\(\int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx\) [96]

Optimal result

Integrand size = 24, antiderivative size = 413 \[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=-\frac {b \cos (a+b x)}{48 d^2 (c+d x)^2}-\frac {b \cos (3 a+3 b x)}{32 d^2 (c+d x)^2}+\frac {5 b \cos (5 a+5 b x)}{96 d^2 (c+d x)^2}-\frac {b^3 \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{48 d^4}-\frac {9 b^3 \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{32 d^4}+\frac {125 b^3 \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {5 b c}{d}+5 b x\right )}{96 d^4}-\frac {\sin (a+b x)}{24 d (c+d x)^3}+\frac {b^2 \sin (a+b x)}{48 d^3 (c+d x)}-\frac {\sin (3 a+3 b x)}{48 d (c+d x)^3}+\frac {3 b^2 \sin (3 a+3 b x)}{32 d^3 (c+d x)}+\frac {\sin (5 a+5 b x)}{48 d (c+d x)^3}-\frac {25 b^2 \sin (5 a+5 b x)}{96 d^3 (c+d x)}+\frac {b^3 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{48 d^4}+\frac {9 b^3 \sin \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{32 d^4}-\frac {125 b^3 \sin \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b c}{d}+5 b x\right )}{96 d^4} \] Output:

-1/48*b*cos(b*x+a)/d^2/(d*x+c)^2-1/32*b*cos(3*b*x+3*a)/d^2/(d*x+c)^2+5/96* 
b*cos(5*b*x+5*a)/d^2/(d*x+c)^2-1/48*b^3*cos(a-b*c/d)*Ci(b*c/d+b*x)/d^4-9/3 
2*b^3*cos(3*a-3*b*c/d)*Ci(3*b*c/d+3*b*x)/d^4+125/96*b^3*cos(5*a-5*b*c/d)*C 
i(5*b*c/d+5*b*x)/d^4-1/24*sin(b*x+a)/d/(d*x+c)^3+1/48*b^2*sin(b*x+a)/d^3/( 
d*x+c)-1/48*sin(3*b*x+3*a)/d/(d*x+c)^3+3/32*b^2*sin(3*b*x+3*a)/d^3/(d*x+c) 
+1/48*sin(5*b*x+5*a)/d/(d*x+c)^3-25/96*b^2*sin(5*b*x+5*a)/d^3/(d*x+c)+1/48 
*b^3*sin(a-b*c/d)*Si(b*c/d+b*x)/d^4+9/32*b^3*sin(3*a-3*b*c/d)*Si(3*b*c/d+3 
*b*x)/d^4-125/96*b^3*sin(5*a-5*b*c/d)*Si(5*b*c/d+5*b*x)/d^4
 

Mathematica [A] (verified)

Time = 1.62 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=\frac {-d \cos (3 b x) \left (3 b d (c+d x) \cos (3 a)-\left (-2 d^2+9 b^2 (c+d x)^2\right ) \sin (3 a)\right )+d \cos (5 b x) \left (5 b d (c+d x) \cos (5 a)-\left (-2 d^2+25 b^2 (c+d x)^2\right ) \sin (5 a)\right )+d \left (\left (-2 d^2+9 b^2 (c+d x)^2\right ) \cos (3 a)+3 b d (c+d x) \sin (3 a)\right ) \sin (3 b x)-d \left (\left (-2 d^2+25 b^2 (c+d x)^2\right ) \cos (5 a)+5 b d (c+d x) \sin (5 a)\right ) \sin (5 b x)-2 \left (d \cos (b x) \left (b d (c+d x) \cos (a)-\left (-2 d^2+b^2 (c+d x)^2\right ) \sin (a)\right )-d \left (\left (-2 d^2+b^2 (c+d x)^2\right ) \cos (a)+b d (c+d x) \sin (a)\right ) \sin (b x)+b^3 (c+d x)^3 \left (\cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right )-\sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )\right )\right )-27 b^3 (c+d x)^3 \left (\cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b (c+d x)}{d}\right )-\sin \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )\right )+125 b^3 (c+d x)^3 \left (\cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {5 b (c+d x)}{d}\right )-\sin \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b (c+d x)}{d}\right )\right )}{96 d^4 (c+d x)^3} \] Input:

Integrate[(Cos[a + b*x]^2*Sin[a + b*x]^3)/(c + d*x)^4,x]
 

Output:

(-(d*Cos[3*b*x]*(3*b*d*(c + d*x)*Cos[3*a] - (-2*d^2 + 9*b^2*(c + d*x)^2)*S 
in[3*a])) + d*Cos[5*b*x]*(5*b*d*(c + d*x)*Cos[5*a] - (-2*d^2 + 25*b^2*(c + 
 d*x)^2)*Sin[5*a]) + d*((-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*a] + 3*b*d*(c + 
 d*x)*Sin[3*a])*Sin[3*b*x] - d*((-2*d^2 + 25*b^2*(c + d*x)^2)*Cos[5*a] + 5 
*b*d*(c + d*x)*Sin[5*a])*Sin[5*b*x] - 2*(d*Cos[b*x]*(b*d*(c + d*x)*Cos[a] 
- (-2*d^2 + b^2*(c + d*x)^2)*Sin[a]) - d*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a 
] + b*d*(c + d*x)*Sin[a])*Sin[b*x] + b^3*(c + d*x)^3*(Cos[a - (b*c)/d]*Cos 
Integral[b*(c/d + x)] - Sin[a - (b*c)/d]*SinIntegral[b*(c/d + x)])) - 27*b 
^3*(c + d*x)^3*(Cos[3*a - (3*b*c)/d]*CosIntegral[(3*b*(c + d*x))/d] - Sin[ 
3*a - (3*b*c)/d]*SinIntegral[(3*b*(c + d*x))/d]) + 125*b^3*(c + d*x)^3*(Co 
s[5*a - (5*b*c)/d]*CosIntegral[(5*b*(c + d*x))/d] - Sin[5*a - (5*b*c)/d]*S 
inIntegral[(5*b*(c + d*x))/d]))/(96*d^4*(c + d*x)^3)
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 413, 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.

\(\displaystyle \int \frac {\sin ^3(a+b x) \cos ^2(a+b x)}{(c+d x)^4} \, dx\)

\(\Big \downarrow \) 4906

\(\displaystyle \int \left (\frac {\sin (a+b x)}{8 (c+d x)^4}+\frac {\sin (3 a+3 b x)}{16 (c+d x)^4}-\frac {\sin (5 a+5 b x)}{16 (c+d x)^4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^3 \cos \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{48 d^4}-\frac {9 b^3 \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{32 d^4}+\frac {125 b^3 \cos \left (5 a-\frac {5 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {5 b c}{d}+5 b x\right )}{96 d^4}+\frac {b^3 \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{48 d^4}+\frac {9 b^3 \sin \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{32 d^4}-\frac {125 b^3 \sin \left (5 a-\frac {5 b c}{d}\right ) \text {Si}\left (\frac {5 b c}{d}+5 b x\right )}{96 d^4}+\frac {b^2 \sin (a+b x)}{48 d^3 (c+d x)}+\frac {3 b^2 \sin (3 a+3 b x)}{32 d^3 (c+d x)}-\frac {25 b^2 \sin (5 a+5 b x)}{96 d^3 (c+d x)}-\frac {b \cos (a+b x)}{48 d^2 (c+d x)^2}-\frac {b \cos (3 a+3 b x)}{32 d^2 (c+d x)^2}+\frac {5 b \cos (5 a+5 b x)}{96 d^2 (c+d x)^2}-\frac {\sin (a+b x)}{24 d (c+d x)^3}-\frac {\sin (3 a+3 b x)}{48 d (c+d x)^3}+\frac {\sin (5 a+5 b x)}{48 d (c+d x)^3}\)

Input:

Int[(Cos[a + b*x]^2*Sin[a + b*x]^3)/(c + d*x)^4,x]
 

Output:

-1/48*(b*Cos[a + b*x])/(d^2*(c + d*x)^2) - (b*Cos[3*a + 3*b*x])/(32*d^2*(c 
 + d*x)^2) + (5*b*Cos[5*a + 5*b*x])/(96*d^2*(c + d*x)^2) - (b^3*Cos[a - (b 
*c)/d]*CosIntegral[(b*c)/d + b*x])/(48*d^4) - (9*b^3*Cos[3*a - (3*b*c)/d]* 
CosIntegral[(3*b*c)/d + 3*b*x])/(32*d^4) + (125*b^3*Cos[5*a - (5*b*c)/d]*C 
osIntegral[(5*b*c)/d + 5*b*x])/(96*d^4) - Sin[a + b*x]/(24*d*(c + d*x)^3) 
+ (b^2*Sin[a + b*x])/(48*d^3*(c + d*x)) - Sin[3*a + 3*b*x]/(48*d*(c + d*x) 
^3) + (3*b^2*Sin[3*a + 3*b*x])/(32*d^3*(c + d*x)) + Sin[5*a + 5*b*x]/(48*d 
*(c + d*x)^3) - (25*b^2*Sin[5*a + 5*b*x])/(96*d^3*(c + d*x)) + (b^3*Sin[a 
- (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(48*d^4) + (9*b^3*Sin[3*a - (3*b*c) 
/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(32*d^4) - (125*b^3*Sin[5*a - (5*b*c)/ 
d]*SinIntegral[(5*b*c)/d + 5*b*x])/(96*d^4)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4906

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]
 

Maple [A] (verified)

Time = 4.20 (sec) , antiderivative size = 585, normalized size of antiderivative = 1.42

method	result	size
derivativedivides	\(\frac {-\frac {b^{4} \left (-\frac {5 \sin \left (5 b x +5 a \right )}{3 \left (-a d +c b +d \left (b x +a \right )\right )^{3} d}+\frac {-\frac {25 \cos \left (5 b x +5 a \right )}{6 \left (-a d +c b +d \left (b x +a \right )\right )^{2} d}-\frac {25 \left (-\frac {5 \sin \left (5 b x +5 a \right )}{\left (-a d +c b +d \left (b x +a \right )\right ) d}+\frac {-\frac {25 \,\operatorname {Si}\left (-5 b x -5 a -\frac {5 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-5 a d +5 c b}{d}\right )}{d}+\frac {25 \,\operatorname {Ci}\left (5 b x +5 a +\frac {-5 a d +5 c b}{d}\right ) \cos \left (\frac {-5 a d +5 c b}{d}\right )}{d}}{d}\right )}{6 d}}{d}\right )}{80}+\frac {b^{4} \left (-\frac {\sin \left (b x +a \right )}{3 \left (-a d +c b +d \left (b x +a \right )\right )^{3} d}+\frac {-\frac {\cos \left (b x +a \right )}{2 \left (-a d +c b +d \left (b x +a \right )\right )^{2} d}-\frac {-\frac {\sin \left (b x +a \right )}{\left (-a d +c b +d \left (b x +a \right )\right ) d}+\frac {-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}+\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}}{d}}{2 d}}{3 d}\right )}{8}+\frac {b^{4} \left (-\frac {\sin \left (3 b x +3 a \right )}{\left (-a d +c b +d \left (b x +a \right )\right )^{3} d}+\frac {-\frac {3 \cos \left (3 b x +3 a \right )}{2 \left (-a d +c b +d \left (b x +a \right )\right )^{2} d}-\frac {3 \left (-\frac {3 \sin \left (3 b x +3 a \right )}{\left (-a d +c b +d \left (b x +a \right )\right ) d}+\frac {-\frac {9 \,\operatorname {Si}\left (-3 b x -3 a -\frac {3 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-3 a d +3 c b}{d}\right )}{d}+\frac {9 \,\operatorname {Ci}\left (3 b x +3 a +\frac {-3 a d +3 c b}{d}\right ) \cos \left (\frac {-3 a d +3 c b}{d}\right )}{d}}{d}\right )}{2 d}}{d}\right )}{48}}{b}\)	\(585\)
default	\(\frac {-\frac {b^{4} \left (-\frac {5 \sin \left (5 b x +5 a \right )}{3 \left (-a d +c b +d \left (b x +a \right )\right )^{3} d}+\frac {-\frac {25 \cos \left (5 b x +5 a \right )}{6 \left (-a d +c b +d \left (b x +a \right )\right )^{2} d}-\frac {25 \left (-\frac {5 \sin \left (5 b x +5 a \right )}{\left (-a d +c b +d \left (b x +a \right )\right ) d}+\frac {-\frac {25 \,\operatorname {Si}\left (-5 b x -5 a -\frac {5 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-5 a d +5 c b}{d}\right )}{d}+\frac {25 \,\operatorname {Ci}\left (5 b x +5 a +\frac {-5 a d +5 c b}{d}\right ) \cos \left (\frac {-5 a d +5 c b}{d}\right )}{d}}{d}\right )}{6 d}}{d}\right )}{80}+\frac {b^{4} \left (-\frac {\sin \left (b x +a \right )}{3 \left (-a d +c b +d \left (b x +a \right )\right )^{3} d}+\frac {-\frac {\cos \left (b x +a \right )}{2 \left (-a d +c b +d \left (b x +a \right )\right )^{2} d}-\frac {-\frac {\sin \left (b x +a \right )}{\left (-a d +c b +d \left (b x +a \right )\right ) d}+\frac {-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +c b}{d}\right ) \sin \left (\frac {-a d +c b}{d}\right )}{d}+\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +c b}{d}\right ) \cos \left (\frac {-a d +c b}{d}\right )}{d}}{d}}{2 d}}{3 d}\right )}{8}+\frac {b^{4} \left (-\frac {\sin \left (3 b x +3 a \right )}{\left (-a d +c b +d \left (b x +a \right )\right )^{3} d}+\frac {-\frac {3 \cos \left (3 b x +3 a \right )}{2 \left (-a d +c b +d \left (b x +a \right )\right )^{2} d}-\frac {3 \left (-\frac {3 \sin \left (3 b x +3 a \right )}{\left (-a d +c b +d \left (b x +a \right )\right ) d}+\frac {-\frac {9 \,\operatorname {Si}\left (-3 b x -3 a -\frac {3 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-3 a d +3 c b}{d}\right )}{d}+\frac {9 \,\operatorname {Ci}\left (3 b x +3 a +\frac {-3 a d +3 c b}{d}\right ) \cos \left (\frac {-3 a d +3 c b}{d}\right )}{d}}{d}\right )}{2 d}}{d}\right )}{48}}{b}\)	\(585\)
risch	\(\text {Expression too large to display}\)	\(1217\)

Input:

int(cos(b*x+a)^2*sin(b*x+a)^3/(d*x+c)^4,x,method=_RETURNVERBOSE)
 

Output:

1/b*(-1/80*b^4*(-5/3*sin(5*b*x+5*a)/(-a*d+c*b+d*(b*x+a))^3/d+5/3*(-5/2*cos 
(5*b*x+5*a)/(-a*d+c*b+d*(b*x+a))^2/d-5/2*(-5*sin(5*b*x+5*a)/(-a*d+c*b+d*(b 
*x+a))/d+5*(-5*Si(-5*b*x-5*a-5*(-a*d+b*c)/d)*sin(5*(-a*d+b*c)/d)/d+5*Ci(5* 
b*x+5*a+5*(-a*d+b*c)/d)*cos(5*(-a*d+b*c)/d)/d)/d)/d)/d)+1/8*b^4*(-1/3*sin( 
b*x+a)/(-a*d+c*b+d*(b*x+a))^3/d+1/3*(-1/2*cos(b*x+a)/(-a*d+c*b+d*(b*x+a))^ 
2/d-1/2*(-sin(b*x+a)/(-a*d+c*b+d*(b*x+a))/d+(-Si(-b*x-a-(-a*d+b*c)/d)*sin( 
(-a*d+b*c)/d)/d+Ci(b*x+a+(-a*d+b*c)/d)*cos((-a*d+b*c)/d)/d)/d)/d)/d)+1/48* 
b^4*(-sin(3*b*x+3*a)/(-a*d+c*b+d*(b*x+a))^3/d+(-3/2*cos(3*b*x+3*a)/(-a*d+c 
*b+d*(b*x+a))^2/d-3/2*(-3*sin(3*b*x+3*a)/(-a*d+c*b+d*(b*x+a))/d+3*(-3*Si(- 
3*b*x-3*a-3*(-a*d+b*c)/d)*sin(3*(-a*d+b*c)/d)/d+3*Ci(3*b*x+3*a+3*(-a*d+b*c 
)/d)*cos(3*(-a*d+b*c)/d)/d)/d)/d)/d))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.58 \[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx =\text {Too large to display} \] Input:

integrate(cos(b*x+a)^2*sin(b*x+a)^3/(d*x+c)^4,x, algorithm="fricas")
 

Output:

1/96*(80*(b*d^3*x + b*c*d^2)*cos(b*x + a)^5 - 112*(b*d^3*x + b*c*d^2)*cos( 
b*x + a)^3 + 125*(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(-5*(b*c - a*d)/d)*cos_integral(5*(b*d*x + b*c)/d) - 27*(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(-3*(b*c - a*d)/d)*cos_inte 
gral(3*(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(-(b*c - a*d)/d)*cos_integral((b*d*x + b*c)/d) - 125*(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(-5*(b*c - a*d)/d)* 
sin_integral(5*(b*d*x + b*c)/d) + 27*(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(-3*(b*c - a*d)/d)*sin_integral(3*(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(-(b*c - 
 a*d)/d)*sin_integral((b*d*x + b*c)/d) + 32*(b*d^3*x + b*c*d^2)*cos(b*x + 
a) - 16*(2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d + (25*b^2*d^3*x^2 + 5 
0*b^2*c*d^2*x + 25*b^2*c^2*d - 2*d^3)*cos(b*x + a)^4 - (21*b^2*d^3*x^2 + 4 
2*b^2*c*d^2*x + 21*b^2*c^2*d - 2*d^3)*cos(b*x + a)^2)*sin(b*x + a))/(d^7*x 
^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4)
 

Sympy [F]

\[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=\int \frac {\sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \] Input:

integrate(cos(b*x+a)**2*sin(b*x+a)**3/(d*x+c)**4,x)
 

Output:

Integral(sin(a + b*x)**3*cos(a + b*x)**2/(c + d*x)**4, x)
 

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.59 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.28 \[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=-\frac {2 \, b^{4} {\left (i \, E_{4}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) - i \, E_{4}\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^{4} {\left (i \, E_{4}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{4}\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^{4} {\left (-i \, E_{4}\left (\frac {5 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{4}\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^{4} {\left (E_{4}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{4}\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^{4} {\left (E_{4}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{4}\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^{4} {\left (E_{4}\left (\frac {5 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{4}\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 \, {\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} \] Input:

integrate(cos(b*x+a)^2*sin(b*x+a)^3/(d*x+c)^4,x, algorithm="maxima")
 

Output:

-1/32*(2*b^4*(I*exp_integral_e(4, (I*b*c + I*(b*x + a)*d - I*a*d)/d) - I*e 
xp_integral_e(4, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*cos(-(b*c - a*d)/d) 
- b^4*(I*exp_integral_e(4, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) - I*exp_i 
ntegral_e(4, -3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-3*(b*c - a*d)/d) 
 - b^4*(-I*exp_integral_e(4, 5*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + I*exp 
_integral_e(4, -5*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-5*(b*c - a*d)/ 
d) + 2*b^4*(exp_integral_e(4, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + exp_int 
egral_e(4, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d) + b^4* 
(exp_integral_e(4, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e( 
4, -3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-3*(b*c - a*d)/d) - b^4*(ex 
p_integral_e(4, 5*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral_e(4, 
-5*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*sin(-5*(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)
 

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 138.55 (sec) , antiderivative size = 2449286, normalized size of antiderivative = 5930.47 \[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=\text {Too large to display} \] Input:

integrate(cos(b*x+a)^2*sin(b*x+a)^3/(d*x+c)^4,x, algorithm="giac")
 

Output:

1/192*(125*b^3*d^3*x^3*real_part(cos_integral(5*b*x + 5*b*c/d))*tan(5/2*b* 
x)^2*tan(3/2*b*x)^2*tan(1/2*b*x)^2*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 - 27*b^3*d^3*x^3*real_p 
art(cos_integral(3*b*x + 3*b*c/d))*tan(5/2*b*x)^2*tan(3/2*b*x)^2*tan(1/2*b 
*x)^2*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*b^3*d^3*x^3*real_part(cos_integral(b*x + b*c/d)) 
*tan(5/2*b*x)^2*tan(3/2*b*x)^2*tan(1/2*b*x)^2*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*b^3*d^3* 
x^3*real_part(cos_integral(-b*x - b*c/d))*tan(5/2*b*x)^2*tan(3/2*b*x)^2*ta 
n(1/2*b*x)^2*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 - 27*b^3*d^3*x^3*real_part(cos_integral(-3*b* 
x - 3*b*c/d))*tan(5/2*b*x)^2*tan(3/2*b*x)^2*tan(1/2*b*x)^2*tan(5/2*a)^2*ta 
n(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 
 + 125*b^3*d^3*x^3*real_part(cos_integral(-5*b*x - 5*b*c/d))*tan(5/2*b*x)^ 
2*tan(3/2*b*x)^2*tan(1/2*b*x)^2*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*b^3*d^3*x^3*imag_part( 
cos_integral(b*x + b*c/d))*tan(5/2*b*x)^2*tan(3/2*b*x)^2*tan(1/2*b*x)^2*ta 
n(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*b^3*d^3*x^3*imag_part(cos_integral(-b*x - b*c/d))*tan(5/2* 
b*x)^2*tan(3/2*b*x)^2*tan(1/2*b*x)^2*tan(5/2*a)^2*tan(3/2*a)^2*tan(1/2*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^4} \,d x \] Input:

int((cos(a + b*x)^2*sin(a + b*x)^3)/(c + d*x)^4,x)
 

Output:

int((cos(a + b*x)^2*sin(a + b*x)^3)/(c + d*x)^4, x)
 

Reduce [F]

\[ \int \frac {\cos ^2(a+b x) \sin ^3(a+b x)}{(c+d x)^4} \, dx=\text {too large to display} \] Input:

int(cos(b*x+a)^2*sin(b*x+a)^3/(d*x+c)^4,x)
 

Output:

(144*cos(a + b*x)*sin(a + b*x)**4*b*c*d**2 + 144*cos(a + b*x)*sin(a + b*x) 
**4*b*d**3*x + 4500*cos(a + b*x)*sin(a + b*x)**3*b**2*c**2*d + 9000*cos(a 
+ b*x)*sin(a + b*x)**3*b**2*c*d**2*x + 4500*cos(a + b*x)*sin(a + b*x)**3*b 
**2*d**3*x**2 + 12610*cos(a + b*x)*sin(a + b*x)**3*d**3 - 4608*cos(a + b*x 
)*sin(a + b*x)**2*b*c*d**2 - 4608*cos(a + b*x)*sin(a + b*x)**2*b*d**3*x - 
29250*cos(a + b*x)*sin(a + b*x)*b**2*c**2*d - 58500*cos(a + b*x)*sin(a + b 
*x)*b**2*c*d**2*x - 29250*cos(a + b*x)*sin(a + b*x)*b**2*d**3*x**2 - 71840 
*cos(a + b*x)*sin(a + b*x)*d**3 + 48384*cos(a + b*x)*b*c*d**2 + 48384*cos( 
a + b*x)*b*d**3*x - 1360800*int(tan((a + b*x)/2)**5/(tan((a + b*x)/2)**10* 
c**4 + 4*tan((a + b*x)/2)**10*c**3*d*x + 6*tan((a + b*x)/2)**10*c**2*d**2* 
x**2 + 4*tan((a + b*x)/2)**10*c*d**3*x**3 + tan((a + b*x)/2)**10*d**4*x**4 
 + 5*tan((a + b*x)/2)**8*c**4 + 20*tan((a + b*x)/2)**8*c**3*d*x + 30*tan(( 
a + b*x)/2)**8*c**2*d**2*x**2 + 20*tan((a + b*x)/2)**8*c*d**3*x**3 + 5*tan 
((a + b*x)/2)**8*d**4*x**4 + 10*tan((a + b*x)/2)**6*c**4 + 40*tan((a + b*x 
)/2)**6*c**3*d*x + 60*tan((a + b*x)/2)**6*c**2*d**2*x**2 + 40*tan((a + b*x 
)/2)**6*c*d**3*x**3 + 10*tan((a + b*x)/2)**6*d**4*x**4 + 10*tan((a + b*x)/ 
2)**4*c**4 + 40*tan((a + b*x)/2)**4*c**3*d*x + 60*tan((a + b*x)/2)**4*c**2 
*d**2*x**2 + 40*tan((a + b*x)/2)**4*c*d**3*x**3 + 10*tan((a + b*x)/2)**4*d 
**4*x**4 + 5*tan((a + b*x)/2)**2*c**4 + 20*tan((a + b*x)/2)**2*c**3*d*x + 
30*tan((a + b*x)/2)**2*c**2*d**2*x**2 + 20*tan((a + b*x)/2)**2*c*d**3*x...