Integrand size = 16, antiderivative size = 145 \[ \int \frac {\cos ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {\cos ^3(a+b x)}{d (c+d x)}-\frac {3 b \operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{4 d^2}-\frac {3 b \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{4 d^2}-\frac {3 b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{4 d^2}-\frac {3 b \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2} \]
-cos(b*x+a)^3/d/(d*x+c)-3/4*b*cos(a-b*c/d)*Si(b*c/d+b*x)/d^2-3/4*b*cos(3*a -3*b*c/d)*Si(3*b*c/d+3*b*x)/d^2-3/4*b*Ci(3*b*c/d+3*b*x)*sin(3*a-3*b*c/d)/d ^2-3/4*b*Ci(b*c/d+b*x)*sin(a-b*c/d)/d^2
Time = 0.74 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.38 \[ \int \frac {\cos ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {3 d \cos (a+b x)+d \cos (3 (a+b x))+3 b (c+d x) \operatorname {CosIntegral}\left (\frac {3 b (c+d x)}{d}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )+3 b (c+d x) \operatorname {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right ) \sin \left (a-\frac {b c}{d}\right )+3 b c \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )+3 b d x \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )+3 b c \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )+3 b d x \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )}{4 d^2 (c+d x)} \]
-1/4*(3*d*Cos[a + b*x] + d*Cos[3*(a + b*x)] + 3*b*(c + d*x)*CosIntegral[(3 *b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] + 3*b*(c + d*x)*CosIntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + 3*b*c*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + 3*b*d*x*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + 3*b*c*Cos[3*a - (3*b*c )/d]*SinIntegral[(3*b*(c + d*x))/d] + 3*b*d*x*Cos[3*a - (3*b*c)/d]*SinInte gral[(3*b*(c + d*x))/d])/(d^2*(c + d*x))
Time = 0.44 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3042, 3794, 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.
| \(\displaystyle \int \frac {\cos ^3(a+b x)}{(c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )^3}{(c+d x)^2}dx\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle \frac {3 b \int \left (-\frac {\sin (a+b x)}{4 (c+d x)}-\frac {\sin (3 a+3 b x)}{4 (c+d x)}\right )dx}{d}-\frac {\cos ^3(a+b x)}{d (c+d x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b \left (-\frac {\sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}-\frac {\sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{4 d}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{4 d}-\frac {\cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{4 d}\right )}{d}-\frac {\cos ^3(a+b x)}{d (c+d x)}\) |
-(Cos[a + b*x]^3/(d*(c + d*x))) + (3*b*(-1/4*(CosIntegral[(3*b*c)/d + 3*b* x]*Sin[3*a - (3*b*c)/d])/d - (CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d]) /(4*d) - (Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(4*d) - (Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(4*d)))/d
3.1.21.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 ))) Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & & LtQ[m, -1]
Time = 0.98 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.70
| method | result | size |
| derivativedivides | \(\frac {\frac {b^{2} \left (-\frac {3 \cos \left (3 b x +3 a \right )}{\left (-a d +b c +d \left (b x +a \right )\right ) d}-\frac {3 \left (-\frac {3 \,\operatorname {Si}\left (-3 b x -3 a -\frac {3 \left (-a d +b c \right )}{d}\right ) \cos \left (\frac {-3 a d +3 b c}{d}\right )}{d}-\frac {3 \,\operatorname {Ci}\left (3 b x +3 a +\frac {-3 a d +3 b c}{d}\right ) \sin \left (\frac {-3 a d +3 b c}{d}\right )}{d}\right )}{d}\right )}{12}+\frac {3 b^{2} \left (-\frac {\cos \left (b x +a \right )}{\left (-a d +b c +d \left (b x +a \right )\right ) d}-\frac {-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}}{d}\right )}{4}}{b}\) | \(247\) |
| default | \(\frac {\frac {b^{2} \left (-\frac {3 \cos \left (3 b x +3 a \right )}{\left (-a d +b c +d \left (b x +a \right )\right ) d}-\frac {3 \left (-\frac {3 \,\operatorname {Si}\left (-3 b x -3 a -\frac {3 \left (-a d +b c \right )}{d}\right ) \cos \left (\frac {-3 a d +3 b c}{d}\right )}{d}-\frac {3 \,\operatorname {Ci}\left (3 b x +3 a +\frac {-3 a d +3 b c}{d}\right ) \sin \left (\frac {-3 a d +3 b c}{d}\right )}{d}\right )}{d}\right )}{12}+\frac {3 b^{2} \left (-\frac {\cos \left (b x +a \right )}{\left (-a d +b c +d \left (b x +a \right )\right ) d}-\frac {-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}}{d}\right )}{4}}{b}\) | \(247\) |
| risch | \(\frac {3 i b \,{\mathrm e}^{-\frac {3 i \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (3 i x b +3 i a -\frac {3 i \left (a d -b c \right )}{d}\right )}{8 d^{2}}+\frac {3 i b \,{\mathrm e}^{-\frac {i \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (i x b +i a -\frac {i \left (a d -b c \right )}{d}\right )}{8 d^{2}}-\frac {3 i b \,{\mathrm e}^{\frac {i \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (-i x b -i a -\frac {-i a d +i b c}{d}\right )}{8 d^{2}}-\frac {3 i b \,{\mathrm e}^{\frac {3 i \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (-3 i x b -3 i a -\frac {3 \left (-i a d +i b c \right )}{d}\right )}{8 d^{2}}-\frac {3 \left (-2 b x d -2 b c \right ) \cos \left (b x +a \right )}{8 d \left (d x +c \right ) \left (-b x d -b c \right )}-\frac {\left (-2 b x d -2 b c \right ) \cos \left (3 b x +3 a \right )}{8 d \left (d x +c \right ) \left (-b x d -b c \right )}\) | \(281\) |
1/b*(1/12*b^2*(-3*cos(3*b*x+3*a)/(-a*d+b*c+d*(b*x+a))/d-3*(-3*Si(-3*b*x-3* a-3*(-a*d+b*c)/d)*cos(3*(-a*d+b*c)/d)/d-3*Ci(3*b*x+3*a+3*(-a*d+b*c)/d)*sin (3*(-a*d+b*c)/d)/d)/d)+3/4*b^2*(-cos(b*x+a)/(-a*d+b*c+d*(b*x+a))/d-(-Si(-b *x-a-(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d-Ci(b*x+a+(-a*d+b*c)/d)*sin((-a*d+b* c)/d)/d)/d))
Time = 0.32 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.22 \[ \int \frac {\cos ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {4 \, d \cos \left (b x + a\right )^{3} + 3 \, {\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 3 \, {\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, {\left (b d x + b c\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + 3 \, {\left (b d x + b c\right )} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right )}{4 \, {\left (d^{3} x + c d^{2}\right )}} \]
-1/4*(4*d*cos(b*x + a)^3 + 3*(b*d*x + b*c)*cos_integral((b*d*x + b*c)/d)*s in(-(b*c - a*d)/d) + 3*(b*d*x + b*c)*cos_integral(3*(b*d*x + b*c)/d)*sin(- 3*(b*c - a*d)/d) + 3*(b*d*x + b*c)*cos(-3*(b*c - a*d)/d)*sin_integral(3*(b *d*x + b*c)/d) + 3*(b*d*x + b*c)*cos(-(b*c - a*d)/d)*sin_integral((b*d*x + b*c)/d))/(d^3*x + c*d^2)
\[ \int \frac {\cos ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.10 \[ \int \frac {\cos ^3(a+b x)}{(c+d x)^2} \, dx=-\frac {3 \, b^{2} {\left (E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{2}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + b^{2} {\left (E_{2}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{2}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, b^{2} {\left (-i \, E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + i \, E_{2}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) + b^{2} {\left (i \, E_{2}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{2}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} b} \]
-1/8*(3*b^2*(exp_integral_e(2, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + exp_in tegral_e(2, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*cos(-(b*c - a*d)/d) + b^2 *(exp_integral_e(2, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e (2, -3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-3*(b*c - a*d)/d) + 3*b^2* (-I*exp_integral_e(2, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + I*exp_integral_ e(2, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d) + b^2*(I*exp _integral_e(2, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) - I*exp_integral_e(2, -3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-3*(b*c - a*d)/d))/((b*c*d + (b*x + a)*d^2 - a*d^2)*b)
Leaf count of result is larger than twice the leaf count of optimal. 1000 vs. \(2 (137) = 274\).
Time = 0.44 (sec) , antiderivative size = 1000, normalized size of antiderivative = 6.90 \[ \int \frac {\cos ^3(a+b x)}{(c+d x)^2} \, dx=\text {Too large to display} \]
-1/4*(3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*cos_integral(((d *x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*sin(-(b*c - a* d)/d) + 3*b^3*c*cos_integral(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c) ) + b*c - a*d)/d)*sin(-(b*c - a*d)/d) - 3*a*b^2*d*cos_integral(((d*x + c)* (b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*sin(-(b*c - a*d)/d) + 3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*cos_integral(3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*sin(-3*(b*c - a*d) /d) + 3*b^3*c*cos_integral(3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c) ) + b*c - a*d)/d)*sin(-3*(b*c - a*d)/d) - 3*a*b^2*d*cos_integral(3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*sin(-3*(b*c - a*d) /d) - 3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*cos(-(b*c - a*d) /d)*sin_integral(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a *d)/d) - 3*b^3*c*cos(-(b*c - a*d)/d)*sin_integral(-((d*x + c)*(b - b*c/(d* x + c) + a*d/(d*x + c)) + b*c - a*d)/d) + 3*a*b^2*d*cos(-(b*c - a*d)/d)*si n_integral(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d) - 3*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*cos(-3*(b*c - a*d)/ d)*sin_integral(-3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d) - 3*b^3*c*cos(-3*(b*c - a*d)/d)*sin_integral(-3*((d*x + c)*(b - b* c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d) + 3*a*b^2*d*cos(-3*(b*c - a*d )/d)*sin_integral(-3*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b...
Timed out. \[ \int \frac {\cos ^3(a+b x)}{(c+d x)^2} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \]