\(\int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx\) [138]

Optimal result

Integrand size = 22, antiderivative size = 196 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {45 d^3 x}{256 b^3}+\frac {3 (c+d x)^3}{32 b}+\frac {9 d^2 (c+d x) \cos ^2(a+b x)}{32 b^3}+\frac {3 d^2 (c+d x) \cos ^4(a+b x)}{32 b^3}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}-\frac {45 d^3 \cos (a+b x) \sin (a+b x)}{256 b^4}+\frac {9 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{32 b^2}-\frac {3 d^3 \cos ^3(a+b x) \sin (a+b x)}{128 b^4}+\frac {3 d (c+d x)^2 \cos ^3(a+b x) \sin (a+b x)}{16 b^2} \] Output:

-45/256*d^3*x/b^3+3/32*(d*x+c)^3/b+9/32*d^2*(d*x+c)*cos(b*x+a)^2/b^3+3/32* 
d^2*(d*x+c)*cos(b*x+a)^4/b^3-1/4*(d*x+c)^3*cos(b*x+a)^4/b-45/256*d^3*cos(b 
*x+a)*sin(b*x+a)/b^4+9/32*d*(d*x+c)^2*cos(b*x+a)*sin(b*x+a)/b^2-3/128*d^3* 
cos(b*x+a)^3*sin(b*x+a)/b^4+3/16*d*(d*x+c)^2*cos(b*x+a)^3*sin(b*x+a)/b^2
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.69 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx=\frac {-64 b (c+d x) \left (-3 d^2+2 b^2 (c+d x)^2\right ) \cos (2 (a+b x))-4 b (c+d x) \left (-3 d^2+8 b^2 (c+d x)^2\right ) \cos (4 (a+b x))+6 d \left (16 \left (-d^2+2 b^2 (c+d x)^2\right )+\left (-d^2+8 b^2 (c+d x)^2\right ) \cos (2 (a+b x))\right ) \sin (2 (a+b x))}{1024 b^4} \] Input:

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

Output:

(-64*b*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] - 4*b*(c + 
d*x)*(-3*d^2 + 8*b^2*(c + d*x)^2)*Cos[4*(a + b*x)] + 6*d*(16*(-d^2 + 2*b^2 
*(c + d*x)^2) + (-d^2 + 8*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[2*(a + b* 
x)])/(1024*b^4)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.24, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {4905, 3042, 3792, 3042, 3115, 3042, 3115, 24, 3792, 17, 3042, 3115, 24}

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.

\(\displaystyle \int (c+d x)^3 \sin (a+b x) \cos ^3(a+b x) \, dx\)

\(\Big \downarrow \) 4905

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

\(\displaystyle \frac {3 d \left (-\frac {d^2 \int \cos ^4(a+b x)dx}{8 b^2}+\frac {3}{4} \int (c+d x)^2 \cos ^2(a+b x)dx+\frac {d (c+d x) \cos ^4(a+b x)}{8 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{4 b}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 d \left (-\frac {d^2 \int \sin \left (a+b x+\frac {\pi }{2}\right )^4dx}{8 b^2}+\frac {3}{4} \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {d (c+d x) \cos ^4(a+b x)}{8 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{4 b}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {3 d \left (-\frac {d^2 \left (\frac {3}{4} \int \cos ^2(a+b x)dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {d (c+d x) \cos ^4(a+b x)}{8 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{4 b}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 d \left (-\frac {d^2 \left (\frac {3}{4} \int \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {d (c+d x) \cos ^4(a+b x)}{8 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{4 b}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {3 d \left (-\frac {d^2 \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {d (c+d x) \cos ^4(a+b x)}{8 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{4 b}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {3 d \left (\frac {3}{4} \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {d (c+d x) \cos ^4(a+b x)}{8 b^2}-\frac {d^2 \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )}{8 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{4 b}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {3 d \left (\frac {3}{4} \left (-\frac {d^2 \int \cos ^2(a+b x)dx}{2 b^2}+\frac {1}{2} \int (c+d x)^2dx+\frac {d (c+d x) \cos ^2(a+b x)}{2 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {d (c+d x) \cos ^4(a+b x)}{8 b^2}-\frac {d^2 \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )}{8 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{4 b}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {3 d \left (\frac {3}{4} \left (-\frac {d^2 \int \cos ^2(a+b x)dx}{2 b^2}+\frac {d (c+d x) \cos ^2(a+b x)}{2 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )+\frac {d (c+d x) \cos ^4(a+b x)}{8 b^2}-\frac {d^2 \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )}{8 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{4 b}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 d \left (\frac {3}{4} \left (-\frac {d^2 \int \sin \left (a+b x+\frac {\pi }{2}\right )^2dx}{2 b^2}+\frac {d (c+d x) \cos ^2(a+b x)}{2 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )+\frac {d (c+d x) \cos ^4(a+b x)}{8 b^2}-\frac {d^2 \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )}{8 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{4 b}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {3 d \left (\frac {3}{4} \left (-\frac {d^2 \left (\frac {\int 1dx}{2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}+\frac {d (c+d x) \cos ^2(a+b x)}{2 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )+\frac {d (c+d x) \cos ^4(a+b x)}{8 b^2}-\frac {d^2 \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )}{8 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{4 b}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {3 d \left (\frac {3}{4} \left (\frac {d (c+d x) \cos ^2(a+b x)}{2 b^2}-\frac {d^2 \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )}{2 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )+\frac {d (c+d x) \cos ^4(a+b x)}{8 b^2}-\frac {d^2 \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )}{8 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{4 b}\right )}{4 b}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}\)

Input:

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

Output:

-1/4*((c + d*x)^3*Cos[a + b*x]^4)/b + (3*d*((d*(c + d*x)*Cos[a + b*x]^4)/( 
8*b^2) + ((c + d*x)^2*Cos[a + b*x]^3*Sin[a + b*x])/(4*b) - (d^2*((Cos[a + 
b*x]^3*Sin[a + b*x])/(4*b) + (3*(x/2 + (Cos[a + b*x]*Sin[a + b*x])/(2*b))) 
/4))/(8*b^2) + (3*((c + d*x)^3/(6*d) + (d*(c + d*x)*Cos[a + b*x]^2)/(2*b^2 
) + ((c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(2*b) - (d^2*(x/2 + (Cos[a + b 
*x]*Sin[a + b*x])/(2*b)))/(2*b^2)))/4))/(4*b)
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 4905

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[a + b*x]^(n + 1)/(b*(n + 1 
))), x] + Simp[d*(m/(b*(n + 1)))   Int[(c + d*x)^(m - 1)*Cos[a + b*x]^(n + 
1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

Maple [A] (verified)

Time = 1.74 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.76

method	result	size
parallelrisch	\(\frac {-32 \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{2}\right ) \left (d x +c \right ) b \cos \left (2 b x +2 a \right )-8 \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{8}\right ) \left (d x +c \right ) b \cos \left (4 b x +4 a \right )+48 d \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{2}\right ) \sin \left (2 b x +2 a \right )+6 d \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{8}\right ) \sin \left (4 b x +4 a \right )+40 b^{3} c^{3}-51 c \,d^{2} b}{256 b^{4}}\)	\(148\)
risch	\(-\frac {\left (8 b^{2} d^{3} x^{3}+24 b^{2} c \,d^{2} x^{2}+24 b^{2} c^{2} d x +8 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \cos \left (4 b x +4 a \right )}{256 b^{3}}+\frac {3 d \left (8 x^{2} d^{2} b^{2}+16 b^{2} c d x +8 b^{2} c^{2}-d^{2}\right ) \sin \left (4 b x +4 a \right )}{1024 b^{4}}-\frac {\left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \cos \left (2 b x +2 a \right )}{16 b^{3}}+\frac {3 d \left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}-d^{2}\right ) \sin \left (2 b x +2 a \right )}{32 b^{4}}\)	\(234\)
derivativedivides	\(\frac {\frac {a^{3} d^{3} \cos \left (b x +a \right )^{4}}{4 b^{3}}-\frac {3 a^{2} c \,d^{2} \cos \left (b x +a \right )^{4}}{4 b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{4}}{4}+\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{16}+\frac {3 b x}{32}+\frac {3 a}{32}\right )}{b^{3}}+\frac {3 a \,c^{2} d \cos \left (b x +a \right )^{4}}{4 b}-\frac {6 a c \,d^{2} \left (-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{4}}{4}+\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{16}+\frac {3 b x}{32}+\frac {3 a}{32}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (-\frac {\left (b x +a \right )^{2} \cos \left (b x +a \right )^{4}}{4}+\frac {\left (b x +a \right ) \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{2}-\frac {3 \left (b x +a \right )^{2}}{32}+\frac {\left (2 \cos \left (b x +a \right )^{2}+3\right )^{2}}{128}\right )}{b^{3}}-\frac {c^{3} \cos \left (b x +a \right )^{4}}{4}+\frac {3 c^{2} d \left (-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{4}}{4}+\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{16}+\frac {3 b x}{32}+\frac {3 a}{32}\right )}{b}+\frac {3 c \,d^{2} \left (-\frac {\left (b x +a \right )^{2} \cos \left (b x +a \right )^{4}}{4}+\frac {\left (b x +a \right ) \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{2}-\frac {3 \left (b x +a \right )^{2}}{32}+\frac {\left (2 \cos \left (b x +a \right )^{2}+3\right )^{2}}{128}\right )}{b^{2}}+\frac {d^{3} \left (-\frac {\left (b x +a \right )^{3} \cos \left (b x +a \right )^{4}}{4}+\frac {3 \left (b x +a \right )^{2} \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{4}+\frac {3 \left (b x +a \right ) \cos \left (b x +a \right )^{4}}{32}-\frac {3 \left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{128}-\frac {45 b x}{256}-\frac {45 a}{256}+\frac {9 \left (b x +a \right ) \cos \left (b x +a \right )^{2}}{32}-\frac {9 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{64}-\frac {3 \left (b x +a \right )^{3}}{16}\right )}{b^{3}}}{b}\)	\(586\)
default	\(\frac {\frac {a^{3} d^{3} \cos \left (b x +a \right )^{4}}{4 b^{3}}-\frac {3 a^{2} c \,d^{2} \cos \left (b x +a \right )^{4}}{4 b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{4}}{4}+\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{16}+\frac {3 b x}{32}+\frac {3 a}{32}\right )}{b^{3}}+\frac {3 a \,c^{2} d \cos \left (b x +a \right )^{4}}{4 b}-\frac {6 a c \,d^{2} \left (-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{4}}{4}+\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{16}+\frac {3 b x}{32}+\frac {3 a}{32}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (-\frac {\left (b x +a \right )^{2} \cos \left (b x +a \right )^{4}}{4}+\frac {\left (b x +a \right ) \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{2}-\frac {3 \left (b x +a \right )^{2}}{32}+\frac {\left (2 \cos \left (b x +a \right )^{2}+3\right )^{2}}{128}\right )}{b^{3}}-\frac {c^{3} \cos \left (b x +a \right )^{4}}{4}+\frac {3 c^{2} d \left (-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{4}}{4}+\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{16}+\frac {3 b x}{32}+\frac {3 a}{32}\right )}{b}+\frac {3 c \,d^{2} \left (-\frac {\left (b x +a \right )^{2} \cos \left (b x +a \right )^{4}}{4}+\frac {\left (b x +a \right ) \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{2}-\frac {3 \left (b x +a \right )^{2}}{32}+\frac {\left (2 \cos \left (b x +a \right )^{2}+3\right )^{2}}{128}\right )}{b^{2}}+\frac {d^{3} \left (-\frac {\left (b x +a \right )^{3} \cos \left (b x +a \right )^{4}}{4}+\frac {3 \left (b x +a \right )^{2} \left (\frac {\left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{4}+\frac {3 b x}{8}+\frac {3 a}{8}\right )}{4}+\frac {3 \left (b x +a \right ) \cos \left (b x +a \right )^{4}}{32}-\frac {3 \left (\cos \left (b x +a \right )^{3}+\frac {3 \cos \left (b x +a \right )}{2}\right ) \sin \left (b x +a \right )}{128}-\frac {45 b x}{256}-\frac {45 a}{256}+\frac {9 \left (b x +a \right ) \cos \left (b x +a \right )^{2}}{32}-\frac {9 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{64}-\frac {3 \left (b x +a \right )^{3}}{16}\right )}{b^{3}}}{b}\)	\(586\)
orering	\(\frac {15 d \left (8 d^{4} x^{4} b^{4}+32 b^{4} c \,d^{3} x^{3}+48 b^{4} c^{2} d^{2} x^{2}+32 b^{4} c^{3} d x +8 b^{4} c^{4}-2 b^{2} d^{4} x^{2}-4 b^{2} c \,d^{3} x -2 b^{2} c^{2} d^{2}-9 d^{4}\right ) \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )}{64 b^{6} \left (d x +c \right )^{2}}-\frac {5 \left (16 d^{4} x^{4} b^{4}+64 b^{4} c \,d^{3} x^{3}+96 b^{4} c^{2} d^{2} x^{2}+64 b^{4} c^{3} d x +16 b^{4} c^{4}+18 b^{2} d^{4} x^{2}+36 b^{2} c \,d^{3} x +18 b^{2} c^{2} d^{2}-63 d^{4}\right ) \left (3 \left (d x +c \right )^{2} \cos \left (b x +a \right )^{3} \sin \left (b x +a \right ) d -3 \left (d x +c \right )^{3} \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{2} b +\left (d x +c \right )^{3} \cos \left (b x +a \right )^{4} b \right )}{256 \left (d x +c \right )^{4} b^{6}}+\frac {3 d \left (16 x^{2} d^{2} b^{2}+32 b^{2} c d x +16 b^{2} c^{2}-25 d^{2}\right ) \left (6 \left (d x +c \right ) \cos \left (b x +a \right )^{3} \sin \left (b x +a \right ) d^{2}-18 \left (d x +c \right )^{2} \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{2} d b +6 \left (d x +c \right )^{2} \cos \left (b x +a \right )^{4} b d +6 \left (d x +c \right )^{3} \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b^{2}-10 \left (d x +c \right )^{3} \cos \left (b x +a \right )^{3} \sin \left (b x +a \right ) b^{2}\right )}{256 b^{6} \left (d x +c \right )^{3}}-\frac {\left (8 x^{2} d^{2} b^{2}+16 b^{2} c d x +8 b^{2} c^{2}-15 d^{2}\right ) \left (6 d^{3} \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )-54 \left (d x +c \right ) \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{2} d^{2} b +18 \left (d x +c \right ) \cos \left (b x +a \right )^{4} b \,d^{2}+54 \left (d x +c \right )^{2} \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} d \,b^{2}-90 \left (d x +c \right )^{2} \cos \left (b x +a \right )^{3} \sin \left (b x +a \right ) d \,b^{2}-6 \left (d x +c \right )^{3} b^{3} \sin \left (b x +a \right )^{4}+48 \left (d x +c \right )^{3} \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{2} b^{3}-10 \left (d x +c \right )^{3} \cos \left (b x +a \right )^{4} b^{3}\right )}{512 \left (d x +c \right )^{2} b^{6}}\)	\(697\)

Input:

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

Output:

1/256*(-32*((d*x+c)^2*b^2-3/2*d^2)*(d*x+c)*b*cos(2*b*x+2*a)-8*((d*x+c)^2*b 
^2-3/8*d^2)*(d*x+c)*b*cos(4*b*x+4*a)+48*d*((d*x+c)^2*b^2-1/2*d^2)*sin(2*b* 
x+2*a)+6*d*((d*x+c)^2*b^2-1/8*d^2)*sin(4*b*x+4*a)+40*b^3*c^3-51*c*d^2*b)/b 
^4
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.21 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx=\frac {24 \, b^{3} d^{3} x^{3} + 72 \, b^{3} c d^{2} x^{2} - 8 \, {\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 8 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (8 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 72 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{2} + 9 \, {\left (8 \, b^{3} c^{2} d - 5 \, b d^{3}\right )} x + 3 \, {\left (2 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{3} + 3 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - 5 \, d^{3}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{256 \, b^{4}} \] Input:

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

Output:

1/256*(24*b^3*d^3*x^3 + 72*b^3*c*d^2*x^2 - 8*(8*b^3*d^3*x^3 + 24*b^3*c*d^2 
*x^2 + 8*b^3*c^3 - 3*b*c*d^2 + 3*(8*b^3*c^2*d - b*d^3)*x)*cos(b*x + a)^4 + 
 72*(b*d^3*x + b*c*d^2)*cos(b*x + a)^2 + 9*(8*b^3*c^2*d - 5*b*d^3)*x + 3*( 
2*(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)^3 + 3* 
(8*b^2*d^3*x^2 + 16*b^2*c*d^2*x + 8*b^2*c^2*d - 5*d^3)*cos(b*x + a))*sin(b 
*x + a))/b^4
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (197) = 394\).

Time = 0.64 (sec) , antiderivative size = 602, normalized size of antiderivative = 3.07 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx=\begin {cases} - \frac {c^{3} \cos ^{4}{\left (a + b x \right )}}{4 b} + \frac {9 c^{2} d x \sin ^{4}{\left (a + b x \right )}}{32 b} + \frac {9 c^{2} d x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16 b} - \frac {15 c^{2} d x \cos ^{4}{\left (a + b x \right )}}{32 b} + \frac {9 c d^{2} x^{2} \sin ^{4}{\left (a + b x \right )}}{32 b} + \frac {9 c d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16 b} - \frac {15 c d^{2} x^{2} \cos ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 d^{3} x^{3} \sin ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 d^{3} x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16 b} - \frac {5 d^{3} x^{3} \cos ^{4}{\left (a + b x \right )}}{32 b} + \frac {9 c^{2} d \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{32 b^{2}} + \frac {15 c^{2} d \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{32 b^{2}} + \frac {9 c d^{2} x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{16 b^{2}} + \frac {15 c d^{2} x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{16 b^{2}} + \frac {9 d^{3} x^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{32 b^{2}} + \frac {15 d^{3} x^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{32 b^{2}} - \frac {9 c d^{2} \sin ^{4}{\left (a + b x \right )}}{64 b^{3}} + \frac {15 c d^{2} \cos ^{4}{\left (a + b x \right )}}{64 b^{3}} - \frac {45 d^{3} x \sin ^{4}{\left (a + b x \right )}}{256 b^{3}} - \frac {9 d^{3} x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{128 b^{3}} + \frac {51 d^{3} x \cos ^{4}{\left (a + b x \right )}}{256 b^{3}} - \frac {45 d^{3} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{256 b^{4}} - \frac {51 d^{3} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{256 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sin {\left (a \right )} \cos ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-c**3*cos(a + b*x)**4/(4*b) + 9*c**2*d*x*sin(a + b*x)**4/(32*b) 
 + 9*c**2*d*x*sin(a + b*x)**2*cos(a + b*x)**2/(16*b) - 15*c**2*d*x*cos(a + 
 b*x)**4/(32*b) + 9*c*d**2*x**2*sin(a + b*x)**4/(32*b) + 9*c*d**2*x**2*sin 
(a + b*x)**2*cos(a + b*x)**2/(16*b) - 15*c*d**2*x**2*cos(a + b*x)**4/(32*b 
) + 3*d**3*x**3*sin(a + b*x)**4/(32*b) + 3*d**3*x**3*sin(a + b*x)**2*cos(a 
 + b*x)**2/(16*b) - 5*d**3*x**3*cos(a + b*x)**4/(32*b) + 9*c**2*d*sin(a + 
b*x)**3*cos(a + b*x)/(32*b**2) + 15*c**2*d*sin(a + b*x)*cos(a + b*x)**3/(3 
2*b**2) + 9*c*d**2*x*sin(a + b*x)**3*cos(a + b*x)/(16*b**2) + 15*c*d**2*x* 
sin(a + b*x)*cos(a + b*x)**3/(16*b**2) + 9*d**3*x**2*sin(a + b*x)**3*cos(a 
 + b*x)/(32*b**2) + 15*d**3*x**2*sin(a + b*x)*cos(a + b*x)**3/(32*b**2) - 
9*c*d**2*sin(a + b*x)**4/(64*b**3) + 15*c*d**2*cos(a + b*x)**4/(64*b**3) - 
 45*d**3*x*sin(a + b*x)**4/(256*b**3) - 9*d**3*x*sin(a + b*x)**2*cos(a + b 
*x)**2/(128*b**3) + 51*d**3*x*cos(a + b*x)**4/(256*b**3) - 45*d**3*sin(a + 
 b*x)**3*cos(a + b*x)/(256*b**4) - 51*d**3*sin(a + b*x)*cos(a + b*x)**3/(2 
56*b**4), Ne(b, 0)), ((c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/ 
4)*sin(a)*cos(a)**3, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (178) = 356\).

Time = 0.05 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.80 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/1024*(256*c^3*cos(b*x + a)^4 - 768*a*c^2*d*cos(b*x + a)^4/b + 768*a^2*c 
*d^2*cos(b*x + a)^4/b^2 - 256*a^3*d^3*cos(b*x + a)^4/b^3 + 24*(4*(b*x + a) 
*cos(4*b*x + 4*a) + 16*(b*x + a)*cos(2*b*x + 2*a) - sin(4*b*x + 4*a) - 8*s 
in(2*b*x + 2*a))*c^2*d/b - 48*(4*(b*x + a)*cos(4*b*x + 4*a) + 16*(b*x + a) 
*cos(2*b*x + 2*a) - sin(4*b*x + 4*a) - 8*sin(2*b*x + 2*a))*a*c*d^2/b^2 + 2 
4*(4*(b*x + a)*cos(4*b*x + 4*a) + 16*(b*x + a)*cos(2*b*x + 2*a) - sin(4*b* 
x + 4*a) - 8*sin(2*b*x + 2*a))*a^2*d^3/b^3 + 12*((8*(b*x + a)^2 - 1)*cos(4 
*b*x + 4*a) + 16*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 4*(b*x + a)*sin(4* 
b*x + 4*a) - 32*(b*x + a)*sin(2*b*x + 2*a))*c*d^2/b^2 - 12*((8*(b*x + a)^2 
 - 1)*cos(4*b*x + 4*a) + 16*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 4*(b*x 
+ a)*sin(4*b*x + 4*a) - 32*(b*x + a)*sin(2*b*x + 2*a))*a*d^3/b^3 + (4*(8*( 
b*x + a)^3 - 3*b*x - 3*a)*cos(4*b*x + 4*a) + 64*(2*(b*x + a)^3 - 3*b*x - 3 
*a)*cos(2*b*x + 2*a) - 3*(8*(b*x + a)^2 - 1)*sin(4*b*x + 4*a) - 96*(2*(b*x 
 + a)^2 - 1)*sin(2*b*x + 2*a))*d^3/b^3)/b
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.23 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {{\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 24 \, b^{3} c^{2} d x + 8 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \cos \left (4 \, b x + 4 \, a\right )}{256 \, b^{4}} - \frac {{\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{2} d x + 2 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{4}} + \frac {3 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (4 \, b x + 4 \, a\right )}{1024 \, b^{4}} + \frac {3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{32 \, b^{4}} \] Input:

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

Output:

-1/256*(8*b^3*d^3*x^3 + 24*b^3*c*d^2*x^2 + 24*b^3*c^2*d*x + 8*b^3*c^3 - 3* 
b*d^3*x - 3*b*c*d^2)*cos(4*b*x + 4*a)/b^4 - 1/16*(2*b^3*d^3*x^3 + 6*b^3*c* 
d^2*x^2 + 6*b^3*c^2*d*x + 2*b^3*c^3 - 3*b*d^3*x - 3*b*c*d^2)*cos(2*b*x + 2 
*a)/b^4 + 3/1024*(8*b^2*d^3*x^2 + 16*b^2*c*d^2*x + 8*b^2*c^2*d - d^3)*sin( 
4*b*x + 4*a)/b^4 + 3/32*(2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d - d^3 
)*sin(2*b*x + 2*a)/b^4
 

Mupad [B] (verification not implemented)

Time = 19.13 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.87 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {24\,d^3\,\sin \left (2\,a+2\,b\,x\right )+\frac {3\,d^3\,\sin \left (4\,a+4\,b\,x\right )}{4}+32\,b^3\,c^3\,\cos \left (2\,a+2\,b\,x\right )+8\,b^3\,c^3\,\cos \left (4\,a+4\,b\,x\right )-48\,b^2\,c^2\,d\,\sin \left (2\,a+2\,b\,x\right )-6\,b^2\,c^2\,d\,\sin \left (4\,a+4\,b\,x\right )+32\,b^3\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )+8\,b^3\,d^3\,x^3\,\cos \left (4\,a+4\,b\,x\right )-48\,b^2\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )-6\,b^2\,d^3\,x^2\,\sin \left (4\,a+4\,b\,x\right )-48\,b\,c\,d^2\,\cos \left (2\,a+2\,b\,x\right )-3\,b\,c\,d^2\,\cos \left (4\,a+4\,b\,x\right )-48\,b\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )-3\,b\,d^3\,x\,\cos \left (4\,a+4\,b\,x\right )+96\,b^3\,c^2\,d\,x\,\cos \left (2\,a+2\,b\,x\right )+24\,b^3\,c^2\,d\,x\,\cos \left (4\,a+4\,b\,x\right )-96\,b^2\,c\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )-12\,b^2\,c\,d^2\,x\,\sin \left (4\,a+4\,b\,x\right )+96\,b^3\,c\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )+24\,b^3\,c\,d^2\,x^2\,\cos \left (4\,a+4\,b\,x\right )}{256\,b^4} \] Input:

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

Output:

-(24*d^3*sin(2*a + 2*b*x) + (3*d^3*sin(4*a + 4*b*x))/4 + 32*b^3*c^3*cos(2* 
a + 2*b*x) + 8*b^3*c^3*cos(4*a + 4*b*x) - 48*b^2*c^2*d*sin(2*a + 2*b*x) - 
6*b^2*c^2*d*sin(4*a + 4*b*x) + 32*b^3*d^3*x^3*cos(2*a + 2*b*x) + 8*b^3*d^3 
*x^3*cos(4*a + 4*b*x) - 48*b^2*d^3*x^2*sin(2*a + 2*b*x) - 6*b^2*d^3*x^2*si 
n(4*a + 4*b*x) - 48*b*c*d^2*cos(2*a + 2*b*x) - 3*b*c*d^2*cos(4*a + 4*b*x) 
- 48*b*d^3*x*cos(2*a + 2*b*x) - 3*b*d^3*x*cos(4*a + 4*b*x) + 96*b^3*c^2*d* 
x*cos(2*a + 2*b*x) + 24*b^3*c^2*d*x*cos(4*a + 4*b*x) - 96*b^2*c*d^2*x*sin( 
2*a + 2*b*x) - 12*b^2*c*d^2*x*sin(4*a + 4*b*x) + 96*b^3*c*d^2*x^2*cos(2*a 
+ 2*b*x) + 24*b^3*c*d^2*x^2*cos(4*a + 4*b*x))/(256*b^4)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.25 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx=\frac {120 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{2} c^{2} d +120 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{2} d^{3} x^{2}+384 \sin \left (b x +a \right )^{2} b^{3} c^{2} d x +384 \sin \left (b x +a \right )^{2} b^{3} c \,d^{2} x^{2}+128 \sin \left (b x +a \right )^{2} b^{3} d^{3} x^{3}-120 \sin \left (b x +a \right )^{2} b \,d^{3} x -96 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b^{2} c \,d^{2} x -120 b^{3} c \,d^{2} x^{2}-128 b^{3} c^{3}-48 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b^{2} c^{2} d -48 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b^{2} d^{3} x^{2}-192 \sin \left (b x +a \right )^{4} b^{3} c^{2} d x -192 \sin \left (b x +a \right )^{4} b^{3} c \,d^{2} x^{2}+6 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} d^{3}-64 \sin \left (b x +a \right )^{4} b^{3} c^{3}-40 b^{3} d^{3} x^{3}+120 b c \,d^{2}+240 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{2} c \,d^{2} x -51 \cos \left (b x +a \right ) \sin \left (b x +a \right ) d^{3}+51 b \,d^{3} x +128 \sin \left (b x +a \right )^{2} b^{3} c^{3}-64 \sin \left (b x +a \right )^{4} b^{3} d^{3} x^{3}+24 \sin \left (b x +a \right )^{4} b c \,d^{2}+24 \sin \left (b x +a \right )^{4} b \,d^{3} x -120 \sin \left (b x +a \right )^{2} b c \,d^{2}-120 b^{3} c^{2} d x}{256 b^{4}} \] Input:

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

Output:

( - 48*cos(a + b*x)*sin(a + b*x)**3*b**2*c**2*d - 96*cos(a + b*x)*sin(a + 
b*x)**3*b**2*c*d**2*x - 48*cos(a + b*x)*sin(a + b*x)**3*b**2*d**3*x**2 + 6 
*cos(a + b*x)*sin(a + b*x)**3*d**3 + 120*cos(a + b*x)*sin(a + b*x)*b**2*c* 
*2*d + 240*cos(a + b*x)*sin(a + b*x)*b**2*c*d**2*x + 120*cos(a + b*x)*sin( 
a + b*x)*b**2*d**3*x**2 - 51*cos(a + b*x)*sin(a + b*x)*d**3 - 64*sin(a + b 
*x)**4*b**3*c**3 - 192*sin(a + b*x)**4*b**3*c**2*d*x - 192*sin(a + b*x)**4 
*b**3*c*d**2*x**2 - 64*sin(a + b*x)**4*b**3*d**3*x**3 + 24*sin(a + b*x)**4 
*b*c*d**2 + 24*sin(a + b*x)**4*b*d**3*x + 128*sin(a + b*x)**2*b**3*c**3 + 
384*sin(a + b*x)**2*b**3*c**2*d*x + 384*sin(a + b*x)**2*b**3*c*d**2*x**2 + 
 128*sin(a + b*x)**2*b**3*d**3*x**3 - 120*sin(a + b*x)**2*b*c*d**2 - 120*s 
in(a + b*x)**2*b*d**3*x - 128*b**3*c**3 - 120*b**3*c**2*d*x - 120*b**3*c*d 
**2*x**2 - 40*b**3*d**3*x**3 + 120*b*c*d**2 + 51*b*d**3*x)/(256*b**4)