Integrand size = 24, antiderivative size = 185 \[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {5 b c}{d}+5 b x\right ) \sin \left (5 a-\frac {5 b c}{d}\right )}{16 d}+\frac {\operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{16 d}+\frac {\operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{8 d}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{8 d}+\frac {\cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{16 d}-\frac {\cos \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b c}{d}+5 b x\right )}{16 d} \]
1/8*cos(a-b*c/d)*Si(b*c/d+b*x)/d+1/16*cos(3*a-3*b*c/d)*Si(3*b*c/d+3*b*x)/d -1/16*cos(5*a-5*b*c/d)*Si(5*b*c/d+5*b*x)/d-1/16*Ci(5*b*c/d+5*b*x)*sin(5*a- 5*b*c/d)/d+1/16*Ci(3*b*c/d+3*b*x)*sin(3*a-3*b*c/d)/d+1/8*Ci(b*c/d+b*x)*sin (a-b*c/d)/d
Time = 0.42 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=\frac {-\operatorname {CosIntegral}\left (\frac {5 b (c+d x)}{d}\right ) \sin \left (5 a-\frac {5 b c}{d}\right )+\operatorname {CosIntegral}\left (\frac {3 b (c+d x)}{d}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )+2 \operatorname {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right ) \sin \left (a-\frac {b c}{d}\right )+2 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )+\cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )-\cos \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b (c+d x)}{d}\right )}{16 d} \]
(-(CosIntegral[(5*b*(c + d*x))/d]*Sin[5*a - (5*b*c)/d]) + CosIntegral[(3*b *(c + d*x))/d]*Sin[3*a - (3*b*c)/d] + 2*CosIntegral[b*(c/d + x)]*Sin[a - ( b*c)/d] + 2*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + Cos[3*a - (3*b*c)/ d]*SinIntegral[(3*b*(c + d*x))/d] - Cos[5*a - (5*b*c)/d]*SinIntegral[(5*b* (c + d*x))/d])/(16*d)
Time = 0.52 (sec) , antiderivative size = 185, 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.
| \(\displaystyle \int \frac {\sin ^3(a+b x) \cos ^2(a+b x)}{c+d x} \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {\sin (a+b x)}{8 (c+d x)}+\frac {\sin (3 a+3 b x)}{16 (c+d x)}-\frac {\sin (5 a+5 b x)}{16 (c+d x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sin \left (5 a-\frac {5 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {5 b c}{d}+5 b x\right )}{16 d}+\frac {\sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{16 d}+\frac {\sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{8 d}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{8 d}+\frac {\cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{16 d}-\frac {\cos \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b c}{d}+5 b x\right )}{16 d}\) |
-1/16*(CosIntegral[(5*b*c)/d + 5*b*x]*Sin[5*a - (5*b*c)/d])/d + (CosIntegr al[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(16*d) + (CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(8*d) + (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b *x])/(8*d) + (Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(16*d) - (Cos[5*a - (5*b*c)/d]*SinIntegral[(5*b*c)/d + 5*b*x])/(16*d)
3.1.93.3.1 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 = 1.24 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.39
| method | result | size |
| derivativedivides | \(\frac {\frac {b \left (-\frac {3 \,\operatorname {Si}\left (-3 x b -3 a -\frac {3 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-3 a d +3 c b}{d}\right )}{d}-\frac {3 \,\operatorname {Ci}\left (3 x b +3 a +\frac {-3 a d +3 c b}{d}\right ) \sin \left (\frac {-3 a d +3 c b}{d}\right )}{d}\right )}{48}+\frac {b \left (-\frac {\operatorname {Si}\left (-x b -a -\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}-\frac {\operatorname {Ci}\left (x b +a +\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}\right )}{8}-\frac {b \left (-\frac {5 \,\operatorname {Si}\left (-5 x b -5 a -\frac {5 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-5 a d +5 c b}{d}\right )}{d}-\frac {5 \,\operatorname {Ci}\left (5 x b +5 a +\frac {-5 a d +5 c b}{d}\right ) \sin \left (\frac {-5 a d +5 c b}{d}\right )}{d}\right )}{80}}{b}\) | \(258\) |
| default | \(\frac {\frac {b \left (-\frac {3 \,\operatorname {Si}\left (-3 x b -3 a -\frac {3 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-3 a d +3 c b}{d}\right )}{d}-\frac {3 \,\operatorname {Ci}\left (3 x b +3 a +\frac {-3 a d +3 c b}{d}\right ) \sin \left (\frac {-3 a d +3 c b}{d}\right )}{d}\right )}{48}+\frac {b \left (-\frac {\operatorname {Si}\left (-x b -a -\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}-\frac {\operatorname {Ci}\left (x b +a +\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}\right )}{8}-\frac {b \left (-\frac {5 \,\operatorname {Si}\left (-5 x b -5 a -\frac {5 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-5 a d +5 c b}{d}\right )}{d}-\frac {5 \,\operatorname {Ci}\left (5 x b +5 a +\frac {-5 a d +5 c b}{d}\right ) \sin \left (\frac {-5 a d +5 c b}{d}\right )}{d}\right )}{80}}{b}\) | \(258\) |
| risch | \(\frac {i {\mathrm e}^{-\frac {5 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (5 i b x +5 i a -\frac {5 i \left (a d -c b \right )}{d}\right )}{32 d}-\frac {i {\mathrm e}^{-\frac {3 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (3 i b x +3 i a -\frac {3 i \left (a d -c b \right )}{d}\right )}{32 d}-\frac {i {\mathrm e}^{-\frac {i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (i b x +i a -\frac {i \left (a d -c b \right )}{d}\right )}{16 d}+\frac {i {\mathrm e}^{\frac {i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-i b x -i a -\frac {-i a d +i c b}{d}\right )}{16 d}+\frac {i {\mathrm e}^{\frac {3 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-3 i b x -3 i a -\frac {3 \left (-i a d +i c b \right )}{d}\right )}{32 d}-\frac {i {\mathrm e}^{\frac {5 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-5 i b x -5 i a -\frac {5 \left (-i a d +i c b \right )}{d}\right )}{32 d}\) | \(290\) |
1/b*(1/48*b*(-3*Si(-3*x*b-3*a-3*(-a*d+b*c)/d)*cos(3*(-a*d+b*c)/d)/d-3*Ci(3 *x*b+3*a+3*(-a*d+b*c)/d)*sin(3*(-a*d+b*c)/d)/d)+1/8*b*(-Si(-x*b-a-(-a*d+b* c)/d)*cos((-a*d+b*c)/d)/d-Ci(x*b+a+(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d)-1/80 *b*(-5*Si(-5*x*b-5*a-5*(-a*d+b*c)/d)*cos(5*(-a*d+b*c)/d)/d-5*Ci(5*x*b+5*a+ 5*(-a*d+b*c)/d)*sin(5*(-a*d+b*c)/d)/d))
Time = 0.25 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.98 \[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=\frac {2 \, \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + \operatorname {Ci}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - \operatorname {Ci}\left (\frac {5 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) - \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {5 \, {\left (b d x + b c\right )}}{d}\right ) + \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + 2 \, \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right )}{16 \, d} \]
1/16*(2*cos_integral((b*d*x + b*c)/d)*sin(-(b*c - a*d)/d) + cos_integral(3 *(b*d*x + b*c)/d)*sin(-3*(b*c - a*d)/d) - cos_integral(5*(b*d*x + b*c)/d)* sin(-5*(b*c - a*d)/d) - cos(-5*(b*c - a*d)/d)*sin_integral(5*(b*d*x + b*c) /d) + cos(-3*(b*c - a*d)/d)*sin_integral(3*(b*d*x + b*c)/d) + 2*cos(-(b*c - a*d)/d)*sin_integral((b*d*x + b*c)/d))/d
\[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=\int \frac {\sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.24 \[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=-\frac {2 \, b {\left (i \, E_{1}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) - i \, E_{1}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) - b {\left (i \, E_{1}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{1}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - b {\left (-i \, E_{1}\left (\frac {5 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{1}\left (-\frac {5 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, b {\left (E_{1}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{1}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) + b {\left (E_{1}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - b {\left (E_{1}\left (\frac {5 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {5 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {5 \, {\left (b c - a d\right )}}{d}\right )}{32 \, b d} \]
-1/32*(2*b*(I*exp_integral_e(1, (I*b*c + I*(b*x + a)*d - I*a*d)/d) - I*exp _integral_e(1, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*cos(-(b*c - a*d)/d) - b*(I*exp_integral_e(1, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) - I*exp_integ ral_e(1, -3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-3*(b*c - a*d)/d) - b *(-I*exp_integral_e(1, 5*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_integ ral_e(1, -5*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-5*(b*c - a*d)/d) + 2 *b*(exp_integral_e(1, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + exp_integral_e( 1, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d) + b*(exp_integ ral_e(1, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(1, -3*(-I* b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-3*(b*c - a*d)/d) - b*(exp_integral_e (1, 5*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(1, -5*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-5*(b*c - a*d)/d))/(b*d)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.76 (sec) , antiderivative size = 46675, normalized size of antiderivative = 252.30 \[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=\text {Too large to display} \]
-1/32*(imag_part(cos_integral(5*b*x + 5*b*c/d))*tan(5/2*a)^2*tan(3/2*a)^2* tan(1/2*a)^2*tan(5/2*b*c/d)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - imag_par t(cos_integral(3*b*x + 3*b*c/d))*tan(5/2*a)^2*tan(3/2*a)^2*tan(1/2*a)^2*ta n(5/2*b*c/d)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 2*imag_part(cos_integra l(b*x + b*c/d))*tan(5/2*a)^2*tan(3/2*a)^2*tan(1/2*a)^2*tan(5/2*b*c/d)^2*ta n(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 2*imag_part(cos_integral(-b*x - b*c/d))* tan(5/2*a)^2*tan(3/2*a)^2*tan(1/2*a)^2*tan(5/2*b*c/d)^2*tan(3/2*b*c/d)^2*t an(1/2*b*c/d)^2 + imag_part(cos_integral(-3*b*x - 3*b*c/d))*tan(5/2*a)^2*t an(3/2*a)^2*tan(1/2*a)^2*tan(5/2*b*c/d)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^ 2 - imag_part(cos_integral(-5*b*x - 5*b*c/d))*tan(5/2*a)^2*tan(3/2*a)^2*ta n(1/2*a)^2*tan(5/2*b*c/d)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 + 2*sin_inte gral(5*(b*d*x + b*c)/d)*tan(5/2*a)^2*tan(3/2*a)^2*tan(1/2*a)^2*tan(5/2*b*c /d)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 2*sin_integral(3*(b*d*x + b*c)/d )*tan(5/2*a)^2*tan(3/2*a)^2*tan(1/2*a)^2*tan(5/2*b*c/d)^2*tan(3/2*b*c/d)^2 *tan(1/2*b*c/d)^2 - 4*sin_integral((b*d*x + b*c)/d)*tan(5/2*a)^2*tan(3/2*a )^2*tan(1/2*a)^2*tan(5/2*b*c/d)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d)^2 - 4*re al_part(cos_integral(b*x + b*c/d))*tan(5/2*a)^2*tan(3/2*a)^2*tan(1/2*a)^2* tan(5/2*b*c/d)^2*tan(3/2*b*c/d)^2*tan(1/2*b*c/d) - 4*real_part(cos_integra l(-b*x - b*c/d))*tan(5/2*a)^2*tan(3/2*a)^2*tan(1/2*a)^2*tan(5/2*b*c/d)^2*t an(3/2*b*c/d)^2*tan(1/2*b*c/d) - 2*real_part(cos_integral(3*b*x + 3*b*c...
Timed out. \[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{c+d x} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3}{c+d\,x} \,d x \]