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

Optimal result

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

[Out]

-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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4490, 3392, 32, 2715, 8} \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {3 d^3 \sin (a+b x) \cos ^3(a+b x)}{128 b^4}-\frac {45 d^3 \sin (a+b x) \cos (a+b x)}{256 b^4}+\frac {3 d^2 (c+d x) \cos ^4(a+b x)}{32 b^3}+\frac {9 d^2 (c+d x) \cos ^2(a+b x)}{32 b^3}+\frac {3 d (c+d x)^2 \sin (a+b x) \cos ^3(a+b x)}{16 b^2}+\frac {9 d (c+d x)^2 \sin (a+b x) \cos (a+b x)}{32 b^2}-\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}-\frac {45 d^3 x}{256 b^3}+\frac {3 (c+d x)^3}{32 b} \]

[In]

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

[Out]

(-45*d^3*x)/(256*b^3) + (3*(c + d*x)^3)/(32*b) + (9*d^2*(c + d*x)*Cos[a + b*x]^2)/(32*b^3) + (3*d^2*(c + d*x)*
Cos[a + b*x]^4)/(32*b^3) - ((c + d*x)^3*Cos[a + b*x]^4)/(4*b) - (45*d^3*Cos[a + b*x]*Sin[a + b*x])/(256*b^4) +
(9*d*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x])/(32*b^2) - (3*d^3*Cos[a + b*x]^3*Sin[a + b*x])/(128*b^4) + (3*d*(
c + d*x)^2*Cos[a + b*x]^3*Sin[a + b*x])/(16*b^2)

Rule 8

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

Rule 32

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

Rule 2715

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] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 3392

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

Rule 4490

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] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^3 \cos ^4(a+b x)}{4 b}+\frac {(3 d) \int (c+d x)^2 \cos ^4(a+b x) \, dx}{4 b} \\ & = \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 {3 d (c+d x)^2 \cos ^3(a+b x) \sin (a+b x)}{16 b^2}+\frac {(9 d) \int (c+d x)^2 \cos ^2(a+b x) \, dx}{16 b}-\frac {\left (3 d^3\right ) \int \cos ^4(a+b x) \, dx}{32 b^3} \\ & = \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 {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}+\frac {(9 d) \int (c+d x)^2 \, dx}{32 b}-\frac {\left (9 d^3\right ) \int \cos ^2(a+b x) \, dx}{128 b^3}-\frac {\left (9 d^3\right ) \int \cos ^2(a+b x) \, dx}{32 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}-\frac {\left (9 d^3\right ) \int 1 \, dx}{256 b^3}-\frac {\left (9 d^3\right ) \int 1 \, dx}{64 b^3} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.69 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx=\frac {-64 b (c+d x) \left (-3 d^2+2 b^2 (c+d x)^2\right ) \cos (2 (a+b x))-4 b (c+d x) \left (-3 d^2+8 b^2 (c+d x)^2\right ) \cos (4 (a+b x))+6 d \left (16 \left (-d^2+2 b^2 (c+d x)^2\right )+\left (-d^2+8 b^2 (c+d x)^2\right ) \cos (2 (a+b x))\right ) \sin (2 (a+b x))}{1024 b^4} \]

[In]

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

[Out]

(-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)*Co
s[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)

Maple [A] (verified)

Time = 1.59 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.76

method	result	size
parallelrisch	\(\frac {-32 b \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{2}\right ) \left (d x +c \right ) \cos \left (2 x b +2 a \right )-8 b \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{8}\right ) \left (d x +c \right ) \cos \left (4 x b +4 a \right )+48 d \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{2}\right ) \sin \left (2 x b +2 a \right )+6 \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{8}\right ) d \sin \left (4 x b +4 a \right )+40 b^{3} c^{3}-51 c \,d^{2} b}{256 b^{4}}\)	\(148\)
risch	\(-\frac {\left (8 b^{2} d^{3} x^{3}+24 b^{2} c \,d^{2} x^{2}+24 b^{2} c^{2} d x +8 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \cos \left (4 x b +4 a \right )}{256 b^{3}}+\frac {3 d \left (8 x^{2} d^{2} b^{2}+16 b^{2} c d x +8 b^{2} c^{2}-d^{2}\right ) \sin \left (4 x b +4 a \right )}{1024 b^{4}}-\frac {\left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \cos \left (2 x b +2 a \right )}{16 b^{3}}+\frac {3 d \left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}-d^{2}\right ) \sin \left (2 x b +2 a \right )}{32 b^{4}}\)	\(234\)
derivativedivides	\(\frac {\frac {a^{3} d^{3} \cos \left (x b +a \right )^{4}}{4 b^{3}}-\frac {3 a^{2} c \,d^{2} \cos \left (x b +a \right )^{4}}{4 b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{4}}{4}+\frac {\left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{16}+\frac {3 x b}{32}+\frac {3 a}{32}\right )}{b^{3}}+\frac {3 a \,c^{2} d \cos \left (x b +a \right )^{4}}{4 b}-\frac {6 a c \,d^{2} \left (-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{4}}{4}+\frac {\left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{16}+\frac {3 x b}{32}+\frac {3 a}{32}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (-\frac {\left (x b +a \right )^{2} \cos \left (x b +a \right )^{4}}{4}+\frac {\left (x b +a \right ) \left (\frac {\left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )}{2}-\frac {3 \left (x b +a \right )^{2}}{32}+\frac {\left (2 \cos \left (x b +a \right )^{2}+3\right )^{2}}{128}\right )}{b^{3}}-\frac {c^{3} \cos \left (x b +a \right )^{4}}{4}+\frac {3 c^{2} d \left (-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{4}}{4}+\frac {\left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{16}+\frac {3 x b}{32}+\frac {3 a}{32}\right )}{b}+\frac {3 c \,d^{2} \left (-\frac {\left (x b +a \right )^{2} \cos \left (x b +a \right )^{4}}{4}+\frac {\left (x b +a \right ) \left (\frac {\left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )}{2}-\frac {3 \left (x b +a \right )^{2}}{32}+\frac {\left (2 \cos \left (x b +a \right )^{2}+3\right )^{2}}{128}\right )}{b^{2}}+\frac {d^{3} \left (-\frac {\left (x b +a \right )^{3} \cos \left (x b +a \right )^{4}}{4}+\frac {3 \left (x b +a \right )^{2} \left (\frac {\left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )}{4}+\frac {3 \left (x b +a \right ) \cos \left (x b +a \right )^{4}}{32}-\frac {3 \left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{128}-\frac {45 x b}{256}-\frac {45 a}{256}+\frac {9 \left (x b +a \right ) \cos \left (x b +a \right )^{2}}{32}-\frac {9 \cos \left (x b +a \right ) \sin \left (x b +a \right )}{64}-\frac {3 \left (x b +a \right )^{3}}{16}\right )}{b^{3}}}{b}\)	\(586\)
default	\(\frac {\frac {a^{3} d^{3} \cos \left (x b +a \right )^{4}}{4 b^{3}}-\frac {3 a^{2} c \,d^{2} \cos \left (x b +a \right )^{4}}{4 b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{4}}{4}+\frac {\left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{16}+\frac {3 x b}{32}+\frac {3 a}{32}\right )}{b^{3}}+\frac {3 a \,c^{2} d \cos \left (x b +a \right )^{4}}{4 b}-\frac {6 a c \,d^{2} \left (-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{4}}{4}+\frac {\left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{16}+\frac {3 x b}{32}+\frac {3 a}{32}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (-\frac {\left (x b +a \right )^{2} \cos \left (x b +a \right )^{4}}{4}+\frac {\left (x b +a \right ) \left (\frac {\left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )}{2}-\frac {3 \left (x b +a \right )^{2}}{32}+\frac {\left (2 \cos \left (x b +a \right )^{2}+3\right )^{2}}{128}\right )}{b^{3}}-\frac {c^{3} \cos \left (x b +a \right )^{4}}{4}+\frac {3 c^{2} d \left (-\frac {\left (x b +a \right ) \cos \left (x b +a \right )^{4}}{4}+\frac {\left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{16}+\frac {3 x b}{32}+\frac {3 a}{32}\right )}{b}+\frac {3 c \,d^{2} \left (-\frac {\left (x b +a \right )^{2} \cos \left (x b +a \right )^{4}}{4}+\frac {\left (x b +a \right ) \left (\frac {\left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )}{2}-\frac {3 \left (x b +a \right )^{2}}{32}+\frac {\left (2 \cos \left (x b +a \right )^{2}+3\right )^{2}}{128}\right )}{b^{2}}+\frac {d^{3} \left (-\frac {\left (x b +a \right )^{3} \cos \left (x b +a \right )^{4}}{4}+\frac {3 \left (x b +a \right )^{2} \left (\frac {\left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{4}+\frac {3 x b}{8}+\frac {3 a}{8}\right )}{4}+\frac {3 \left (x b +a \right ) \cos \left (x b +a \right )^{4}}{32}-\frac {3 \left (\cos \left (x b +a \right )^{3}+\frac {3 \cos \left (x b +a \right )}{2}\right ) \sin \left (x b +a \right )}{128}-\frac {45 x b}{256}-\frac {45 a}{256}+\frac {9 \left (x b +a \right ) \cos \left (x b +a \right )^{2}}{32}-\frac {9 \cos \left (x b +a \right ) \sin \left (x b +a \right )}{64}-\frac {3 \left (x b +a \right )^{3}}{16}\right )}{b^{3}}}{b}\)	\(586\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.21 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx=\frac {24 \, b^{3} d^{3} x^{3} + 72 \, b^{3} c d^{2} x^{2} - 8 \, {\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 8 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (8 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 72 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right )^{2} + 9 \, {\left (8 \, b^{3} c^{2} d - 5 \, b d^{3}\right )} x + 3 \, {\left (2 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{3} + 3 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - 5 \, d^{3}\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{256 \, b^{4}} \]

[In]

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

[Out]

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 = 0.63 (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} \]

[In]

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

[Out]

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*s
in(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/(3
2*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/(32*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*c
os(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/(256*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 = 0.25 (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=-\frac {256 \, c^{3} \cos \left (b x + a\right )^{4} - \frac {768 \, a c^{2} d \cos \left (b x + a\right )^{4}}{b} + \frac {768 \, a^{2} c d^{2} \cos \left (b x + a\right )^{4}}{b^{2}} - \frac {256 \, a^{3} d^{3} \cos \left (b x + a\right )^{4}}{b^{3}} + \frac {24 \, {\left (4 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) + 16 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (4 \, b x + 4 \, a\right ) - 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} c^{2} d}{b} - \frac {48 \, {\left (4 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) + 16 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (4 \, b x + 4 \, a\right ) - 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a c d^{2}}{b^{2}} + \frac {24 \, {\left (4 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) + 16 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (4 \, b x + 4 \, a\right ) - 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} d^{3}}{b^{3}} + \frac {12 \, {\left ({\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) + 16 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - 32 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c d^{2}}{b^{2}} - \frac {12 \, {\left ({\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) + 16 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - 32 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{3}}{b^{3}} + \frac {{\left (4 \, {\left (8 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \cos \left (4 \, b x + 4 \, a\right ) + 64 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right ) - 96 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{3}}{b^{3}}}{1024 \, b} \]

[In]

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

[Out]

-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
*sin(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 + 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*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)

none

Time = 0.33 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.23 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {{\left (8 \, b^{3} d^{3} x^{3} + 24 \, b^{3} c d^{2} x^{2} + 24 \, b^{3} c^{2} d x + 8 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \cos \left (4 \, b x + 4 \, a\right )}{256 \, b^{4}} - \frac {{\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{2} d x + 2 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{4}} + \frac {3 \, {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (4 \, b x + 4 \, a\right )}{1024 \, b^{4}} + \frac {3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{32 \, b^{4}} \]

[In]

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

[Out]

-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 = 25.02 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.87 \[ \int (c+d x)^3 \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {24\,d^3\,\sin \left (2\,a+2\,b\,x\right )+\frac {3\,d^3\,\sin \left (4\,a+4\,b\,x\right )}{4}+32\,b^3\,c^3\,\cos \left (2\,a+2\,b\,x\right )+8\,b^3\,c^3\,\cos \left (4\,a+4\,b\,x\right )-48\,b^2\,c^2\,d\,\sin \left (2\,a+2\,b\,x\right )-6\,b^2\,c^2\,d\,\sin \left (4\,a+4\,b\,x\right )+32\,b^3\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )+8\,b^3\,d^3\,x^3\,\cos \left (4\,a+4\,b\,x\right )-48\,b^2\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )-6\,b^2\,d^3\,x^2\,\sin \left (4\,a+4\,b\,x\right )-48\,b\,c\,d^2\,\cos \left (2\,a+2\,b\,x\right )-3\,b\,c\,d^2\,\cos \left (4\,a+4\,b\,x\right )-48\,b\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )-3\,b\,d^3\,x\,\cos \left (4\,a+4\,b\,x\right )+96\,b^3\,c^2\,d\,x\,\cos \left (2\,a+2\,b\,x\right )+24\,b^3\,c^2\,d\,x\,\cos \left (4\,a+4\,b\,x\right )-96\,b^2\,c\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )-12\,b^2\,c\,d^2\,x\,\sin \left (4\,a+4\,b\,x\right )+96\,b^3\,c\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )+24\,b^3\,c\,d^2\,x^2\,\cos \left (4\,a+4\,b\,x\right )}{256\,b^4} \]

[In]

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

[Out]

-(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*sin(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*si
n(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)