\(\int (c+d x)^3 \sin ^3(a+b x) \, dx\) [17]

Optimal result

Integrand size = 16, antiderivative size = 175 \[ \int (c+d x)^3 \sin ^3(a+b x) \, dx=\frac {40 d^2 (c+d x) \cos (a+b x)}{9 b^3}-\frac {2 (c+d x)^3 \cos (a+b x)}{3 b}-\frac {40 d^3 \sin (a+b x)}{9 b^4}+\frac {2 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {2 d^2 (c+d x) \cos (a+b x) \sin ^2(a+b x)}{9 b^3}-\frac {(c+d x)^3 \cos (a+b x) \sin ^2(a+b x)}{3 b}-\frac {2 d^3 \sin ^3(a+b x)}{27 b^4}+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2} \] Output:

40/9*d^2*(d*x+c)*cos(b*x+a)/b^3-2/3*(d*x+c)^3*cos(b*x+a)/b-40/9*d^3*sin(b* 
x+a)/b^4+2*d*(d*x+c)^2*sin(b*x+a)/b^2+2/9*d^2*(d*x+c)*cos(b*x+a)*sin(b*x+a 
)^2/b^3-1/3*(d*x+c)^3*cos(b*x+a)*sin(b*x+a)^2/b-2/27*d^3*sin(b*x+a)^3/b^4+ 
1/3*d*(d*x+c)^2*sin(b*x+a)^3/b^2
 

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.73 \[ \int (c+d x)^3 \sin ^3(a+b x) \, dx=\frac {-162 b (c+d x) \left (-6 d^2+b^2 (c+d x)^2\right ) \cos (a+b x)+6 b (c+d x) \left (-2 d^2+3 b^2 (c+d x)^2\right ) \cos (3 (a+b x))-4 d \left (242 d^2-117 b^2 (c+d x)^2+\left (-2 d^2+9 b^2 (c+d x)^2\right ) \cos (2 (a+b x))\right ) \sin (a+b x)}{216 b^4} \] Input:

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

Output:

(-162*b*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + 6*b*(c + d*x)* 
(-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] - 4*d*(242*d^2 - 117*b^2*(c 
+ d*x)^2 + (-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[a + b*x])/(2 
16*b^4)
 

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.25, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3792, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3117, 3791, 3042, 3777, 3042, 3117}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 3791

\(\displaystyle -\frac {2 d^2 \left (\frac {2}{3} \int (c+d x) \sin (a+b x)dx+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 d^2 \left (\frac {2}{3} \int (c+d x) \sin (a+b x)dx+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {2 d^2 \left (\frac {2}{3} \left (\frac {d \int \cos (a+b x)dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 d^2 \left (\frac {2}{3} \left (\frac {d \int \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

\(\Big \downarrow \) 3117

\(\displaystyle -\frac {2 d^2 \left (\frac {2}{3} \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\)

Input:

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

Output:

-1/3*((c + d*x)^3*Cos[a + b*x]*Sin[a + b*x]^2)/b + (d*(c + d*x)^2*Sin[a + 
b*x]^3)/(3*b^2) - (2*d^2*(-1/3*((c + d*x)*Cos[a + b*x]*Sin[a + b*x]^2)/b + 
 (d*Sin[a + b*x]^3)/(9*b^2) + (2*(-(((c + d*x)*Cos[a + b*x])/b) + (d*Sin[a 
 + b*x])/b^2))/3))/(3*b^2) + (2*(-(((c + d*x)^3*Cos[a + b*x])/b) + (3*d*(( 
(c + d*x)^2*Sin[a + b*x])/b - (2*d*(-(((c + d*x)*Cos[a + b*x])/b) + (d*Sin 
[a + b*x])/b^2))/b))/b))/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3791

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

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]
 

Maple [A] (verified)

Time = 2.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.81

method	result	size
parallelrisch	\(\frac {3 \left (d x +c \right ) \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{3}\right ) b \cos \left (3 b x +3 a \right )-3 d \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{9}\right ) \sin \left (3 b x +3 a \right )-27 \left (d x +c \right ) b \left (\left (d x +c \right )^{2} b^{2}-6 d^{2}\right ) \cos \left (b x +a \right )+81 d \left (\left (d x +c \right )^{2} b^{2}-2 d^{2}\right ) \sin \left (b x +a \right )-24 b^{3} c^{3}+160 c \,d^{2} b}{36 b^{4}}\)	\(142\)
risch	\(-\frac {3 \left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \cos \left (b x +a \right )}{4 b^{3}}+\frac {9 d \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 d^{2}\right ) \sin \left (b x +a \right )}{4 b^{4}}+\frac {\left (3 b^{2} d^{3} x^{3}+9 b^{2} c \,d^{2} x^{2}+9 b^{2} c^{2} d x +3 b^{2} c^{3}-2 d^{3} x -2 c \,d^{2}\right ) \cos \left (3 b x +3 a \right )}{36 b^{3}}-\frac {d \left (9 x^{2} d^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}-2 d^{2}\right ) \sin \left (3 b x +3 a \right )}{108 b^{4}}\)	\(224\)
derivativedivides	\(\frac {\frac {a^{3} d^{3} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3 b^{3}}-\frac {a^{2} c \,d^{2} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\left (b x +a \right ) \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {\sin \left (b x +a \right )^{3}}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b^{3}}+\frac {a \,c^{2} d \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{b}-\frac {6 a c \,d^{2} \left (-\frac {\left (b x +a \right ) \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {\sin \left (b x +a \right )^{3}}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (-\frac {\left (b x +a \right )^{2} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {4 \cos \left (b x +a \right )}{3}+\frac {4 \left (b x +a \right ) \sin \left (b x +a \right )}{3}+\frac {2 \left (b x +a \right ) \sin \left (b x +a \right )^{3}}{9}+\frac {2 \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{27}\right )}{b^{3}}-\frac {c^{3} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {3 c^{2} d \left (-\frac {\left (b x +a \right ) \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {\sin \left (b x +a \right )^{3}}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b}+\frac {3 c \,d^{2} \left (-\frac {\left (b x +a \right )^{2} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {4 \cos \left (b x +a \right )}{3}+\frac {4 \left (b x +a \right ) \sin \left (b x +a \right )}{3}+\frac {2 \left (b x +a \right ) \sin \left (b x +a \right )^{3}}{9}+\frac {2 \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{27}\right )}{b^{2}}+\frac {d^{3} \left (-\frac {\left (b x +a \right )^{3} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+2 \left (b x +a \right )^{2} \sin \left (b x +a \right )-\frac {40 \sin \left (b x +a \right )}{9}+4 \left (b x +a \right ) \cos \left (b x +a \right )+\frac {\left (b x +a \right )^{2} \sin \left (b x +a \right )^{3}}{3}+\frac {2 \left (b x +a \right ) \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{9}-\frac {2 \sin \left (b x +a \right )^{3}}{27}\right )}{b^{3}}}{b}\)	\(560\)
default	\(\frac {\frac {a^{3} d^{3} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3 b^{3}}-\frac {a^{2} c \,d^{2} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\left (b x +a \right ) \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {\sin \left (b x +a \right )^{3}}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b^{3}}+\frac {a \,c^{2} d \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{b}-\frac {6 a c \,d^{2} \left (-\frac {\left (b x +a \right ) \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {\sin \left (b x +a \right )^{3}}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (-\frac {\left (b x +a \right )^{2} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {4 \cos \left (b x +a \right )}{3}+\frac {4 \left (b x +a \right ) \sin \left (b x +a \right )}{3}+\frac {2 \left (b x +a \right ) \sin \left (b x +a \right )^{3}}{9}+\frac {2 \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{27}\right )}{b^{3}}-\frac {c^{3} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {3 c^{2} d \left (-\frac {\left (b x +a \right ) \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {\sin \left (b x +a \right )^{3}}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b}+\frac {3 c \,d^{2} \left (-\frac {\left (b x +a \right )^{2} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {4 \cos \left (b x +a \right )}{3}+\frac {4 \left (b x +a \right ) \sin \left (b x +a \right )}{3}+\frac {2 \left (b x +a \right ) \sin \left (b x +a \right )^{3}}{9}+\frac {2 \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{27}\right )}{b^{2}}+\frac {d^{3} \left (-\frac {\left (b x +a \right )^{3} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+2 \left (b x +a \right )^{2} \sin \left (b x +a \right )-\frac {40 \sin \left (b x +a \right )}{9}+4 \left (b x +a \right ) \cos \left (b x +a \right )+\frac {\left (b x +a \right )^{2} \sin \left (b x +a \right )^{3}}{3}+\frac {2 \left (b x +a \right ) \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{9}-\frac {2 \sin \left (b x +a \right )^{3}}{27}\right )}{b^{3}}}{b}\)	\(560\)
orering	\(\frac {20 d \left (9 d^{4} x^{4} b^{4}+36 b^{4} c \,d^{3} x^{3}+54 b^{4} c^{2} d^{2} x^{2}+36 b^{4} c^{3} d x +9 b^{4} c^{4}-22 b^{2} d^{4} x^{2}-44 b^{2} c \,d^{3} x -22 b^{2} c^{2} d^{2}-72 d^{4}\right ) \sin \left (b x +a \right )^{3}}{27 b^{6} \left (d x +c \right )^{2}}-\frac {10 \left (3 d^{4} x^{4} b^{4}+12 b^{4} c \,d^{3} x^{3}+18 b^{4} c^{2} d^{2} x^{2}+12 b^{4} c^{3} d x +3 b^{4} c^{4}-2 b^{2} d^{4} x^{2}-4 b^{2} c \,d^{3} x -2 b^{2} c^{2} d^{2}-84 d^{4}\right ) \left (3 \left (d x +c \right )^{2} \sin \left (b x +a \right )^{3} d +3 \left (d x +c \right )^{3} \sin \left (b x +a \right )^{2} b \cos \left (b x +a \right )\right )}{27 \left (d x +c \right )^{4} b^{6}}+\frac {4 d \left (9 x^{2} d^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}-50 d^{2}\right ) \left (6 \left (d x +c \right ) \sin \left (b x +a \right )^{3} d^{2}+18 \left (d x +c \right )^{2} \sin \left (b x +a \right )^{2} d b \cos \left (b x +a \right )+6 \left (d x +c \right )^{3} \sin \left (b x +a \right ) b^{2} \cos \left (b x +a \right )^{2}-3 \left (d x +c \right )^{3} \sin \left (b x +a \right )^{3} b^{2}\right )}{27 b^{6} \left (d x +c \right )^{3}}-\frac {\left (3 x^{2} d^{2} b^{2}+6 b^{2} c d x +3 b^{2} c^{2}-20 d^{2}\right ) \left (6 d^{3} \sin \left (b x +a \right )^{3}+54 \left (d x +c \right ) \sin \left (b x +a \right )^{2} d^{2} b \cos \left (b x +a \right )+54 \left (d x +c \right )^{2} \sin \left (b x +a \right ) d \,b^{2} \cos \left (b x +a \right )^{2}-27 \left (d x +c \right )^{2} \sin \left (b x +a \right )^{3} d \,b^{2}+6 \left (d x +c \right )^{3} b^{3} \cos \left (b x +a \right )^{3}-21 \left (d x +c \right )^{3} \sin \left (b x +a \right )^{2} b^{3} \cos \left (b x +a \right )\right )}{27 \left (d x +c \right )^{2} b^{6}}\)	\(578\)
norman	\(\frac {\frac {8 c \,d^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{b^{3}}+\frac {-12 b^{2} c^{3}+80 c \,d^{2}}{9 b^{3}}-\frac {2 d^{3} x^{3}}{3 b}+\frac {\left (-12 b^{2} c^{3}+56 c \,d^{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{3 b^{3}}-\frac {2 c \,d^{2} x^{2}}{b}+\frac {4 d \left (9 b^{2} c^{2}-20 d^{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{9 b^{4}}+\frac {4 d \left (9 b^{2} c^{2}-20 d^{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}}{9 b^{4}}-\frac {2 d \left (9 b^{2} c^{2}-20 d^{2}\right ) x}{9 b^{3}}+\frac {16 d \left (18 b^{2} c^{2}-31 d^{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{27 b^{4}}+\frac {4 d^{3} x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{2}}+\frac {32 d^{3} x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{3 b^{2}}+\frac {4 d^{3} x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}}{b^{2}}-\frac {2 d^{3} x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b}+\frac {2 d^{3} x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{b}+\frac {2 d^{3} x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{3 b}-\frac {6 c \,d^{2} x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b}+\frac {6 c \,d^{2} x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{b}+\frac {2 c \,d^{2} x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{b}+\frac {8 c \,d^{2} x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{2}}+\frac {64 c \,d^{2} x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{3 b^{2}}+\frac {8 c \,d^{2} x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}}{b^{2}}+\frac {2 d \left (9 b^{2} c^{2}-20 d^{2}\right ) x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{9 b^{3}}-\frac {2 d \left (9 b^{2} c^{2}-8 d^{2}\right ) x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{3 b^{3}}+\frac {2 d \left (9 b^{2} c^{2}-8 d^{2}\right ) x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{3 b^{3}}}{\left (1+\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right )^{3}}\)	\(580\)

Input:

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

Output:

1/36*(3*(d*x+c)*((d*x+c)^2*b^2-2/3*d^2)*b*cos(3*b*x+3*a)-3*d*((d*x+c)^2*b^ 
2-2/9*d^2)*sin(3*b*x+3*a)-27*(d*x+c)*b*((d*x+c)^2*b^2-6*d^2)*cos(b*x+a)+81 
*d*((d*x+c)^2*b^2-2*d^2)*sin(b*x+a)-24*b^3*c^3+160*c*d^2*b)/b^4
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.30 \[ \int (c+d x)^3 \sin ^3(a+b x) \, dx=\frac {3 \, {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} - 2 \, b c d^{2} + {\left (9 \, b^{3} c^{2} d - 2 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} - 9 \, {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} - 14 \, b c d^{2} + {\left (9 \, b^{3} c^{2} d - 14 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right ) + {\left (63 \, b^{2} d^{3} x^{2} + 126 \, b^{2} c d^{2} x + 63 \, b^{2} c^{2} d - 122 \, d^{3} - {\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )}{27 \, b^{4}} \] Input:

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

Output:

1/27*(3*(3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^2 + 3*b^3*c^3 - 2*b*c*d^2 + (9*b^3* 
c^2*d - 2*b*d^3)*x)*cos(b*x + a)^3 - 9*(3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^2 + 
3*b^3*c^3 - 14*b*c*d^2 + (9*b^3*c^2*d - 14*b*d^3)*x)*cos(b*x + a) + (63*b^ 
2*d^3*x^2 + 126*b^2*c*d^2*x + 63*b^2*c^2*d - 122*d^3 - (9*b^2*d^3*x^2 + 18 
*b^2*c*d^2*x + 9*b^2*c^2*d - 2*d^3)*cos(b*x + a)^2)*sin(b*x + a))/b^4
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (173) = 346\).

Time = 0.46 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.83 \[ \int (c+d x)^3 \sin ^3(a+b x) \, dx=\begin {cases} - \frac {c^{3} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 c^{3} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {3 c^{2} d x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 c^{2} d x \cos ^{3}{\left (a + b x \right )}}{b} - \frac {3 c d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 c d^{2} x^{2} \cos ^{3}{\left (a + b x \right )}}{b} - \frac {d^{3} x^{3} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 d^{3} x^{3} \cos ^{3}{\left (a + b x \right )}}{3 b} + \frac {7 c^{2} d \sin ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 c^{2} d \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b^{2}} + \frac {14 c d^{2} x \sin ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {4 c d^{2} x \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b^{2}} + \frac {7 d^{3} x^{2} \sin ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 d^{3} x^{2} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b^{2}} + \frac {14 c d^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{3}} + \frac {40 c d^{2} \cos ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {14 d^{3} x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{3}} + \frac {40 d^{3} x \cos ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {122 d^{3} \sin ^{3}{\left (a + b x \right )}}{27 b^{4}} - \frac {40 d^{3} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{9 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sin ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-c**3*sin(a + b*x)**2*cos(a + b*x)/b - 2*c**3*cos(a + b*x)**3/( 
3*b) - 3*c**2*d*x*sin(a + b*x)**2*cos(a + b*x)/b - 2*c**2*d*x*cos(a + b*x) 
**3/b - 3*c*d**2*x**2*sin(a + b*x)**2*cos(a + b*x)/b - 2*c*d**2*x**2*cos(a 
 + b*x)**3/b - d**3*x**3*sin(a + b*x)**2*cos(a + b*x)/b - 2*d**3*x**3*cos( 
a + b*x)**3/(3*b) + 7*c**2*d*sin(a + b*x)**3/(3*b**2) + 2*c**2*d*sin(a + b 
*x)*cos(a + b*x)**2/b**2 + 14*c*d**2*x*sin(a + b*x)**3/(3*b**2) + 4*c*d**2 
*x*sin(a + b*x)*cos(a + b*x)**2/b**2 + 7*d**3*x**2*sin(a + b*x)**3/(3*b**2 
) + 2*d**3*x**2*sin(a + b*x)*cos(a + b*x)**2/b**2 + 14*c*d**2*sin(a + b*x) 
**2*cos(a + b*x)/(3*b**3) + 40*c*d**2*cos(a + b*x)**3/(9*b**3) + 14*d**3*x 
*sin(a + b*x)**2*cos(a + b*x)/(3*b**3) + 40*d**3*x*cos(a + b*x)**3/(9*b**3 
) - 122*d**3*sin(a + b*x)**3/(27*b**4) - 40*d**3*sin(a + b*x)*cos(a + b*x) 
**2/(9*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)**3, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (161) = 322\).

Time = 0.06 (sec) , antiderivative size = 541, normalized size of antiderivative = 3.09 \[ \int (c+d x)^3 \sin ^3(a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

1/108*(36*(cos(b*x + a)^3 - 3*cos(b*x + a))*c^3 - 108*(cos(b*x + a)^3 - 3* 
cos(b*x + a))*a*c^2*d/b + 108*(cos(b*x + a)^3 - 3*cos(b*x + a))*a^2*c*d^2/ 
b^2 - 36*(cos(b*x + a)^3 - 3*cos(b*x + a))*a^3*d^3/b^3 + 9*(3*(b*x + a)*co 
s(3*b*x + 3*a) - 27*(b*x + a)*cos(b*x + a) - sin(3*b*x + 3*a) + 27*sin(b*x 
 + a))*c^2*d/b - 18*(3*(b*x + a)*cos(3*b*x + 3*a) - 27*(b*x + a)*cos(b*x + 
 a) - sin(3*b*x + 3*a) + 27*sin(b*x + a))*a*c*d^2/b^2 + 9*(3*(b*x + a)*cos 
(3*b*x + 3*a) - 27*(b*x + a)*cos(b*x + a) - sin(3*b*x + 3*a) + 27*sin(b*x 
+ a))*a^2*d^3/b^3 + 3*((9*(b*x + a)^2 - 2)*cos(3*b*x + 3*a) - 81*((b*x + a 
)^2 - 2)*cos(b*x + a) - 6*(b*x + a)*sin(3*b*x + 3*a) + 162*(b*x + a)*sin(b 
*x + a))*c*d^2/b^2 - 3*((9*(b*x + a)^2 - 2)*cos(3*b*x + 3*a) - 81*((b*x + 
a)^2 - 2)*cos(b*x + a) - 6*(b*x + a)*sin(3*b*x + 3*a) + 162*(b*x + a)*sin( 
b*x + a))*a*d^3/b^3 + (3*(3*(b*x + a)^3 - 2*b*x - 2*a)*cos(3*b*x + 3*a) - 
81*((b*x + a)^3 - 6*b*x - 6*a)*cos(b*x + a) - (9*(b*x + a)^2 - 2)*sin(3*b* 
x + 3*a) + 243*((b*x + a)^2 - 2)*sin(b*x + a))*d^3/b^3)/b
 

Giac [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.32 \[ \int (c+d x)^3 \sin ^3(a+b x) \, dx=\frac {{\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 9 \, b^{3} c^{2} d x + 3 \, b^{3} c^{3} - 2 \, b d^{3} x - 2 \, b c d^{2}\right )} \cos \left (3 \, b x + 3 \, a\right )}{36 \, b^{4}} - \frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \cos \left (b x + a\right )}{4 \, b^{4}} - \frac {{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (3 \, b x + 3 \, a\right )}{108 \, b^{4}} + \frac {9 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (b x + a\right )}{4 \, b^{4}} \] Input:

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

Output:

1/36*(3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^2 + 9*b^3*c^2*d*x + 3*b^3*c^3 - 2*b*d^ 
3*x - 2*b*c*d^2)*cos(3*b*x + 3*a)/b^4 - 3/4*(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 - 6*b*d^3*x - 6*b*c*d^2)*cos(b*x + a)/b^4 - 1/1 
08*(9*b^2*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d - 2*d^3)*sin(3*b*x + 3*a) 
/b^4 + 9/4*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d - 2*d^3)*sin(b*x + a)/ 
b^4
 

Mupad [B] (verification not implemented)

Time = 35.81 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.09 \[ \int (c+d x)^3 \sin ^3(a+b x) \, dx=\frac {2\,{\cos \left (a+b\,x\right )}^3\,\left (20\,c\,d^2-3\,b^2\,c^3\right )}{9\,b^3}-\frac {{\sin \left (a+b\,x\right )}^3\,\left (122\,d^3-63\,b^2\,c^2\,d\right )}{27\,b^4}+\frac {\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (14\,c\,d^2-3\,b^2\,c^3\right )}{3\,b^3}-\frac {2\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (20\,d^3-9\,b^2\,c^2\,d\right )}{9\,b^4}+\frac {2\,x\,{\cos \left (a+b\,x\right )}^3\,\left (20\,d^3-9\,b^2\,c^2\,d\right )}{9\,b^3}-\frac {2\,d^3\,x^3\,{\cos \left (a+b\,x\right )}^3}{3\,b}+\frac {7\,d^3\,x^2\,{\sin \left (a+b\,x\right )}^3}{3\,b^2}+\frac {14\,c\,d^2\,x\,{\sin \left (a+b\,x\right )}^3}{3\,b^2}+\frac {x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (14\,d^3-9\,b^2\,c^2\,d\right )}{3\,b^3}-\frac {2\,c\,d^2\,x^2\,{\cos \left (a+b\,x\right )}^3}{b}-\frac {d^3\,x^3\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{b}+\frac {2\,d^3\,x^2\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{b^2}-\frac {3\,c\,d^2\,x^2\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{b}+\frac {4\,c\,d^2\,x\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{b^2} \] Input:

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

Output:

(2*cos(a + b*x)^3*(20*c*d^2 - 3*b^2*c^3))/(9*b^3) - (sin(a + b*x)^3*(122*d 
^3 - 63*b^2*c^2*d))/(27*b^4) + (cos(a + b*x)*sin(a + b*x)^2*(14*c*d^2 - 3* 
b^2*c^3))/(3*b^3) - (2*cos(a + b*x)^2*sin(a + b*x)*(20*d^3 - 9*b^2*c^2*d)) 
/(9*b^4) + (2*x*cos(a + b*x)^3*(20*d^3 - 9*b^2*c^2*d))/(9*b^3) - (2*d^3*x^ 
3*cos(a + b*x)^3)/(3*b) + (7*d^3*x^2*sin(a + b*x)^3)/(3*b^2) + (14*c*d^2*x 
*sin(a + b*x)^3)/(3*b^2) + (x*cos(a + b*x)*sin(a + b*x)^2*(14*d^3 - 9*b^2* 
c^2*d))/(3*b^3) - (2*c*d^2*x^2*cos(a + b*x)^3)/b - (d^3*x^3*cos(a + b*x)*s 
in(a + b*x)^2)/b + (2*d^3*x^2*cos(a + b*x)^2*sin(a + b*x))/b^2 - (3*c*d^2* 
x^2*cos(a + b*x)*sin(a + b*x)^2)/b + (4*c*d^2*x*cos(a + b*x)^2*sin(a + b*x 
))/b^2
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.25 \[ \int (c+d x)^3 \sin ^3(a+b x) \, dx=\frac {-9 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{3} c^{3}-27 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{3} c^{2} d x -27 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{3} c \,d^{2} x^{2}-9 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{3} d^{3} x^{3}+6 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b c \,d^{2}+6 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b \,d^{3} x -18 \cos \left (b x +a \right ) b^{3} c^{3}-54 \cos \left (b x +a \right ) b^{3} c^{2} d x -54 \cos \left (b x +a \right ) b^{3} c \,d^{2} x^{2}-18 \cos \left (b x +a \right ) b^{3} d^{3} x^{3}+120 \cos \left (b x +a \right ) b c \,d^{2}+120 \cos \left (b x +a \right ) b \,d^{3} x +9 \sin \left (b x +a \right )^{3} b^{2} c^{2} d +18 \sin \left (b x +a \right )^{3} b^{2} c \,d^{2} x +9 \sin \left (b x +a \right )^{3} b^{2} d^{3} x^{2}-2 \sin \left (b x +a \right )^{3} d^{3}+54 \sin \left (b x +a \right ) b^{2} c^{2} d +108 \sin \left (b x +a \right ) b^{2} c \,d^{2} x +54 \sin \left (b x +a \right ) b^{2} d^{3} x^{2}-120 \sin \left (b x +a \right ) d^{3}+54 a \,b^{2} c^{2} d -48 a \,d^{3}-18 b^{3} c^{3}+48 b c \,d^{2}}{27 b^{4}} \] Input:

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

Output:

( - 9*cos(a + b*x)*sin(a + b*x)**2*b**3*c**3 - 27*cos(a + b*x)*sin(a + b*x 
)**2*b**3*c**2*d*x - 27*cos(a + b*x)*sin(a + b*x)**2*b**3*c*d**2*x**2 - 9* 
cos(a + b*x)*sin(a + b*x)**2*b**3*d**3*x**3 + 6*cos(a + b*x)*sin(a + b*x)* 
*2*b*c*d**2 + 6*cos(a + b*x)*sin(a + b*x)**2*b*d**3*x - 18*cos(a + b*x)*b* 
*3*c**3 - 54*cos(a + b*x)*b**3*c**2*d*x - 54*cos(a + b*x)*b**3*c*d**2*x**2 
 - 18*cos(a + b*x)*b**3*d**3*x**3 + 120*cos(a + b*x)*b*c*d**2 + 120*cos(a 
+ b*x)*b*d**3*x + 9*sin(a + b*x)**3*b**2*c**2*d + 18*sin(a + b*x)**3*b**2* 
c*d**2*x + 9*sin(a + b*x)**3*b**2*d**3*x**2 - 2*sin(a + b*x)**3*d**3 + 54* 
sin(a + b*x)*b**2*c**2*d + 108*sin(a + b*x)*b**2*c*d**2*x + 54*sin(a + b*x 
)*b**2*d**3*x**2 - 120*sin(a + b*x)*d**3 + 54*a*b**2*c**2*d - 48*a*d**3 - 
18*b**3*c**3 + 48*b*c*d**2)/(27*b**4)