Integrand size = 22, antiderivative size = 221 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=-\frac {b \cos (a+b x)}{8 d^2 (c+d x)}-\frac {3 b \cos (3 a+3 b x)}{8 d^2 (c+d x)}-\frac {9 b^2 \operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{8 d^3}-\frac {b^2 \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{8 d^3}-\frac {\sin (a+b x)}{8 d (c+d x)^2}-\frac {\sin (3 a+3 b x)}{8 d (c+d x)^2}-\frac {b^2 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{8 d^3}-\frac {9 b^2 \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3} \]
-1/8*b*cos(b*x+a)/d^2/(d*x+c)-3/8*b*cos(3*b*x+3*a)/d^2/(d*x+c)-1/8*b^2*cos (a-b*c/d)*Si(b*c/d+b*x)/d^3-9/8*b^2*cos(3*a-3*b*c/d)*Si(3*b*c/d+3*b*x)/d^3 -9/8*b^2*Ci(3*b*c/d+3*b*x)*sin(3*a-3*b*c/d)/d^3-1/8*b^2*Ci(b*c/d+b*x)*sin( a-b*c/d)/d^3-1/8*sin(b*x+a)/d/(d*x+c)^2-1/8*sin(3*b*x+3*a)/d/(d*x+c)^2
Time = 2.69 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.82 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=-\frac {9 b^2 \operatorname {CosIntegral}\left (\frac {3 b (c+d x)}{d}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )+b^2 \operatorname {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right ) \sin \left (a-\frac {b c}{d}\right )+\frac {d (b (c+d x) \cos (a+b x)+d \sin (a+b x))}{(c+d x)^2}+\frac {d (3 b (c+d x) \cos (3 (a+b x))+d \sin (3 (a+b x)))}{(c+d x)^2}+b^2 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )+9 b^2 \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )}{8 d^3} \]
-1/8*(9*b^2*CosIntegral[(3*b*(c + d*x))/d]*Sin[3*a - (3*b*c)/d] + b^2*CosI ntegral[b*(c/d + x)]*Sin[a - (b*c)/d] + (d*(b*(c + d*x)*Cos[a + b*x] + d*S in[a + b*x]))/(c + d*x)^2 + (d*(3*b*(c + d*x)*Cos[3*(a + b*x)] + d*Sin[3*( a + b*x)]))/(c + d*x)^2 + b^2*Cos[a - (b*c)/d]*SinIntegral[b*(c/d + x)] + 9*b^2*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d])/d^3
Time = 0.54 (sec) , antiderivative size = 221, 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.
| \(\displaystyle \int \frac {\sin (a+b x) \cos ^2(a+b x)}{(c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \int \left (\frac {\sin (a+b x)}{4 (c+d x)^3}+\frac {\sin (3 a+3 b x)}{4 (c+d x)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {9 b^2 \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3}-\frac {b^2 \sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{8 d^3}-\frac {b^2 \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{8 d^3}-\frac {9 b^2 \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{8 d^3}-\frac {b \cos (a+b x)}{8 d^2 (c+d x)}-\frac {3 b \cos (3 a+3 b x)}{8 d^2 (c+d x)}-\frac {\sin (a+b x)}{8 d (c+d x)^2}-\frac {\sin (3 a+3 b x)}{8 d (c+d x)^2}\) |
-1/8*(b*Cos[a + b*x])/(d^2*(c + d*x)) - (3*b*Cos[3*a + 3*b*x])/(8*d^2*(c + d*x)) - (9*b^2*CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(8*d^ 3) - (b^2*CosIntegral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(8*d^3) - Sin[a + b *x]/(8*d*(c + d*x)^2) - Sin[3*a + 3*b*x]/(8*d*(c + d*x)^2) - (b^2*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(8*d^3) - (9*b^2*Cos[3*a - (3*b*c)/d] *SinIntegral[(3*b*c)/d + 3*b*x])/(8*d^3)
3.1.77.3.1 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 = 1.51 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.44
| method | result | size |
| derivativedivides | \(\frac {\frac {b^{3} \left (-\frac {3 \sin \left (3 x b +3 a \right )}{2 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {9 \cos \left (3 x b +3 a \right )}{2 \left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {9 \left (-\frac {3 \,\operatorname {Si}\left (-3 x b -3 a -\frac {3 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-3 a d +3 c b}{d}\right )}{d}-\frac {3 \,\operatorname {Ci}\left (3 x b +3 a +\frac {-3 a d +3 c b}{d}\right ) \sin \left (\frac {-3 a d +3 c b}{d}\right )}{d}\right )}{2 d}}{d}\right )}{12}+\frac {b^{3} \left (-\frac {\sin \left (x b +a \right )}{2 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {\cos \left (x b +a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {-\frac {\operatorname {Si}\left (-x b -a -\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}-\frac {\operatorname {Ci}\left (x b +a +\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}}{d}}{2 d}\right )}{4}}{b}\) | \(318\) |
| default | \(\frac {\frac {b^{3} \left (-\frac {3 \sin \left (3 x b +3 a \right )}{2 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {9 \cos \left (3 x b +3 a \right )}{2 \left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {9 \left (-\frac {3 \,\operatorname {Si}\left (-3 x b -3 a -\frac {3 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-3 a d +3 c b}{d}\right )}{d}-\frac {3 \,\operatorname {Ci}\left (3 x b +3 a +\frac {-3 a d +3 c b}{d}\right ) \sin \left (\frac {-3 a d +3 c b}{d}\right )}{d}\right )}{2 d}}{d}\right )}{12}+\frac {b^{3} \left (-\frac {\sin \left (x b +a \right )}{2 \left (-a d +c b +d \left (x b +a \right )\right )^{2} d}+\frac {-\frac {\cos \left (x b +a \right )}{\left (-a d +c b +d \left (x b +a \right )\right ) d}-\frac {-\frac {\operatorname {Si}\left (-x b -a -\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}-\frac {\operatorname {Ci}\left (x b +a +\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}}{d}}{2 d}\right )}{4}}{b}\) | \(318\) |
| risch | \(\frac {9 i b^{2} {\mathrm e}^{-\frac {3 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (3 i b x +3 i a -\frac {3 i \left (a d -c b \right )}{d}\right )}{16 d^{3}}+\frac {i b^{2} {\mathrm e}^{-\frac {i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (i b x +i a -\frac {i \left (a d -c b \right )}{d}\right )}{16 d^{3}}-\frac {i b^{2} {\mathrm e}^{\frac {i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-i b x -i a -\frac {-i a d +i c b}{d}\right )}{16 d^{3}}-\frac {9 i b^{2} {\mathrm e}^{\frac {3 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-3 i b x -3 i a -\frac {3 \left (-i a d +i c b \right )}{d}\right )}{16 d^{3}}+\frac {i \left (2 i b^{3} d^{3} x^{3}+6 i b^{3} c \,d^{2} x^{2}+6 i b^{3} c^{2} d x +2 i b^{3} c^{3}\right ) \cos \left (x b +a \right )}{16 d^{2} \left (d x +c \right )^{2} \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {\left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}\right ) \sin \left (x b +a \right )}{16 d \left (d x +c \right )^{2} \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {i \left (6 i b^{3} d^{3} x^{3}+18 i b^{3} c \,d^{2} x^{2}+18 i b^{3} c^{2} d x +6 i b^{3} c^{3}\right ) \cos \left (3 x b +3 a \right )}{16 d^{2} \left (d x +c \right )^{2} \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {\left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}\right ) \sin \left (3 x b +3 a \right )}{16 d \left (d x +c \right )^{2} \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}\) | \(546\) |
1/b*(1/12*b^3*(-3/2*sin(3*b*x+3*a)/(-a*d+c*b+d*(b*x+a))^2/d+3/2*(-3*cos(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)*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)/d)/d )+1/4*b^3*(-1/2*sin(b*x+a)/(-a*d+c*b+d*(b*x+a))^2/d+1/2*(-cos(b*x+a)/(-a*d +c*b+d*(b*x+a))/d-(-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)/d)/d))
Time = 0.27 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.38 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=-\frac {4 \, d^{2} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right ) + 12 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 9 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \operatorname {Ci}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 9 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) - 8 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )}{8 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \]
-1/8*(4*d^2*cos(b*x + a)^2*sin(b*x + a) + 12*(b*d^2*x + b*c*d)*cos(b*x + a )^3 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos_integral((b*d*x + b*c)/d)* sin(-(b*c - a*d)/d) + 9*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos_integral (3*(b*d*x + b*c)/d)*sin(-3*(b*c - a*d)/d) + 9*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(-3*(b*c - a*d)/d)*sin_integral(3*(b*d*x + b*c)/d) + (b^2*d^2 *x^2 + 2*b^2*c*d*x + b^2*c^2)*cos(-(b*c - a*d)/d)*sin_integral((b*d*x + b* c)/d) - 8*(b*d^2*x + b*c*d)*cos(b*x + a))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
\[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=\int \frac {\sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \]
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.52 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=-\frac {b^{3} {\left (i \, E_{3}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) - i \, E_{3}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + b^{3} {\left (-i \, E_{3}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{3}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + b^{3} {\left (E_{3}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{3}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) + b^{3} {\left (E_{3}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{3}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + {\left (b x + a\right )}^{2} d^{3} + a^{2} d^{3} + 2 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}\right )} b} \]
-1/8*(b^3*(I*exp_integral_e(3, (I*b*c + I*(b*x + a)*d - I*a*d)/d) - I*exp_ integral_e(3, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*cos(-(b*c - a*d)/d) + b ^3*(-I*exp_integral_e(3, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp_int egral_e(3, -3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-3*(b*c - a*d)/d) + b^3*(exp_integral_e(3, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + exp_integral_ e(3, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d) + b^3*(exp_i ntegral_e(3, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(3, -3* (-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-3*(b*c - a*d)/d))/((b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + a^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*b )
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.48 (sec) , antiderivative size = 118262, normalized size of antiderivative = 535.12 \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=\text {Too large to display} \]
-1/16*(9*b^2*d^2*x^2*imag_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^2*d^2*x^2*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 ^2*d^2*x^2*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 - 9*b^2*d ^2*x^2*imag_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 + 18*b^2* d^2*x^2*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)^2*tan(1/2*b*c/d)^2 + 2*b^2*d^2*x^2*si n_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)^2 + 2*b^2*d^2*x^2*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^2*d^2*x^2*real_part(cos_integra l(-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*t an(3/2*b*c/d)^2*tan(1/2*b*c/d) + 18*b^2*d^2*x^2*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)*tan(1/2*b*c/d)^2 + 18*b^2*d^2*x^2*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)*tan(1/2*b*c/d)^2 - 2*b^2*d^2*x^2*real_part(cos_integral(b*x ...
Timed out. \[ \int \frac {\cos ^2(a+b x) \sin (a+b x)}{(c+d x)^3} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{{\left (c+d\,x\right )}^3} \,d x \]