Integrand size = 22, antiderivative size = 287 \[ \int \frac {\cos (a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=-\frac {b \cos (2 a+2 b x)}{12 d^2 (c+d x)^2}+\frac {b \cos (4 a+4 b x)}{12 d^2 (c+d x)^2}-\frac {b^3 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4}+\frac {4 b^3 \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{3 d^4}-\frac {\sin (2 a+2 b x)}{12 d (c+d x)^3}+\frac {b^2 \sin (2 a+2 b x)}{6 d^3 (c+d x)}+\frac {\sin (4 a+4 b x)}{24 d (c+d x)^3}-\frac {b^2 \sin (4 a+4 b x)}{3 d^3 (c+d x)}+\frac {b^3 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4}-\frac {4 b^3 \sin \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{3 d^4} \]
4/3*b^3*Ci(4*b*c/d+4*b*x)*cos(4*a-4*b*c/d)/d^4-1/3*b^3*Ci(2*b*c/d+2*b*x)*c os(2*a-2*b*c/d)/d^4-1/12*b*cos(2*b*x+2*a)/d^2/(d*x+c)^2+1/12*b*cos(4*b*x+4 *a)/d^2/(d*x+c)^2-4/3*b^3*Si(4*b*c/d+4*b*x)*sin(4*a-4*b*c/d)/d^4+1/3*b^3*S i(2*b*c/d+2*b*x)*sin(2*a-2*b*c/d)/d^4-1/12*sin(2*b*x+2*a)/d/(d*x+c)^3+1/6* b^2*sin(2*b*x+2*a)/d^3/(d*x+c)+1/24*sin(4*b*x+4*a)/d/(d*x+c)^3-1/3*b^2*sin (4*b*x+4*a)/d^3/(d*x+c)
Time = 2.28 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.10 \[ \int \frac {\cos (a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=\frac {-2 d \cos (2 b x) \left (b d (c+d x) \cos (2 a)+\left (d^2-2 b^2 (c+d x)^2\right ) \sin (2 a)\right )+d \cos (4 b x) \left (2 b d (c+d x) \cos (4 a)+\left (d^2-8 b^2 (c+d x)^2\right ) \sin (4 a)\right )+2 d \left (\left (-d^2+2 b^2 (c+d x)^2\right ) \cos (2 a)+b d (c+d x) \sin (2 a)\right ) \sin (2 b x)-d \left (\left (-d^2+8 b^2 (c+d x)^2\right ) \cos (4 a)+2 b d (c+d x) \sin (4 a)\right ) \sin (4 b x)-8 b^3 (c+d x)^3 \left (\cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right )-\sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )\right )+32 b^3 (c+d x)^3 \left (\cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b (c+d x)}{d}\right )-\sin \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b (c+d x)}{d}\right )\right )}{24 d^4 (c+d x)^3} \]
(-2*d*Cos[2*b*x]*(b*d*(c + d*x)*Cos[2*a] + (d^2 - 2*b^2*(c + d*x)^2)*Sin[2 *a]) + d*Cos[4*b*x]*(2*b*d*(c + d*x)*Cos[4*a] + (d^2 - 8*b^2*(c + d*x)^2)* Sin[4*a]) + 2*d*((-d^2 + 2*b^2*(c + d*x)^2)*Cos[2*a] + b*d*(c + d*x)*Sin[2 *a])*Sin[2*b*x] - d*((-d^2 + 8*b^2*(c + d*x)^2)*Cos[4*a] + 2*b*d*(c + d*x) *Sin[4*a])*Sin[4*b*x] - 8*b^3*(c + d*x)^3*(Cos[2*a - (2*b*c)/d]*CosIntegra l[(2*b*(c + d*x))/d] - Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*(c + d*x))/d] ) + 32*b^3*(c + d*x)^3*(Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*(c + d*x))/d ] - Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d]))/(24*d^4*(c + d*x )^3)
Time = 0.64 (sec) , antiderivative size = 287, 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.
| \(\displaystyle \int \frac {\sin ^3(a+b x) \cos (a+b x)}{(c+d x)^4} \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {\sin (2 a+2 b x)}{4 (c+d x)^4}-\frac {\sin (4 a+4 b x)}{8 (c+d x)^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^3 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4}+\frac {4 b^3 \cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{3 d^4}+\frac {b^3 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4}-\frac {4 b^3 \sin \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{3 d^4}+\frac {b^2 \sin (2 a+2 b x)}{6 d^3 (c+d x)}-\frac {b^2 \sin (4 a+4 b x)}{3 d^3 (c+d x)}-\frac {b \cos (2 a+2 b x)}{12 d^2 (c+d x)^2}+\frac {b \cos (4 a+4 b x)}{12 d^2 (c+d x)^2}-\frac {\sin (2 a+2 b x)}{12 d (c+d x)^3}+\frac {\sin (4 a+4 b x)}{24 d (c+d x)^3}\) |
-1/12*(b*Cos[2*a + 2*b*x])/(d^2*(c + d*x)^2) + (b*Cos[4*a + 4*b*x])/(12*d^ 2*(c + d*x)^2) - (b^3*Cos[2*a - (2*b*c)/d]*CosIntegral[(2*b*c)/d + 2*b*x]) /(3*d^4) + (4*b^3*Cos[4*a - (4*b*c)/d]*CosIntegral[(4*b*c)/d + 4*b*x])/(3* d^4) - Sin[2*a + 2*b*x]/(12*d*(c + d*x)^3) + (b^2*Sin[2*a + 2*b*x])/(6*d^3 *(c + d*x)) + Sin[4*a + 4*b*x]/(24*d*(c + d*x)^3) - (b^2*Sin[4*a + 4*b*x]) /(3*d^3*(c + d*x)) + (b^3*Sin[2*a - (2*b*c)/d]*SinIntegral[(2*b*c)/d + 2*b *x])/(3*d^4) - (4*b^3*Sin[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x]) /(3*d^4)
3.1.30.3.1 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 = 2.12 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.41
| method | result | size |
| derivativedivides | \(\frac {\frac {b^{4} \left (-\frac {2 \sin \left (2 x b +2 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}+\frac {-\frac {2 \cos \left (2 x b +2 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {2 \left (-\frac {2 \sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {4 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}}{d}\right )}{3 d}}{d}\right )}{8}-\frac {b^{4} \left (-\frac {4 \sin \left (4 x b +4 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}+\frac {-\frac {8 \cos \left (4 x b +4 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {8 \left (-\frac {4 \sin \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {16 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}+\frac {16 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}}{d}\right )}{3 d}}{d}\right )}{32}}{b}\) | \(404\) |
| default | \(\frac {\frac {b^{4} \left (-\frac {2 \sin \left (2 x b +2 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}+\frac {-\frac {2 \cos \left (2 x b +2 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {2 \left (-\frac {2 \sin \left (2 x b +2 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {4 \,\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}}{d}\right )}{3 d}}{d}\right )}{8}-\frac {b^{4} \left (-\frac {4 \sin \left (4 x b +4 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}+\frac {-\frac {8 \cos \left (4 x b +4 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}-\frac {8 \left (-\frac {4 \sin \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}+\frac {-\frac {16 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}+\frac {16 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}}{d}\right )}{3 d}}{d}\right )}{32}}{b}\) | \(404\) |
| risch | \(\frac {b^{3} {\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 )}{6 d^{4}}-\frac {2 b^{3} {\mathrm e}^{-\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (4 i b x +4 i a -\frac {4 i \left (a d -c b \right )}{d}\right )}{3 d^{4}}+\frac {b^{3} {\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 )}{6 d^{4}}-\frac {2 b^{3} {\mathrm e}^{\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-4 i b x -4 i a -\frac {4 \left (-i a d +i c b \right )}{d}\right )}{3 d^{4}}-\frac {i \left (4 i b^{4} d^{5} x^{4}+16 i b^{4} c \,d^{4} x^{3}+24 i b^{4} c^{2} d^{3} x^{2}+16 i b^{4} c^{3} d^{2} x +4 i b^{4} c^{4} d \right ) \cos \left (4 x b +4 a \right )}{48 d^{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 ) \left (d x +c \right )^{3}}+\frac {\left (-16 b^{5} d^{5} x^{5}-80 b^{5} c \,d^{4} x^{4}-160 b^{5} c^{2} d^{3} x^{3}-160 b^{5} c^{3} d^{2} x^{2}-80 b^{5} c^{4} d x +2 b^{3} d^{5} x^{3}-16 b^{5} c^{5}+6 b^{3} c \,d^{4} x^{2}+6 b^{3} c^{2} d^{3} x +2 b^{3} c^{3} d^{2}\right ) \sin \left (4 x b +4 a \right )}{48 d^{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 ) \left (d x +c \right )^{3}}+\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 (2 x b +2 a \right )}{24 d^{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 ) \left (d x +c \right )^{3}}-\frac {\left (-4 b^{5} d^{5} x^{5}-20 b^{5} c \,d^{4} x^{4}-40 b^{5} c^{2} d^{3} x^{3}-40 b^{5} c^{3} d^{2} x^{2}-20 b^{5} c^{4} d x +2 b^{3} d^{5} x^{3}-4 b^{5} c^{5}+6 b^{3} c \,d^{4} x^{2}+6 b^{3} c^{2} d^{3} x +2 b^{3} c^{3} d^{2}\right ) \sin \left (2 x b +2 a \right )}{24 d^{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 ) \left (d x +c \right )^{3}}\) | \(816\) |
1/b*(1/8*b^4*(-2/3*sin(2*b*x+2*a)/(-a*d+c*b+d*(b*x+a))^3/d+2/3*(-cos(2*b*x +2*a)/(-a*d+c*b+d*(b*x+a))^2/d-(-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)/d)/d)-1/32*b^4*(-4/3*sin(4*b*x+4*a )/(-a*d+c*b+d*(b*x+a))^3/d+4/3*(-2*cos(4*b*x+4*a)/(-a*d+c*b+d*(b*x+a))^2/d -2*(-4*sin(4*b*x+4*a)/(-a*d+c*b+d*(b*x+a))/d+4*(-4*Si(-4*x*b-4*a-4*(-a*d+b *c)/d)*sin(4*(-a*d+b*c)/d)/d+4*Ci(4*x*b+4*a+4*(-a*d+b*c)/d)*cos(4*(-a*d+b* c)/d)/d)/d)/d)/d))
Time = 0.27 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.65 \[ \int \frac {\cos (a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=\frac {b d^{3} x + 4 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{4} + b c d^{2} - 5 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{2} + 8 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) - 2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - 8 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) + 2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - 2 \, {\left ({\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{3} - {\left (5 \, b^{2} d^{3} x^{2} + 10 \, b^{2} c d^{2} x + 5 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{6 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \]
1/6*(b*d^3*x + 4*(b*d^3*x + b*c*d^2)*cos(b*x + a)^4 + b*c*d^2 - 5*(b*d^3*x + b*c*d^2)*cos(b*x + a)^2 + 8*(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(-4*(b*c - a*d)/d)*cos_integral(4*(b*d*x + b*c)/d) - 2*( 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(-2*(b*c - a*d )/d)*cos_integral(2*(b*d*x + b*c)/d) - 8*(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(-4*(b*c - a*d)/d)*sin_integral(4*(b*d*x + b*c )/d) + 2*(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(-2* (b*c - a*d)/d)*sin_integral(2*(b*d*x + b*c)/d) - 2*((8*b^2*d^3*x^2 + 16*b^ 2*c*d^2*x + 8*b^2*c^2*d - d^3)*cos(b*x + a)^3 - (5*b^2*d^3*x^2 + 10*b^2*c* d^2*x + 5*b^2*c^2*d - d^3)*cos(b*x + a))*sin(b*x + a))/(d^7*x^3 + 3*c*d^6* x^2 + 3*c^2*d^5*x + c^3*d^4)
\[ \int \frac {\cos (a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=\int \frac {\sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \]
Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.37 \[ \int \frac {\cos (a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=-\frac {2 \, b^{4} {\left (-i \, E_{4}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{4}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - b^{4} {\left (-i \, E_{4}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{4}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, b^{4} {\left (E_{4}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{4}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - b^{4} {\left (E_{4}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{4}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right )}{16 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + {\left (b x + a\right )}^{3} d^{4} - a^{3} d^{4} + 3 \, {\left (b c d^{3} - a d^{4}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} {\left (b x + a\right )}\right )} b} \]
-1/16*(2*b^4*(-I*exp_integral_e(4, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_integral_e(4, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-2*(b*c - a*d)/d) - b^4*(-I*exp_integral_e(4, 4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_integral_e(4, -4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-4*(b*c - a*d)/d) + 2*b^4*(exp_integral_e(4, 2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d ) + exp_integral_e(4, -2*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-2*(b*c - a*d)/d) - b^4*(exp_integral_e(4, 4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(4, -4*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-4*(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 = 4.53 (sec) , antiderivative size = 157526, normalized size of antiderivative = 548.87 \[ \int \frac {\cos (a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=\text {Too large to display} \]
1/12*(8*b^3*d^3*x^3*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2* tan(b*x)^2*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*b^3*d^3*x^3 *real_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a) ^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d)^2 - 2*b^3*d^3*x^3*real_part(cos_inte gral(-2*b*x - 2*b*c/d))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2*tan(2* b*c/d)^2*tan(b*c/d)^2 + 8*b^3*d^3*x^3*real_part(cos_integral(-4*b*x - 4*b* c/d))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d )^2 + 4*b^3*d^3*x^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan(2*b*x)^2* tan(b*x)^2*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d) - 4*b^3*d^3*x^3*i mag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^ 2*tan(a)^2*tan(2*b*c/d)^2*tan(b*c/d) + 8*b^3*d^3*x^3*sin_integral(2*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)^2*tan( b*c/d) - 16*b^3*d^3*x^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x )^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^2 + 16*b^3*d^3* x^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(b*x)^2*tan( 2*a)^2*tan(a)^2*tan(2*b*c/d)*tan(b*c/d)^2 - 32*b^3*d^3*x^3*sin_integral(4* (b*d*x + b*c)/d)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2*tan(2*b*c/d)* tan(b*c/d)^2 - 4*b^3*d^3*x^3*imag_part(cos_integral(2*b*x + 2*b*c/d))*tan( 2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)*tan(2*b*c/d)^2*tan(b*c/d)^2 + 4*b^3* d^3*x^3*imag_part(cos_integral(-2*b*x - 2*b*c/d))*tan(2*b*x)^2*tan(b*x)...
Timed out. \[ \int \frac {\cos (a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=\int \frac {\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^4} \,d x \]