\(\int (c+d x)^3 \cos ^2(a+b x) \sin (a+b x) \, dx\) [72]

Optimal result

Integrand size = 22, antiderivative size = 151 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin (a+b x) \, dx=\frac {4 d^2 (c+d x) \cos (a+b x)}{3 b^3}+\frac {2 d^2 (c+d x) \cos ^3(a+b x)}{9 b^3}-\frac {(c+d x)^3 \cos ^3(a+b x)}{3 b}-\frac {14 d^3 \sin (a+b x)}{9 b^4}+\frac {2 d (c+d x)^2 \sin (a+b x)}{3 b^2}+\frac {d (c+d x)^2 \cos ^2(a+b x) \sin (a+b x)}{3 b^2}+\frac {2 d^3 \sin ^3(a+b x)}{27 b^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4490, 3392, 3377, 2717, 2713} \[ \int (c+d x)^3 \cos ^2(a+b x) \sin (a+b x) \, dx=\frac {2 d^3 \sin ^3(a+b x)}{27 b^4}-\frac {14 d^3 \sin (a+b x)}{9 b^4}+\frac {2 d^2 (c+d x) \cos ^3(a+b x)}{9 b^3}+\frac {4 d^2 (c+d x) \cos (a+b x)}{3 b^3}+\frac {2 d (c+d x)^2 \sin (a+b x)}{3 b^2}+\frac {d (c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b^2}-\frac {(c+d x)^3 \cos ^3(a+b x)}{3 b} \]

[In]

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

[Out]

(4*d^2*(c + d*x)*Cos[a + b*x])/(3*b^3) + (2*d^2*(c + d*x)*Cos[a + b*x]^3)/(9*b^3) - ((c + d*x)^3*Cos[a + b*x]^
3)/(3*b) - (14*d^3*Sin[a + b*x])/(9*b^4) + (2*d*(c + d*x)^2*Sin[a + b*x])/(3*b^2) + (d*(c + d*x)^2*Cos[a + b*x
]^2*Sin[a + b*x])/(3*b^2) + (2*d^3*Sin[a + b*x]^3)/(27*b^4)

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2717

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

Rule 3377

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

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 ^3(a+b x)}{3 b}+\frac {d \int (c+d x)^2 \cos ^3(a+b x) \, dx}{b} \\ & = \frac {2 d^2 (c+d x) \cos ^3(a+b x)}{9 b^3}-\frac {(c+d x)^3 \cos ^3(a+b x)}{3 b}+\frac {d (c+d x)^2 \cos ^2(a+b x) \sin (a+b x)}{3 b^2}+\frac {(2 d) \int (c+d x)^2 \cos (a+b x) \, dx}{3 b}-\frac {\left (2 d^3\right ) \int \cos ^3(a+b x) \, dx}{9 b^3} \\ & = \frac {2 d^2 (c+d x) \cos ^3(a+b x)}{9 b^3}-\frac {(c+d x)^3 \cos ^3(a+b x)}{3 b}+\frac {2 d (c+d x)^2 \sin (a+b x)}{3 b^2}+\frac {d (c+d x)^2 \cos ^2(a+b x) \sin (a+b x)}{3 b^2}-\frac {\left (4 d^2\right ) \int (c+d x) \sin (a+b x) \, dx}{3 b^2}+\frac {\left (2 d^3\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (a+b x)\right )}{9 b^4} \\ & = \frac {4 d^2 (c+d x) \cos (a+b x)}{3 b^3}+\frac {2 d^2 (c+d x) \cos ^3(a+b x)}{9 b^3}-\frac {(c+d x)^3 \cos ^3(a+b x)}{3 b}-\frac {2 d^3 \sin (a+b x)}{9 b^4}+\frac {2 d (c+d x)^2 \sin (a+b x)}{3 b^2}+\frac {d (c+d x)^2 \cos ^2(a+b x) \sin (a+b x)}{3 b^2}+\frac {2 d^3 \sin ^3(a+b x)}{27 b^4}-\frac {\left (4 d^3\right ) \int \cos (a+b x) \, dx}{3 b^3} \\ & = \frac {4 d^2 (c+d x) \cos (a+b x)}{3 b^3}+\frac {2 d^2 (c+d x) \cos ^3(a+b x)}{9 b^3}-\frac {(c+d x)^3 \cos ^3(a+b x)}{3 b}-\frac {14 d^3 \sin (a+b x)}{9 b^4}+\frac {2 d (c+d x)^2 \sin (a+b x)}{3 b^2}+\frac {d (c+d x)^2 \cos ^2(a+b x) \sin (a+b x)}{3 b^2}+\frac {2 d^3 \sin ^3(a+b x)}{27 b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.84 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin (a+b x) \, dx=\frac {-27 b (c+d x) \left (-6 d^2+b^2 (c+d x)^2\right ) \cos (a+b x)-3 b (c+d x) \left (-2 d^2+3 b^2 (c+d x)^2\right ) \cos (3 (a+b x))+2 d \left (-82 d^2+45 b^2 (c+d x)^2+\left (-2 d^2+9 b^2 (c+d x)^2\right ) \cos (2 (a+b x))\right ) \sin (a+b x)}{108 b^4} \]

[In]

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

[Out]

(-27*b*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] - 3*b*(c + d*x)*(-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[3*(a
+ b*x)] + 2*d*(-82*d^2 + 45*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])/(1
08*b^4)

Maple [A] (verified)

Time = 1.44 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.48

method	result	size
risch	\(-\frac {\left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \cos \left (x b +a \right )}{4 b^{3}}+\frac {3 d \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 d^{2}\right ) \sin \left (x b +a \right )}{4 b^{4}}-\frac {\left (3 b^{2} d^{3} x^{3}+9 b^{2} c \,d^{2} x^{2}+9 b^{2} c^{2} d x +3 b^{2} c^{3}-2 d^{3} x -2 c \,d^{2}\right ) \cos \left (3 x b +3 a \right )}{36 b^{3}}+\frac {d \left (9 x^{2} d^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}-2 d^{2}\right ) \sin \left (3 x b +3 a \right )}{108 b^{4}}\)	\(224\)
parallelrisch	\(\frac {3 b x \left (\left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) b^{2}-\frac {14 d^{2}}{9}\right ) d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}+6 d \left (\left (d x +c \right )^{2} b^{2}-\frac {14 d^{2}}{9}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}-6 b \left (\left (x^{2} d^{2}+c d x +c^{2}\right ) \left (\frac {d x}{2}+c \right ) b^{2}+\frac {d^{3} x}{3}-2 c \,d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}+4 \left (\left (d x +c \right )^{2} b^{2}-\frac {38 d^{2}}{9}\right ) d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+9 b \left (x \left (\frac {1}{3} x^{2} d^{2}+c d x +c^{2}\right ) b^{2}+\frac {16 d \left (\frac {d x}{8}+c \right )}{9}\right ) d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+6 d \left (\left (d x +c \right )^{2} b^{2}-\frac {14 d^{2}}{9}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )-2 b \left (\left (x^{2} d^{2}+c d x +c^{2}\right ) b^{2}-\frac {14 d^{2}}{3}\right ) \left (\frac {d x}{2}+c \right )}{3 b^{4} \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{3}}\)	\(287\)
derivativedivides	\(\frac {\frac {a^{3} d^{3} \cos \left (x b +a \right )^{3}}{3 b^{3}}-\frac {a^{2} c \,d^{2} \cos \left (x b +a \right )^{3}}{b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\cos \left (x b +a \right )^{3} \left (x b +a \right )}{3}+\frac {\left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{9}\right )}{b^{3}}+\frac {a \,c^{2} d \cos \left (x b +a \right )^{3}}{b}-\frac {6 a c \,d^{2} \left (-\frac {\cos \left (x b +a \right )^{3} \left (x b +a \right )}{3}+\frac {\left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{9}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (-\frac {\left (x b +a \right )^{2} \cos \left (x b +a \right )^{3}}{3}+\frac {2 \left (x b +a \right ) \left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{9}+\frac {2 \cos \left (x b +a \right )^{3}}{27}+\frac {4 \cos \left (x b +a \right )}{9}\right )}{b^{3}}-\frac {c^{3} \cos \left (x b +a \right )^{3}}{3}+\frac {3 c^{2} d \left (-\frac {\cos \left (x b +a \right )^{3} \left (x b +a \right )}{3}+\frac {\left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{9}\right )}{b}+\frac {3 c \,d^{2} \left (-\frac {\left (x b +a \right )^{2} \cos \left (x b +a \right )^{3}}{3}+\frac {2 \left (x b +a \right ) \left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{9}+\frac {2 \cos \left (x b +a \right )^{3}}{27}+\frac {4 \cos \left (x b +a \right )}{9}\right )}{b^{2}}+\frac {d^{3} \left (-\frac {\left (x b +a \right )^{3} \cos \left (x b +a \right )^{3}}{3}+\frac {\left (x b +a \right )^{2} \left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{3}-\frac {4 \sin \left (x b +a \right )}{3}+\frac {4 \left (x b +a \right ) \cos \left (x b +a \right )}{3}+\frac {2 \cos \left (x b +a \right )^{3} \left (x b +a \right )}{9}-\frac {2 \left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{27}\right )}{b^{3}}}{b}\)	\(447\)
default	\(\frac {\frac {a^{3} d^{3} \cos \left (x b +a \right )^{3}}{3 b^{3}}-\frac {a^{2} c \,d^{2} \cos \left (x b +a \right )^{3}}{b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\cos \left (x b +a \right )^{3} \left (x b +a \right )}{3}+\frac {\left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{9}\right )}{b^{3}}+\frac {a \,c^{2} d \cos \left (x b +a \right )^{3}}{b}-\frac {6 a c \,d^{2} \left (-\frac {\cos \left (x b +a \right )^{3} \left (x b +a \right )}{3}+\frac {\left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{9}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (-\frac {\left (x b +a \right )^{2} \cos \left (x b +a \right )^{3}}{3}+\frac {2 \left (x b +a \right ) \left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{9}+\frac {2 \cos \left (x b +a \right )^{3}}{27}+\frac {4 \cos \left (x b +a \right )}{9}\right )}{b^{3}}-\frac {c^{3} \cos \left (x b +a \right )^{3}}{3}+\frac {3 c^{2} d \left (-\frac {\cos \left (x b +a \right )^{3} \left (x b +a \right )}{3}+\frac {\left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{9}\right )}{b}+\frac {3 c \,d^{2} \left (-\frac {\left (x b +a \right )^{2} \cos \left (x b +a \right )^{3}}{3}+\frac {2 \left (x b +a \right ) \left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{9}+\frac {2 \cos \left (x b +a \right )^{3}}{27}+\frac {4 \cos \left (x b +a \right )}{9}\right )}{b^{2}}+\frac {d^{3} \left (-\frac {\left (x b +a \right )^{3} \cos \left (x b +a \right )^{3}}{3}+\frac {\left (x b +a \right )^{2} \left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{3}-\frac {4 \sin \left (x b +a \right )}{3}+\frac {4 \left (x b +a \right ) \cos \left (x b +a \right )}{3}+\frac {2 \cos \left (x b +a \right )^{3} \left (x b +a \right )}{9}-\frac {2 \left (2+\cos \left (x b +a \right )^{2}\right ) \sin \left (x b +a \right )}{27}\right )}{b^{3}}}{b}\)	\(447\)
norman	\(\frac {\frac {\left (-2 b^{2} c^{3}+4 c \,d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{b^{3}}+\frac {d^{3} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{b}+\frac {c \,d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{b}+\frac {-6 b^{2} c^{3}+28 c \,d^{2}}{9 b^{3}}-\frac {d^{3} x^{3}}{3 b}+\frac {16 c \,d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{3 b^{3}}+\frac {2 d \left (9 b^{2} c^{2}-14 d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{9 b^{4}}+\frac {2 d \left (9 b^{2} c^{2}-14 d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{9 b^{4}}-\frac {c \,d^{2} x^{2}}{b}+\frac {4 d \left (9 b^{2} c^{2}-38 d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{27 b^{4}}-\frac {d \left (9 b^{2} c^{2}-14 d^{2}\right ) x}{9 b^{3}}-\frac {d^{3} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{b}+\frac {d^{3} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{3 b}+\frac {2 d^{3} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{b^{2}}+\frac {4 d^{3} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{3 b^{2}}+\frac {2 d^{3} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{b^{2}}+\frac {3 c \,d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{b}-\frac {3 c \,d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{b}+\frac {4 c \,d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{b^{2}}+\frac {8 c \,d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{3 b^{2}}+\frac {4 c \,d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{b^{2}}+\frac {d \left (9 b^{2} c^{2}-14 d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{9 b^{3}}+\frac {d \left (9 b^{2} c^{2}+2 d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{3 b^{3}}-\frac {d \left (9 b^{2} c^{2}+2 d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{3 b^{3}}}{\left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{3}}\)	\(577\)

[In]

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

[Out]

-1/4/b^3*(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)*cos(b*x+a)+3/4*d*(b^2*d^2*x^2+2*b
^2*c*d*x+b^2*c^2-2*d^2)/b^4*sin(b*x+a)-1/36/b^3*(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)*cos(3*b*x+3*a)+1/108*d*(9*b^2*d^2*x^2+18*b^2*c*d*x+9*b^2*c^2-2*d^2)/b^4*sin(3*b*x+3*a)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.21 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin (a+b x) \, dx=-\frac {3 \, {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} - 2 \, b c d^{2} + {\left (9 \, b^{3} c^{2} d - 2 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} - 36 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right ) - {\left (18 \, b^{2} d^{3} x^{2} + 36 \, b^{2} c d^{2} x + 18 \, b^{2} c^{2} d - 40 \, d^{3} + {\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )}{27 \, b^{4}} \]

[In]

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

[Out]

-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
- 36*(b*d^3*x + b*c*d^2)*cos(b*x + a) - (18*b^2*d^3*x^2 + 36*b^2*c*d^2*x + 18*b^2*c^2*d - 40*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. 391 vs. \(2 (150) = 300\).

Time = 0.45 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.59 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin (a+b x) \, dx=\begin {cases} - \frac {c^{3} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {c^{2} d x \cos ^{3}{\left (a + b x \right )}}{b} - \frac {c d^{2} x^{2} \cos ^{3}{\left (a + b x \right )}}{b} - \frac {d^{3} x^{3} \cos ^{3}{\left (a + b x \right )}}{3 b} + \frac {2 c^{2} d \sin ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {c^{2} d \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b^{2}} + \frac {4 c d^{2} x \sin ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 c d^{2} x \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b^{2}} + \frac {2 d^{3} x^{2} \sin ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {d^{3} x^{2} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b^{2}} + \frac {4 c d^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{3}} + \frac {14 c d^{2} \cos ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {4 d^{3} x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{3}} + \frac {14 d^{3} x \cos ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {40 d^{3} \sin ^{3}{\left (a + b x \right )}}{27 b^{4}} - \frac {14 d^{3} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{9 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sin {\left (a \right )} \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (137) = 274\).

Time = 0.27 (sec) , antiderivative size = 505, normalized size of antiderivative = 3.34 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin (a+b x) \, dx=-\frac {36 \, c^{3} \cos \left (b x + a\right )^{3} - \frac {108 \, a c^{2} d \cos \left (b x + a\right )^{3}}{b} + \frac {108 \, a^{2} c d^{2} \cos \left (b x + a\right )^{3}}{b^{2}} - \frac {36 \, a^{3} d^{3} \cos \left (b x + a\right )^{3}}{b^{3}} + \frac {9 \, {\left (3 \, {\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) + 9 \, {\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (3 \, b x + 3 \, a\right ) - 9 \, \sin \left (b x + a\right )\right )} c^{2} d}{b} - \frac {18 \, {\left (3 \, {\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) + 9 \, {\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (3 \, b x + 3 \, a\right ) - 9 \, \sin \left (b x + a\right )\right )} a c d^{2}}{b^{2}} + \frac {9 \, {\left (3 \, {\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) + 9 \, {\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (3 \, b x + 3 \, a\right ) - 9 \, \sin \left (b x + a\right )\right )} a^{2} d^{3}}{b^{3}} + \frac {3 \, {\left ({\left (9 \, {\left (b x + a\right )}^{2} - 2\right )} \cos \left (3 \, b x + 3 \, a\right ) + 27 \, {\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) - 6 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 54 \, {\left (b x + a\right )} \sin \left (b x + a\right )\right )} c d^{2}}{b^{2}} - \frac {3 \, {\left ({\left (9 \, {\left (b x + a\right )}^{2} - 2\right )} \cos \left (3 \, b x + 3 \, a\right ) + 27 \, {\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) - 6 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 54 \, {\left (b x + a\right )} \sin \left (b x + a\right )\right )} a d^{3}}{b^{3}} + \frac {{\left (3 \, {\left (3 \, {\left (b x + a\right )}^{3} - 2 \, b x - 2 \, a\right )} \cos \left (3 \, b x + 3 \, a\right ) + 27 \, {\left ({\left (b x + a\right )}^{3} - 6 \, b x - 6 \, a\right )} \cos \left (b x + a\right ) - {\left (9 \, {\left (b x + a\right )}^{2} - 2\right )} \sin \left (3 \, b x + 3 \, a\right ) - 81 \, {\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} d^{3}}{b^{3}}}{108 \, b} \]

[In]

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

[Out]

-1/108*(36*c^3*cos(b*x + a)^3 - 108*a*c^2*d*cos(b*x + a)^3/b + 108*a^2*c*d^2*cos(b*x + a)^3/b^2 - 36*a^3*d^3*c
os(b*x + a)^3/b^3 + 9*(3*(b*x + a)*cos(3*b*x + 3*a) + 9*(b*x + a)*cos(b*x + a) - sin(3*b*x + 3*a) - 9*sin(b*x
+ a))*c^2*d/b - 18*(3*(b*x + a)*cos(3*b*x + 3*a) + 9*(b*x + a)*cos(b*x + a) - sin(3*b*x + 3*a) - 9*sin(b*x + a
))*a*c*d^2/b^2 + 9*(3*(b*x + a)*cos(3*b*x + 3*a) + 9*(b*x + a)*cos(b*x + a) - sin(3*b*x + 3*a) - 9*sin(b*x + a
))*a^2*d^3/b^3 + 3*((9*(b*x + a)^2 - 2)*cos(3*b*x + 3*a) + 27*((b*x + a)^2 - 2)*cos(b*x + a) - 6*(b*x + a)*sin
(3*b*x + 3*a) - 54*(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) + 27*((b*x + a)
^2 - 2)*cos(b*x + a) - 6*(b*x + a)*sin(3*b*x + 3*a) - 54*(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) + 27*((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) - 81*((b*x + a)^2 - 2)*sin(b*x + a))*d^3/b^3)/b

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.53 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin (a+b x) \, dx=-\frac {{\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 9 \, b^{3} c^{2} d x + 3 \, b^{3} c^{3} - 2 \, b d^{3} x - 2 \, b c d^{2}\right )} \cos \left (3 \, b x + 3 \, a\right )}{36 \, b^{4}} - \frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \cos \left (b x + a\right )}{4 \, b^{4}} + \frac {{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (3 \, b x + 3 \, a\right )}{108 \, b^{4}} + \frac {3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (b x + a\right )}{4 \, b^{4}} \]

[In]

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

[Out]

-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 - 1/4*(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 +
1/108*(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 + 3/4*(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 = 24.41 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.92 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin (a+b x) \, dx=\frac {{\cos \left (a+b\,x\right )}^3\,\left (14\,c\,d^2-3\,b^2\,c^3\right )}{9\,b^3}-\frac {2\,{\sin \left (a+b\,x\right )}^3\,\left (20\,d^3-9\,b^2\,c^2\,d\right )}{27\,b^4}-\frac {{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (14\,d^3-9\,b^2\,c^2\,d\right )}{9\,b^4}+\frac {x\,{\cos \left (a+b\,x\right )}^3\,\left (14\,d^3-9\,b^2\,c^2\,d\right )}{9\,b^3}-\frac {d^3\,x^3\,{\cos \left (a+b\,x\right )}^3}{3\,b}+\frac {2\,d^3\,x^2\,{\sin \left (a+b\,x\right )}^3}{3\,b^2}+\frac {4\,c\,d^2\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{3\,b^3}+\frac {4\,d^3\,x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{3\,b^3}+\frac {4\,c\,d^2\,x\,{\sin \left (a+b\,x\right )}^3}{3\,b^2}-\frac {c\,d^2\,x^2\,{\cos \left (a+b\,x\right )}^3}{b}+\frac {d^3\,x^2\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{b^2}+\frac {2\,c\,d^2\,x\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{b^2} \]

[In]

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

[Out]

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