Integrand size = 24, antiderivative size = 235 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^3} \, dx=-\frac {3 b \cos (2 a+2 b x)}{32 d^2 (c+d x)}+\frac {3 b \cos (6 a+6 b x)}{32 d^2 (c+d x)}+\frac {9 b^2 \operatorname {CosIntegral}\left (\frac {6 b c}{d}+6 b x\right ) \sin \left (6 a-\frac {6 b c}{d}\right )}{16 d^3}-\frac {3 b^2 \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{16 d^3}-\frac {3 \sin (2 a+2 b x)}{64 d (c+d x)^2}+\frac {\sin (6 a+6 b x)}{64 d (c+d x)^2}-\frac {3 b^2 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^3}+\frac {9 b^2 \cos \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^3} \]
-3/32*b*cos(2*b*x+2*a)/d^2/(d*x+c)+3/32*b*cos(6*b*x+6*a)/d^2/(d*x+c)-3/16* b^2*cos(2*a-2*b*c/d)*Si(2*b*c/d+2*b*x)/d^3+9/16*b^2*cos(6*a-6*b*c/d)*Si(6* b*c/d+6*b*x)/d^3+9/16*b^2*Ci(6*b*c/d+6*b*x)*sin(6*a-6*b*c/d)/d^3-3/16*b^2* Ci(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d^3-3/64*sin(2*b*x+2*a)/d/(d*x+c)^2+1/6 4*sin(6*b*x+6*a)/d/(d*x+c)^2
Time = 1.11 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.02 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^3} \, dx=\frac {-3 d \cos (2 b x) (2 b (c+d x) \cos (2 a)+d \sin (2 a))+d \cos (6 b x) (6 b (c+d x) \cos (6 a)+d \sin (6 a))+3 d (-d \cos (2 a)+2 b (c+d x) \sin (2 a)) \sin (2 b x)+d (d \cos (6 a)-6 b (c+d x) \sin (6 a)) \sin (6 b x)+6 b^2 (c+d x)^2 \left (6 \operatorname {CosIntegral}\left (\frac {6 b (c+d x)}{d}\right ) \sin \left (6 a-\frac {6 b c}{d}\right )-2 \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )-2 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )+6 \cos \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b (c+d x)}{d}\right )\right )}{64 d^3 (c+d x)^2} \]
(-3*d*Cos[2*b*x]*(2*b*(c + d*x)*Cos[2*a] + d*Sin[2*a]) + d*Cos[6*b*x]*(6*b *(c + d*x)*Cos[6*a] + d*Sin[6*a]) + 3*d*(-(d*Cos[2*a]) + 2*b*(c + d*x)*Sin [2*a])*Sin[2*b*x] + d*(d*Cos[6*a] - 6*b*(c + d*x)*Sin[6*a])*Sin[6*b*x] + 6 *b^2*(c + d*x)^2*(6*CosIntegral[(6*b*(c + d*x))/d]*Sin[6*a - (6*b*c)/d] - 2*CosIntegral[(2*b*(c + d*x))/d]*Sin[2*a - (2*b*c)/d] - 2*Cos[2*a - (2*b*c )/d]*SinIntegral[(2*b*(c + d*x))/d] + 6*Cos[6*a - (6*b*c)/d]*SinIntegral[( 6*b*(c + d*x))/d]))/(64*d^3*(c + d*x)^2)
Time = 0.58 (sec) , antiderivative size = 235, 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.
| \(\displaystyle \int \frac {\sin ^3(a+b x) \cos ^3(a+b x)}{(c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {3 \sin (2 a+2 b x)}{32 (c+d x)^3}-\frac {\sin (6 a+6 b x)}{32 (c+d x)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {9 b^2 \sin \left (6 a-\frac {6 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^3}-\frac {3 b^2 \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^3}-\frac {3 b^2 \cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{16 d^3}+\frac {9 b^2 \cos \left (6 a-\frac {6 b c}{d}\right ) \text {Si}\left (\frac {6 b c}{d}+6 b x\right )}{16 d^3}-\frac {3 b \cos (2 a+2 b x)}{32 d^2 (c+d x)}+\frac {3 b \cos (6 a+6 b x)}{32 d^2 (c+d x)}-\frac {3 \sin (2 a+2 b x)}{64 d (c+d x)^2}+\frac {\sin (6 a+6 b x)}{64 d (c+d x)^2}\) |
(-3*b*Cos[2*a + 2*b*x])/(32*d^2*(c + d*x)) + (3*b*Cos[6*a + 6*b*x])/(32*d^ 2*(c + d*x)) + (9*b^2*CosIntegral[(6*b*c)/d + 6*b*x]*Sin[6*a - (6*b*c)/d]) /(16*d^3) - (3*b^2*CosIntegral[(2*b*c)/d + 2*b*x]*Sin[2*a - (2*b*c)/d])/(1 6*d^3) - (3*Sin[2*a + 2*b*x])/(64*d*(c + d*x)^2) + Sin[6*a + 6*b*x]/(64*d* (c + d*x)^2) - (3*b^2*Cos[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b*x]) /(16*d^3) + (9*b^2*Cos[6*a - (6*b*c)/d]*SinIntegral[(6*b*c)/d + 6*b*x])/(1 6*d^3)
3.2.61.3.1 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 = 2.72 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.40
| method | result | size |
| derivativedivides | \(\frac {-\frac {b^{3} \left (-\frac {3 \sin \left (6 x b +6 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {18 \cos \left (6 x b +6 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {18 \left (-\frac {6 \,\operatorname {Si}\left (-6 x b -6 a -\frac {6 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-6 a d +6 c b}{d}\right )}{d}-\frac {6 \,\operatorname {Ci}\left (6 x b +6 a +\frac {-6 a d +6 c b}{d}\right ) \sin \left (\frac {-6 a d +6 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{192}+\frac {3 b^{3} \left (-\frac {\sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {2 \cos \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {2 \left (-\frac {2 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{64}}{b}\) | \(329\) |
| default | \(\frac {-\frac {b^{3} \left (-\frac {3 \sin \left (6 x b +6 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {18 \cos \left (6 x b +6 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {18 \left (-\frac {6 \,\operatorname {Si}\left (-6 x b -6 a -\frac {6 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-6 a d +6 c b}{d}\right )}{d}-\frac {6 \,\operatorname {Ci}\left (6 x b +6 a +\frac {-6 a d +6 c b}{d}\right ) \sin \left (\frac {-6 a d +6 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{192}+\frac {3 b^{3} \left (-\frac {\sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {2 \cos \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {2 \left (-\frac {2 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{64}}{b}\) | \(329\) |
| risch | \(-\frac {9 i b^{2} {\mathrm e}^{-\frac {6 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (6 i b x +6 i a -\frac {6 i \left (a d -c b \right )}{d}\right )}{32 d^{3}}+\frac {3 i b^{2} {\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{32 d^{3}}-\frac {3 i b^{2} {\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{32 d^{3}}+\frac {9 i b^{2} {\mathrm e}^{\frac {6 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-6 i b x -6 i a -\frac {6 \left (-i a d +i c b \right )}{d}\right )}{32 d^{3}}-\frac {i \left (12 i b^{3} d^{3} x^{3}+36 i b^{3} c \,d^{2} x^{2}+36 i b^{3} c^{2} d x +12 i b^{3} c^{3}\right ) \cos \left (6 x b +6 a \right )}{128 d^{2} \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right ) \left (d x +c \right )^{2}}+\frac {\left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}\right ) \sin \left (6 x b +6 a \right )}{128 d \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right ) \left (d x +c \right )^{2}}+\frac {3 i \left (4 i b^{3} d^{3} x^{3}+12 i b^{3} c \,d^{2} x^{2}+12 i b^{3} c^{2} d x +4 i b^{3} c^{3}\right ) \cos \left (2 x b +2 a \right )}{128 d^{2} \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right ) \left (d x +c \right )^{2}}-\frac {3 \left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}\right ) \sin \left (2 x b +2 a \right )}{128 d \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right ) \left (d x +c \right )^{2}}\) | \(552\) |
1/b*(-1/192*b^3*(-3*sin(6*b*x+6*a)/(-a*d+c*b+d*(b*x+a))^2/d+3*(-6*cos(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)*cos(6*(- a*d+b*c)/d)/d-6*Ci(6*x*b+6*a+6*(-a*d+b*c)/d)*sin(6*(-a*d+b*c)/d)/d)/d)/d)+ 3/64*b^3*(-sin(2*b*x+2*a)/(-a*d+c*b+d*(b*x+a))^2/d+(-2*cos(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)*cos(2*(-a*d+b*c)/d) /d-2*Ci(2*x*b+2*a+2*(-a*d+b*c)/d)*sin(2*(-a*d+b*c)/d)/d)/d)/d))
Time = 0.27 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.48 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^3} \, dx=\frac {48 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{6} - 72 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{4} + 24 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{2} - 3 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 9 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname {Ci}\left (\frac {6 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) + 9 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {6 \, {\left (b d x + b c\right )}}{d}\right ) - 3 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + 8 \, {\left (d^{2} \cos \left (b x + a\right )^{5} - d^{2} \cos \left (b x + a\right )^{3}\right )} \sin \left (b x + a\right )}{16 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
1/16*(48*(b*d^2*x + b*c*d)*cos(b*x + a)^6 - 72*(b*d^2*x + b*c*d)*cos(b*x + a)^4 + 24*(b*d^2*x + b*c*d)*cos(b*x + a)^2 - 3*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos_integral(2*(b*d*x + b*c)/d)*sin(-2*(b*c - a*d)/d) + 9*(b^2 *d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos_integral(6*(b*d*x + b*c)/d)*sin(-6*( b*c - a*d)/d) + 9*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(-6*(b*c - a*d) /d)*sin_integral(6*(b*d*x + b*c)/d) - 3*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c ^2)*cos(-2*(b*c - a*d)/d)*sin_integral(2*(b*d*x + b*c)/d) + 8*(d^2*cos(b*x + a)^5 - d^2*cos(b*x + a)^3)*sin(b*x + a))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3 )
\[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^3} \, dx=\int \frac {\sin ^{3}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \]
Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.46 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^3} \, dx=-\frac {3 \, b^{3} {\left (-i \, E_{3}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{3}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - b^{3} {\left (-i \, E_{3}\left (\frac {6 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{3}\left (-\frac {6 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, b^{3} {\left (E_{3}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{3}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - b^{3} {\left (E_{3}\left (\frac {6 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{3}\left (-\frac {6 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {6 \, {\left (b c - a d\right )}}{d}\right )}{64 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + {\left (b x + a\right )}^{2} d^{3} + a^{2} d^{3} + 2 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}\right )} b} \]
-1/64*(3*b^3*(-I*exp_integral_e(3, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_integral_e(3, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-2*(b*c - a*d)/d) - b^3*(-I*exp_integral_e(3, 6*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_integral_e(3, -6*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-6*(b*c - a*d)/d) + 3*b^3*(exp_integral_e(3, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d ) + exp_integral_e(3, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-2*(b*c - a*d)/d) - b^3*(exp_integral_e(3, 6*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(3, -6*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-6*(b*c - a *d)/d))/((b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*b)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.47 (sec) , antiderivative size = 111694, normalized size of antiderivative = 475.29 \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^3} \, dx=\text {Too large to display} \]
1/32*(9*b^2*d^2*x^2*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)^2*tan(b*c/d)^2 - 3*b^2*d^2*x^2 *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)^2 + 3*b^2*d^2*x^2*imag_part(cos_inte gral(-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 - 9*b^2*d^2*x^2*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)^2*tan(b*c/d )^2 + 18*b^2*d^2*x^2*sin_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)^2*tan(b*c/d)^2 - 6*b^2*d^2*x^2*sin_int egral(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)^2 - 6*b^2*d^2*x^2*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 ) - 6*b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*b*x)^2*t an(b*x)^2*tan(3*a)^2*tan(a)^2*tan(3*b*c/d)^2*tan(b*c/d) + 18*b^2*d^2*x^2*r eal_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 + 18*b^2*d^2*x^2*real_part(cos_integra l(-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^2*d^2*x^2*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)*tan(3*b*c/d)^2*tan(b*c/d)^2 + 6* b^2*d^2*x^2*real_part(cos_integral(-2*b*x - 2*b*c/d))*tan(3*b*x)^2*tan(...
Timed out. \[ \int \frac {\cos ^3(a+b x) \sin ^3(a+b x)}{(c+d x)^3} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^3} \,d x \]