Integrand size = 22, antiderivative size = 270 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=-\frac {b \cos (a+b x)}{24 d^2 (c+d x)^2}-\frac {b \cos (3 a+3 b x)}{8 d^2 (c+d x)^2}-\frac {b^3 \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{24 d^4}-\frac {9 b^3 \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^4}-\frac {\sin (a+b x)}{12 d (c+d x)^3}+\frac {b^2 \sin (a+b x)}{24 d^3 (c+d x)}-\frac {\sin (3 a+3 b x)}{12 d (c+d x)^3}+\frac {3 b^2 \sin (3 a+3 b x)}{8 d^3 (c+d x)}+\frac {b^3 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{24 d^4}+\frac {9 b^3 \sin \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^4} \]
-9/8*b^3*Ci(3*b*c/d+3*b*x)*cos(3*a-3*b*c/d)/d^4-1/24*b^3*Ci(b*c/d+b*x)*cos (a-b*c/d)/d^4-1/24*b*cos(b*x+a)/d^2/(d*x+c)^2-1/8*b*cos(3*b*x+3*a)/d^2/(d* x+c)^2+9/8*b^3*Si(3*b*c/d+3*b*x)*sin(3*a-3*b*c/d)/d^4+1/24*b^3*Si(b*c/d+b* x)*sin(a-b*c/d)/d^4-1/12*sin(b*x+a)/d/(d*x+c)^3+1/24*b^2*sin(b*x+a)/d^3/(d *x+c)-1/12*sin(3*b*x+3*a)/d/(d*x+c)^3+3/8*b^2*sin(3*b*x+3*a)/d^3/(d*x+c)
Time = 1.86 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=-\frac {d \cos (b x) \left (b d (c+d x) \cos (a)-\left (-2 d^2+b^2 (c+d x)^2\right ) \sin (a)\right )+d \cos (3 b x) \left (3 b d (c+d x) \cos (3 a)-\left (-2 d^2+9 b^2 (c+d x)^2\right ) \sin (3 a)\right )-d \left (\left (-2 d^2+b^2 (c+d x)^2\right ) \cos (a)+b d (c+d x) \sin (a)\right ) \sin (b x)-d \left (\left (-2 d^2+9 b^2 (c+d x)^2\right ) \cos (3 a)+3 b d (c+d x) \sin (3 a)\right ) \sin (3 b x)+b^3 (c+d x)^3 \left (\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right )-\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )\right )+27 b^3 (c+d x)^3 \left (\cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b (c+d x)}{d}\right )-\sin \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )\right )}{24 d^4 (c+d x)^3} \]
-1/24*(d*Cos[b*x]*(b*d*(c + d*x)*Cos[a] - (-2*d^2 + b^2*(c + d*x)^2)*Sin[a ]) + d*Cos[3*b*x]*(3*b*d*(c + d*x)*Cos[3*a] - (-2*d^2 + 9*b^2*(c + d*x)^2) *Sin[3*a]) - d*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] + b*d*(c + d*x)*Sin[a])* Sin[b*x] - d*((-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*a] + 3*b*d*(c + d*x)*Sin[ 3*a])*Sin[3*b*x] + b^3*(c + d*x)^3*(Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)] - Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)]) + 27*b^3*(c + d*x)^3*(Cos [3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] - Sin[3*a - (3*b*c)/d]*Si nIntegral[(3*b*(c + d*x))/d]))/(d^4*(c + d*x)^3)
Time = 0.61 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int \frac {\sin (a+b x) \cos ^2(a+b x)}{(c+d x)^4} \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {\sin (a+b x)}{4 (c+d x)^4}+\frac {\sin (3 a+3 b x)}{4 (c+d x)^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^3 \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{24 d^4}-\frac {9 b^3 \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^4}+\frac {b^3 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{24 d^4}+\frac {9 b^3 \sin \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^4}+\frac {b^2 \sin (a+b x)}{24 d^3 (c+d x)}+\frac {3 b^2 \sin (3 a+3 b x)}{8 d^3 (c+d x)}-\frac {b \cos (a+b x)}{24 d^2 (c+d x)^2}-\frac {b \cos (3 a+3 b x)}{8 d^2 (c+d x)^2}-\frac {\sin (a+b x)}{12 d (c+d x)^3}-\frac {\sin (3 a+3 b x)}{12 d (c+d x)^3}\) |
-1/24*(b*Cos[a + b*x])/(d^2*(c + d*x)^2) - (b*Cos[3*a + 3*b*x])/(8*d^2*(c + d*x)^2) - (b^3*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/(24*d^4) - ( 9*b^3*Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*c)/d + 3*b*x])/(8*d^4) - Sin[a + b*x]/(12*d*(c + d*x)^3) + (b^2*Sin[a + b*x])/(24*d^3*(c + d*x)) - Sin[3 *a + 3*b*x]/(12*d*(c + d*x)^3) + (3*b^2*Sin[3*a + 3*b*x])/(8*d^3*(c + d*x) ) + (b^3*Sin[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(24*d^4) + (9*b^3*Si n[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(8*d^4)
3.1.78.3.1 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 = 1.71 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.43
| method | result | size |
| derivativedivides | \(\frac {\frac {b^{4} \left (-\frac {\sin \left (3 x b +3 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{3} d}+\frac {-\frac {3 \cos \left (3 x b +3 a \right )}{2 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {3 \left (-\frac {3 \sin \left (3 x b +3 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {9 \,\operatorname {Si}\left (-3 x b -3 a -\frac {3 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-3 a d +3 c b}{d}\right )}{d}+\frac {9 \,\operatorname {Ci}\left (3 x b +3 a +\frac {-3 a d +3 c b}{d}\right ) \cos \left (\frac {-3 a d +3 c b}{d}\right )}{d}}{d}\right )}{2 d}}{d}\right )}{12}+\frac {b^{4} \left (-\frac {\sin \left (x b +a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}+\frac {-\frac {\cos \left (x b +a \right )}{2 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {-\frac {\sin \left (x b +a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {\operatorname {Si}\left (-x b -a -\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}+\frac {\operatorname {Ci}\left (x b +a +\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}}{d}}{2 d}}{3 d}\right )}{4}}{b}\) | \(386\) |
| default | \(\frac {\frac {b^{4} \left (-\frac {\sin \left (3 x b +3 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{3} d}+\frac {-\frac {3 \cos \left (3 x b +3 a \right )}{2 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {3 \left (-\frac {3 \sin \left (3 x b +3 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {9 \,\operatorname {Si}\left (-3 x b -3 a -\frac {3 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-3 a d +3 c b}{d}\right )}{d}+\frac {9 \,\operatorname {Ci}\left (3 x b +3 a +\frac {-3 a d +3 c b}{d}\right ) \cos \left (\frac {-3 a d +3 c b}{d}\right )}{d}}{d}\right )}{2 d}}{d}\right )}{12}+\frac {b^{4} \left (-\frac {\sin \left (x b +a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}+\frac {-\frac {\cos \left (x b +a \right )}{2 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {-\frac {\sin \left (x b +a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {\operatorname {Si}\left (-x b -a -\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}+\frac {\operatorname {Ci}\left (x b +a +\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}}{d}}{2 d}}{3 d}\right )}{4}}{b}\) | \(386\) |
| risch | \(\frac {9 b^{3} {\mathrm e}^{-\frac {3 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (3 i b x +3 i a -\frac {3 i \left (a d -c b \right )}{d}\right )}{16 d^{4}}+\frac {b^{3} {\mathrm e}^{-\frac {i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (i b x +i a -\frac {i \left (a d -c b \right )}{d}\right )}{48 d^{4}}+\frac {b^{3} {\mathrm e}^{\frac {i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-i b x -i a -\frac {-i a d +i c b}{d}\right )}{48 d^{4}}+\frac {9 b^{3} {\mathrm e}^{\frac {3 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-3 i b x -3 i a -\frac {3 \left (-i a d +i c b \right )}{d}\right )}{16 d^{4}}+\frac {i \left (2 i b^{4} d^{5} x^{4}+8 i b^{4} c \,d^{4} x^{3}+12 i b^{4} c^{2} d^{3} x^{2}+8 i b^{4} c^{3} d^{2} x +2 i b^{4} c^{4} d \right ) \cos \left (x b +a \right )}{48 d^{3} \left (d x +c \right )^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {\left (-2 b^{5} d^{5} x^{5}-10 b^{5} c \,d^{4} x^{4}-20 b^{5} c^{2} d^{3} x^{3}-20 b^{5} c^{3} d^{2} x^{2}-10 b^{5} c^{4} d x +4 b^{3} d^{5} x^{3}-2 b^{5} c^{5}+12 b^{3} c \,d^{4} x^{2}+12 b^{3} c^{2} d^{3} x +4 b^{3} c^{3} d^{2}\right ) \sin \left (x b +a \right )}{48 d^{3} \left (d x +c \right )^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}+\frac {i \left (6 i b^{4} d^{5} x^{4}+24 i b^{4} c \,d^{4} x^{3}+36 i b^{4} c^{2} d^{3} x^{2}+24 i b^{4} c^{3} d^{2} x +6 i b^{4} c^{4} d \right ) \cos \left (3 x b +3 a \right )}{48 d^{3} \left (d x +c \right )^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {\left (-18 b^{5} d^{5} x^{5}-90 b^{5} c \,d^{4} x^{4}-180 b^{5} c^{2} d^{3} x^{3}-180 b^{5} c^{3} d^{2} x^{2}-90 b^{5} c^{4} d x +4 b^{3} d^{5} x^{3}-18 b^{5} c^{5}+12 b^{3} c \,d^{4} x^{2}+12 b^{3} c^{2} d^{3} x +4 b^{3} c^{3} d^{2}\right ) \sin \left (3 x b +3 a \right )}{48 d^{3} \left (d x +c \right )^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}\) | \(810\) |
1/b*(1/12*b^4*(-sin(3*b*x+3*a)/(-a*d+c*b+d*(b*x+a))^3/d+(-3/2*cos(3*b*x+3* a)/(-a*d+c*b+d*(b*x+a))^2/d-3/2*(-3*sin(3*b*x+3*a)/(-a*d+c*b+d*(b*x+a))/d+ 3*(-3*Si(-3*x*b-3*a-3*(-a*d+b*c)/d)*sin(3*(-a*d+b*c)/d)/d+3*Ci(3*x*b+3*a+3 *(-a*d+b*c)/d)*cos(3*(-a*d+b*c)/d)/d)/d)/d)/d)+1/4*b^4*(-1/3*sin(b*x+a)/(- a*d+c*b+d*(b*x+a))^3/d+1/3*(-1/2*cos(b*x+a)/(-a*d+c*b+d*(b*x+a))^2/d-1/2*( -sin(b*x+a)/(-a*d+c*b+d*(b*x+a))/d+(-Si(-x*b-a-(-a*d+b*c)/d)*sin((-a*d+b*c )/d)/d+Ci(x*b+a+(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d)/d)/d)/d))
Time = 0.27 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.64 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=-\frac {12 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{3} + 27 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) - 27 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \sin \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) - 8 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right ) + 4 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - {\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )}{24 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \]
-1/24*(12*(b*d^3*x + b*c*d^2)*cos(b*x + a)^3 + 27*(b^3*d^3*x^3 + 3*b^3*c*d ^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*cos(-3*(b*c - a*d)/d)*cos_integral(3*(b* d*x + b*c)/d) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)* cos(-(b*c - a*d)/d)*cos_integral((b*d*x + b*c)/d) - 27*(b^3*d^3*x^3 + 3*b^ 3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*sin(-3*(b*c - a*d)/d)*sin_integral( 3*(b*d*x + b*c)/d) - (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3* c^3)*sin(-(b*c - a*d)/d)*sin_integral((b*d*x + b*c)/d) - 8*(b*d^3*x + b*c* d^2)*cos(b*x + a) + 4*(2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d - (9*b^ 2*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d - 2*d^3)*cos(b*x + a)^2)*sin(b*x + a))/(d^7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4)
\[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=\int \frac {\sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \]
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.43 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=-\frac {b^{4} {\left (i \, E_{4}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) - i \, E_{4}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + b^{4} {\left (-i \, E_{4}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{4}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + b^{4} {\left (E_{4}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{4}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) + b^{4} {\left (E_{4}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{4}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + {\left (b x + a\right )}^{3} d^{4} - a^{3} d^{4} + 3 \, {\left (b c d^{3} - a d^{4}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} {\left (b x + a\right )}\right )} b} \]
-1/8*(b^4*(I*exp_integral_e(4, (I*b*c + I*(b*x + a)*d - I*a*d)/d) - I*exp_ integral_e(4, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*cos(-(b*c - a*d)/d) + b ^4*(-I*exp_integral_e(4, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_int egral_e(4, -3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-3*(b*c - a*d)/d) + b^4*(exp_integral_e(4, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + exp_integral_ e(4, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d) + b^4*(exp_i ntegral_e(4, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(4, -3* (-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-3*(b*c - a*d)/d))/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*b )
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.75 (sec) , antiderivative size = 166374, normalized size of antiderivative = 616.20 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=\text {Too large to display} \]
-1/48*(27*b^3*d^3*x^3*real_part(cos_integral(3*b*x + 3*b*c/d))*tan(3/2*b*x )^2*tan(1/2*b*x)^2*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/ d)^2 + b^3*d^3*x^3*real_part(cos_integral(b*x + b*c/d))*tan(3/2*b*x)^2*tan (1/2*b*x)^2*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + b^3*d^3*x^3*real_part(cos_integral(-b*x - b*c/d))*tan(3/2*b*x)^2*tan(1/2*b *x)^2*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 27*b^3 *d^3*x^3*real_part(cos_integral(-3*b*x - 3*b*c/d))*tan(3/2*b*x)^2*tan(1/2* b*x)^2*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 2*b^3 *d^3*x^3*imag_part(cos_integral(b*x + b*c/d))*tan(3/2*b*x)^2*tan(1/2*b*x)^ 2*tan(3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) + 2*b^3*d^3*x^ 3*imag_part(cos_integral(-b*x - b*c/d))*tan(3/2*b*x)^2*tan(1/2*b*x)^2*tan( 3/2*a)^2*tan(1/2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) - 4*b^3*d^3*x^3*sin_ integral((b*d*x + b*c)/d)*tan(3/2*b*x)^2*tan(1/2*b*x)^2*tan(3/2*a)^2*tan(1 /2*a)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) - 54*b^3*d^3*x^3*imag_part(cos_int egral(3*b*x + 3*b*c/d))*tan(3/2*b*x)^2*tan(1/2*b*x)^2*tan(3/2*a)^2*tan(1/2 *a)^2*tan(3/2*b*c/d)*tan(1/2*b*c/d)^2 + 54*b^3*d^3*x^3*imag_part(cos_integ ral(-3*b*x - 3*b*c/d))*tan(3/2*b*x)^2*tan(1/2*b*x)^2*tan(3/2*a)^2*tan(1/2* a)^2*tan(3/2*b*c/d)*tan(1/2*b*c/d)^2 - 108*b^3*d^3*x^3*sin_integral(3*(b*d *x + b*c)/d)*tan(3/2*b*x)^2*tan(1/2*b*x)^2*tan(3/2*a)^2*tan(1/2*a)^2*tan(3 /2*b*c/d)*tan(1/2*b*c/d)^2 + 2*b^3*d^3*x^3*imag_part(cos_integral(b*x +...
Timed out. \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^4} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{{\left (c+d\,x\right )}^4} \,d x \]