d*x*real_part(cos_integral(6*b*x + 6*b*c/d))*tan(3*b*x)^2*tan(b *x)^2*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 - 3*b*d*x*real_part( cos_integral(2*b*x + 2*b*c/d))*tan(3*b*x)^2*tan(b*x)^2*tan(3*a)^2*tan(a)^2 *tan(3*b*c/d)^2*tan(b*c/d)^2 - 3*b*d*x*real_part(cos_integral(-2*b*x - 2*b *c/d))*tan(3*b*x)^2*tan(b*x)^2*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/ d)^2 + 3*b*d*x*real_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*b*x)^2*tan( b*x)^2*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d)^2 + 6*b*d*x*imag_part (cos_integral(2*b*x + 2*b*c/d))*tan(3*b*x)^2*tan(b*x)^2*tan(3*a)^2*tan(a)^ 2*tan(3*b*c/d)^2*tan(b*c/d) - 6*b*d*x*imag_part(cos_integral(-2*b*x - 2*b* c/d))*tan(3*b*x)^2*tan(b*x)^2*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d ) + 12*b*d*x*sin_integral(2*(b*d*x + b*c)/d)*tan(3*b*x)^2*tan(b*x)^2*tan(3 *a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d) - 6*b*d*x*imag_part(cos_integral( 6*b*x + 6*b*c/d))*tan(3*b*x)^2*tan(b*x)^2*tan(3*a)^2*tan(a)^2*tan(3*b*c/d) *tan(b*c/d)^2 + 6*b*d*x*imag_part(cos_integral(-6*b*x - 6*b*c/d))*tan(3*b* x)^2*tan(b*x)^2*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)*tan(b*c/d)^2 - 12*b*d*x*s in_integral(6*(b*d*x + b*c)/d)*tan(3*b*x)^2*tan(b*x)^2*tan(3*a)^2*tan(a)^2 *tan(3*b*c/d)*tan(b*c/d)^2 - 6*b*d*x*imag_part(cos_integral(2*b*x + 2*b*c/ d))*tan(3*b*x)^2*tan(b*x)^2*tan(3*a)^2*tan(a)*tan(3*b*c/d)^2*tan(b*c/d)^2 + 6*b*d*x*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*b*x)^2*tan(b*x)^ 2*tan(3*a)^2*tan(a)*tan(3*b*c/d)^2*tan(b*c/d)^2 - 12*b*d*x*sin_integral...
Timed out. \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \]