Integrand size = 24, antiderivative size = 158 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=-\frac {1}{24 d (c+d x)^3}+\frac {\cos (4 a+4 b x)}{24 d (c+d x)^3}-\frac {b^2 \cos (4 a+4 b x)}{3 d^3 (c+d x)}-\frac {4 b^3 \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right ) \sin \left (4 a-\frac {4 b c}{d}\right )}{3 d^4}-\frac {b \sin (4 a+4 b x)}{12 d^2 (c+d x)^2}-\frac {4 b^3 \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{3 d^4} \]
-1/24/d/(d*x+c)^3+1/24*cos(4*b*x+4*a)/d/(d*x+c)^3-1/3*b^2*cos(4*b*x+4*a)/d ^3/(d*x+c)-4/3*b^3*cos(4*a-4*b*c/d)*Si(4*b*c/d+4*b*x)/d^4-4/3*b^3*Ci(4*b*c /d+4*b*x)*sin(4*a-4*b*c/d)/d^4-1/12*b*sin(4*b*x+4*a)/d^2/(d*x+c)^2
Time = 1.80 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=-\frac {32 b^3 \operatorname {CosIntegral}\left (\frac {4 b (c+d x)}{d}\right ) \sin \left (4 a-\frac {4 b c}{d}\right )+\frac {d \left (\left (-d^2+8 b^2 (c+d x)^2\right ) \cos (4 (a+b x))+d (d+2 b (c+d x) \sin (4 (a+b x)))\right )}{(c+d x)^3}+32 b^3 \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b (c+d x)}{d}\right )}{24 d^4} \]
-1/24*(32*b^3*CosIntegral[(4*b*(c + d*x))/d]*Sin[4*a - (4*b*c)/d] + (d*((- d^2 + 8*b^2*(c + d*x)^2)*Cos[4*(a + b*x)] + d*(d + 2*b*(c + d*x)*Sin[4*(a + b*x)])))/(c + d*x)^3 + 32*b^3*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*(c + d*x))/d])/d^4
Time = 0.43 (sec) , antiderivative size = 158, 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.
| \(\displaystyle \int \frac {\sin ^2(a+b x) \cos ^2(a+b x)}{(c+d x)^4} \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {1}{8 (c+d x)^4}-\frac {\cos (4 a+4 b x)}{8 (c+d x)^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 b^3 \sin \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{3 d^4}-\frac {4 b^3 \cos \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{3 d^4}-\frac {b^2 \cos (4 a+4 b x)}{3 d^3 (c+d x)}-\frac {b \sin (4 a+4 b x)}{12 d^2 (c+d x)^2}+\frac {\cos (4 a+4 b x)}{24 d (c+d x)^3}-\frac {1}{24 d (c+d x)^3}\) |
-1/24*1/(d*(c + d*x)^3) + Cos[4*a + 4*b*x]/(24*d*(c + d*x)^3) - (b^2*Cos[4 *a + 4*b*x])/(3*d^3*(c + d*x)) - (4*b^3*CosIntegral[(4*b*c)/d + 4*b*x]*Sin [4*a - (4*b*c)/d])/(3*d^4) - (b*Sin[4*a + 4*b*x])/(12*d^2*(c + d*x)^2) - ( 4*b^3*Cos[4*a - (4*b*c)/d]*SinIntegral[(4*b*c)/d + 4*b*x])/(3*d^4)
3.1.87.3.1 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 = 2.02 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.46
| method | result | size |
| derivativedivides | \(\frac {-\frac {b^{4} \left (-\frac {4 \cos \left (4 x b +4 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}-\frac {4 \left (-\frac {2 \sin \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {8 \cos \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {8 \left (-\frac {4 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}-\frac {4 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{3 d}\right )}{32}-\frac {b^{4}}{24 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}}{b}\) | \(230\) |
| default | \(\frac {-\frac {b^{4} \left (-\frac {4 \cos \left (4 x b +4 a \right )}{3 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}-\frac {4 \left (-\frac {2 \sin \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {8 \cos \left (4 x b +4 a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {8 \left (-\frac {4 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}-\frac {4 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}\right )}{d}}{d}\right )}{3 d}\right )}{32}-\frac {b^{4}}{24 \left (-a d +c b +d \left (x b +a \right )\right )^{3} d}}{b}\) | \(230\) |
| risch | \(-\frac {1}{24 d \left (d x +c \right )^{3}}+\frac {2 i b^{3} {\mathrm e}^{-\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (4 i b x +4 i a -\frac {4 i \left (a d -c b \right )}{d}\right )}{3 d^{4}}-\frac {2 i b^{3} {\mathrm e}^{\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-4 i b x -4 i a -\frac {4 \left (-i a d +i c b \right )}{d}\right )}{3 d^{4}}+\frac {\left (-16 b^{5} d^{5} x^{5}-80 b^{5} c \,d^{4} x^{4}-160 b^{5} c^{2} d^{3} x^{3}-160 b^{5} c^{3} d^{2} x^{2}-80 b^{5} c^{4} d x +2 b^{3} d^{5} x^{3}-16 b^{5} c^{5}+6 b^{3} c \,d^{4} x^{2}+6 b^{3} c^{2} d^{3} x +2 b^{3} c^{3} d^{2}\right ) \cos \left (4 x b +4 a \right )}{48 d^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right ) \left (d x +c \right )^{3}}+\frac {i \left (4 i b^{4} d^{5} x^{4}+16 i b^{4} c \,d^{4} x^{3}+24 i b^{4} c^{2} d^{3} x^{2}+16 i b^{4} c^{3} d^{2} x +4 i b^{4} c^{4} d \right ) \sin \left (4 x b +4 a \right )}{48 d^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right ) \left (d x +c \right )^{3}}\) | \(423\) |
1/b*(-1/32*b^4*(-4/3*cos(4*b*x+4*a)/(-a*d+c*b+d*(b*x+a))^3/d-4/3*(-2*sin(4 *b*x+4*a)/(-a*d+c*b+d*(b*x+a))^2/d+2*(-4*cos(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)*cos(4*(-a*d+b*c)/d)/d-4*Ci(4*x*b+ 4*a+4*(-a*d+b*c)/d)*sin(4*(-a*d+b*c)/d)/d)/d)/d)/d)-1/24*b^4/(-a*d+c*b+d*( b*x+a))^3/d)
Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (146) = 292\).
Time = 0.27 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.21 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=-\frac {b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d + {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{4} - {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{2} + 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \operatorname {Ci}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (2 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{3} - {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{3 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \]
-1/3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d + (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)^4 - (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)^2 + 4*(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_integral(4*(b*d*x + b*c)/d)*sin(-4*(b*c - a* d)/d) + 4*(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)*sin_integral(4*(b*d*x + b*c)/d) + (2*(b*d^3*x + b*c*d^2)*c os(b*x + a)^3 - (b*d^3*x + b*c*d^2)*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)
\[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=\int \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \]
Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.63 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=\frac {3 \, b^{4} {\left (E_{4}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{4}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, b^{4} {\left (i \, E_{4}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{4}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, b^{4}}{48 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + {\left (b x + a\right )}^{3} d^{4} - a^{3} d^{4} + 3 \, {\left (b c d^{3} - a d^{4}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} {\left (b x + a\right )}\right )} b} \]
1/48*(3*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))*cos(-4*(b*c - a*d)/ d) + 3*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))*sin(-4*(b*c - a* d)/d) - 2*b^4)/((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 = 0.57 (sec) , antiderivative size = 8508, normalized size of antiderivative = 53.85 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=\text {Too large to display} \]
-1/12*(8*b^3*d^3*x^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2 *tan(2*a)^2*tan(2*b*c/d)^2 - 8*b^3*d^3*x^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 16*b^3*d^3*x^3*sin_int egral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 16*b^3*d ^3*x^3*real_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*ta n(2*b*c/d) + 16*b^3*d^3*x^3*real_part(cos_integral(-4*b*x - 4*b*c/d))*tan( 2*b*x)^2*tan(2*a)^2*tan(2*b*c/d) - 16*b^3*d^3*x^3*real_part(cos_integral(4 *b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^2 - 16*b^3*d^3*x^3*rea l_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d)^ 2 + 24*b^3*c*d^2*x^2*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2 *tan(2*a)^2*tan(2*b*c/d)^2 - 24*b^3*c*d^2*x^2*imag_part(cos_integral(-4*b* x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 + 48*b^3*c*d^2*x^2*si n_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)^2*tan(2*b*c/d)^2 - 8*b ^3*d^3*x^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b*x)^2*tan(2*a)^ 2 + 8*b^3*d^3*x^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*t an(2*a)^2 - 16*b^3*d^3*x^3*sin_integral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*ta n(2*a)^2 + 32*b^3*d^3*x^3*imag_part(cos_integral(4*b*x + 4*b*c/d))*tan(2*b *x)^2*tan(2*a)*tan(2*b*c/d) - 32*b^3*d^3*x^3*imag_part(cos_integral(-4*b*x - 4*b*c/d))*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 64*b^3*d^3*x^3*sin_integ ral(4*(b*d*x + b*c)/d)*tan(2*b*x)^2*tan(2*a)*tan(2*b*c/d) + 48*b^3*c*d^...
Timed out. \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{(c+d x)^4} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^4} \,d x \]