\(\int (c+d x)^3 \cos (a+b x) \sin ^2(a+b x) \, dx\) [15]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.16 (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 = {4489, 3392, 3377, 2718, 2713} \[ \int (c+d x)^3 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {2 d^3 \cos ^3(a+b x)}{27 b^4}-\frac {14 d^3 \cos (a+b x)}{9 b^4}-\frac {2 d^2 (c+d x) \sin ^3(a+b x)}{9 b^3}-\frac {4 d^2 (c+d x) \sin (a+b x)}{3 b^3}+\frac {2 d (c+d x)^2 \cos (a+b x)}{3 b^2}+\frac {d (c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b^2}+\frac {(c+d x)^3 \sin ^3(a+b x)}{3 b} \]

[In]

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

[Out]

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

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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 4489

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[(c + d
*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))), x] - Dist[d*(m/(b*(n + 1))), Int[(c + d*x)^(m - 1)*Sin[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 \sin ^3(a+b x)}{3 b}-\frac {d \int (c+d x)^2 \sin ^3(a+b x) \, dx}{b} \\ & = \frac {d (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{3 b^2}-\frac {2 d^2 (c+d x) \sin ^3(a+b x)}{9 b^3}+\frac {(c+d x)^3 \sin ^3(a+b x)}{3 b}-\frac {(2 d) \int (c+d x)^2 \sin (a+b x) \, dx}{3 b}+\frac {\left (2 d^3\right ) \int \sin ^3(a+b x) \, dx}{9 b^3} \\ & = \frac {2 d (c+d x)^2 \cos (a+b x)}{3 b^2}+\frac {d (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{3 b^2}-\frac {2 d^2 (c+d x) \sin ^3(a+b x)}{9 b^3}+\frac {(c+d x)^3 \sin ^3(a+b x)}{3 b}-\frac {\left (4 d^2\right ) \int (c+d x) \cos (a+b x) \, dx}{3 b^2}-\frac {\left (2 d^3\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{9 b^4} \\ & = -\frac {2 d^3 \cos (a+b x)}{9 b^4}+\frac {2 d (c+d x)^2 \cos (a+b x)}{3 b^2}+\frac {2 d^3 \cos ^3(a+b x)}{27 b^4}-\frac {4 d^2 (c+d x) \sin (a+b x)}{3 b^3}+\frac {d (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{3 b^2}-\frac {2 d^2 (c+d x) \sin ^3(a+b x)}{9 b^3}+\frac {(c+d x)^3 \sin ^3(a+b x)}{3 b}+\frac {\left (4 d^3\right ) \int \sin (a+b x) \, dx}{3 b^3} \\ & = -\frac {14 d^3 \cos (a+b x)}{9 b^4}+\frac {2 d (c+d x)^2 \cos (a+b x)}{3 b^2}+\frac {2 d^3 \cos ^3(a+b x)}{27 b^4}-\frac {4 d^2 (c+d x) \sin (a+b x)}{3 b^3}+\frac {d (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{3 b^2}-\frac {2 d^2 (c+d x) \sin ^3(a+b x)}{9 b^3}+\frac {(c+d x)^3 \sin ^3(a+b x)}{3 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.80 \[ \int (c+d x)^3 \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {-81 d \left (-2 d^2+b^2 (c+d x)^2\right ) \cos (a+b x)+d \left (-2 d^2+9 b^2 (c+d x)^2\right ) \cos (3 (a+b x))+6 b (c+d x) \left (26 d^2-3 b^2 (c+d x)^2+\left (-2 d^2+3 b^2 (c+d x)^2\right ) \cos (2 (a+b x))\right ) \sin (a+b x)}{108 b^4} \]

[In]

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

[Out]

-1/108*(-81*d*(-2*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + d*(-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] + 6*b*
(c + d*x)*(26*d^2 - 3*b^2*(c + d*x)^2 + (-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*Sin[a + b*x])/b^4

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.48

method	result	size
risch	\(\frac {3 d \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 d^{2}\right ) \cos \left (x b +a \right )}{4 b^{4}}+\frac {\left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \sin \left (x b +a \right )}{4 b^{3}}-\frac {d \left (9 x^{2} d^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}-2 d^{2}\right ) \cos \left (3 x b +3 a \right )}{108 b^{4}}-\frac {\left (3 b^{2} d^{3} x^{3}+9 b^{2} c \,d^{2} x^{2}+9 b^{2} c^{2} d x +3 b^{2} c^{3}-2 d^{3} x -2 c \,d^{2}\right ) \sin \left (3 x b +3 a \right )}{36 b^{3}}\)	\(224\)
parallelrisch	\(\frac {-36 b^{2} x \,d^{2} \left (\frac {d x}{2}+c \right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}-72 b \,d^{2} \left (d x +c \right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}+\left (\left (-54 x^{2} b^{2}-72\right ) d^{3}-108 b^{2} c \,d^{2} x \right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}+72 b \left (d x +c \right ) \left (\left (x^{2} b^{2}-\frac {8}{3}\right ) d^{2}+2 b^{2} c d x +b^{2} c^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+\left (\left (54 x^{2} b^{2}-168\right ) d^{3}+108 b^{2} c \,d^{2} x +108 b^{2} c^{2} d \right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-72 b \,d^{2} \left (d x +c \right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+\left (18 x^{2} b^{2}-80\right ) d^{3}+36 b^{2} c \,d^{2} x +36 b^{2} c^{2} d}{27 b^{4} \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{3}}\)	\(254\)
norman	\(\frac {\frac {36 b^{2} c^{2} d -80 d^{3}}{27 b^{4}}+\frac {2 d^{3} x^{2}}{3 b^{2}}-\frac {8 d^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{3 b^{4}}+\frac {\left (36 b^{2} c^{2} d -56 d^{3}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{9 b^{4}}-\frac {8 c \,d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{3 b^{3}}-\frac {8 c \,d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{3 b^{3}}+\frac {4 c \,d^{2} x}{3 b^{2}}+\frac {8 c \left (3 b^{2} c^{2}-8 d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{9 b^{3}}+\frac {8 d^{3} x^{3} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{3 b}-\frac {8 d^{3} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{3 b^{3}}-\frac {8 d^{3} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{3 b^{3}}+\frac {2 d^{3} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{b^{2}}-\frac {2 d^{3} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{b^{2}}-\frac {2 d^{3} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{3 b^{2}}+\frac {8 c \,d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{b}+\frac {4 c \,d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{b^{2}}-\frac {4 c \,d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{b^{2}}-\frac {4 c \,d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{3 b^{2}}+\frac {8 d \left (9 b^{2} c^{2}-8 d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{9 b^{3}}}{\left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{3}}\)	\(422\)
derivativedivides	\(\frac {-\frac {a^{3} d^{3} \sin \left (x b +a \right )^{3}}{3 b^{3}}+\frac {a^{2} c \,d^{2} \sin \left (x b +a \right )^{3}}{b^{2}}+\frac {3 a^{2} d^{3} \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b^{3}}-\frac {a \,c^{2} d \sin \left (x b +a \right )^{3}}{b}-\frac {6 a c \,d^{2} \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (\frac {\left (x b +a \right )^{2} \sin \left (x b +a \right )^{3}}{3}+\frac {2 \left (x b +a \right ) \left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}-\frac {2 \sin \left (x b +a \right )^{3}}{27}-\frac {4 \sin \left (x b +a \right )}{9}\right )}{b^{3}}+\frac {c^{3} \sin \left (x b +a \right )^{3}}{3}+\frac {3 c^{2} d \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b}+\frac {3 c \,d^{2} \left (\frac {\left (x b +a \right )^{2} \sin \left (x b +a \right )^{3}}{3}+\frac {2 \left (x b +a \right ) \left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}-\frac {2 \sin \left (x b +a \right )^{3}}{27}-\frac {4 \sin \left (x b +a \right )}{9}\right )}{b^{2}}+\frac {d^{3} \left (\frac {\left (x b +a \right )^{3} \sin \left (x b +a \right )^{3}}{3}+\frac {\left (x b +a \right )^{2} \left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{3}-\frac {4 \cos \left (x b +a \right )}{3}-\frac {4 \left (x b +a \right ) \sin \left (x b +a \right )}{3}-\frac {2 \left (x b +a \right ) \sin \left (x b +a \right )^{3}}{9}-\frac {2 \left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{27}\right )}{b^{3}}}{b}\)	\(447\)
default	\(\frac {-\frac {a^{3} d^{3} \sin \left (x b +a \right )^{3}}{3 b^{3}}+\frac {a^{2} c \,d^{2} \sin \left (x b +a \right )^{3}}{b^{2}}+\frac {3 a^{2} d^{3} \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b^{3}}-\frac {a \,c^{2} d \sin \left (x b +a \right )^{3}}{b}-\frac {6 a c \,d^{2} \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (\frac {\left (x b +a \right )^{2} \sin \left (x b +a \right )^{3}}{3}+\frac {2 \left (x b +a \right ) \left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}-\frac {2 \sin \left (x b +a \right )^{3}}{27}-\frac {4 \sin \left (x b +a \right )}{9}\right )}{b^{3}}+\frac {c^{3} \sin \left (x b +a \right )^{3}}{3}+\frac {3 c^{2} d \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b}+\frac {3 c \,d^{2} \left (\frac {\left (x b +a \right )^{2} \sin \left (x b +a \right )^{3}}{3}+\frac {2 \left (x b +a \right ) \left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}-\frac {2 \sin \left (x b +a \right )^{3}}{27}-\frac {4 \sin \left (x b +a \right )}{9}\right )}{b^{2}}+\frac {d^{3} \left (\frac {\left (x b +a \right )^{3} \sin \left (x b +a \right )^{3}}{3}+\frac {\left (x b +a \right )^{2} \left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{3}-\frac {4 \cos \left (x b +a \right )}{3}-\frac {4 \left (x b +a \right ) \sin \left (x b +a \right )}{3}-\frac {2 \left (x b +a \right ) \sin \left (x b +a \right )^{3}}{9}-\frac {2 \left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{27}\right )}{b^{3}}}{b}\)	\(447\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.50 \[ \int (c+d x)^3 \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 14 \, d^{3}\right )} \cos \left (b x + a\right ) - 3 \, {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} - 14 \, b c d^{2} - {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} - 2 \, b c d^{2} + {\left (9 \, b^{3} c^{2} d - 2 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + {\left (9 \, b^{3} c^{2} d - 14 \, b d^{3}\right )} x\right )} \sin \left (b x + a\right )}{27 \, b^{4}} \]

[In]

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

[Out]

-1/27*((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)^3 - 3*(9*b^2*d^3*x^2 + 18*b^2*c*d^2
*x + 9*b^2*c^2*d - 14*d^3)*cos(b*x + a) - 3*(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 - (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)^2 + (9*b^3*c^2*d
- 14*b*d^3)*x)*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 = 0.46 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.59 \[ \int (c+d x)^3 \cos (a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {c^{3} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {c^{2} d x \sin ^{3}{\left (a + b x \right )}}{b} + \frac {c d^{2} x^{2} \sin ^{3}{\left (a + b x \right )}}{b} + \frac {d^{3} x^{3} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {c^{2} d \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b^{2}} + \frac {2 c^{2} d \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 c d^{2} x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b^{2}} + \frac {4 c d^{2} x \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {d^{3} x^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b^{2}} + \frac {2 d^{3} x^{2} \cos ^{3}{\left (a + b x \right )}}{3 b^{2}} - \frac {14 c d^{2} \sin ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {4 c d^{2} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {14 d^{3} x \sin ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {4 d^{3} x \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {14 d^{3} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{9 b^{4}} - \frac {40 d^{3} \cos ^{3}{\left (a + b x \right )}}{27 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sin ^{2}{\left (a \right )} \cos {\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (137) = 274\).

Time = 0.24 (sec) , antiderivative size = 499, normalized size of antiderivative = 3.30 \[ \int (c+d x)^3 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {36 \, c^{3} \sin \left (b x + a\right )^{3} - \frac {108 \, a c^{2} d \sin \left (b x + a\right )^{3}}{b} + \frac {108 \, a^{2} c d^{2} \sin \left (b x + a\right )^{3}}{b^{2}} - \frac {36 \, a^{3} d^{3} \sin \left (b x + a\right )^{3}}{b^{3}} - \frac {9 \, {\left (3 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 9 \, {\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) - 9 \, \cos \left (b x + a\right )\right )} c^{2} d}{b} + \frac {18 \, {\left (3 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 9 \, {\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) - 9 \, \cos \left (b x + a\right )\right )} a c d^{2}}{b^{2}} - \frac {9 \, {\left (3 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 9 \, {\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) - 9 \, \cos \left (b x + a\right )\right )} a^{2} d^{3}}{b^{3}} - \frac {3 \, {\left (6 \, {\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) - 54 \, {\left (b x + 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 ) - 27 \, {\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} c d^{2}}{b^{2}} + \frac {3 \, {\left (6 \, {\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) - 54 \, {\left (b x + 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 ) - 27 \, {\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} a d^{3}}{b^{3}} - \frac {{\left ({\left (9 \, {\left (b x + a\right )}^{2} - 2\right )} \cos \left (3 \, b x + 3 \, a\right ) - 81 \, {\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) + 3 \, {\left (3 \, {\left (b x + a\right )}^{3} - 2 \, b x - 2 \, a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 27 \, {\left ({\left (b x + a\right )}^{3} - 6 \, b x - 6 \, a\right )} \sin \left (b x + a\right )\right )} d^{3}}{b^{3}}}{108 \, b} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.53 \[ \int (c+d x)^3 \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (3 \, b x + 3 \, a\right )}{108 \, b^{4}} + \frac {3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )}{4 \, b^{4}} - \frac {{\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 9 \, b^{3} c^{2} d x + 3 \, b^{3} c^{3} - 2 \, b d^{3} x - 2 \, b c d^{2}\right )} \sin \left (3 \, b x + 3 \, a\right )}{36 \, b^{4}} + \frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \sin \left (b x + a\right )}{4 \, b^{4}} \]

[In]

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

[Out]

-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)*cos(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)*cos(b*x + a)/b^4 - 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)*sin(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)*sin(b*x + a)/b^4

Mupad [B] (verification not implemented)

Time = 23.32 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.91 \[ \int (c+d x)^3 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {2\,d^3\,x^2\,{\cos \left (a+b\,x\right )}^3}{3\,b^2}-\frac {{\sin \left (a+b\,x\right )}^3\,\left (14\,c\,d^2-3\,b^2\,c^3\right )}{9\,b^3}-\frac {\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (14\,d^3-9\,b^2\,c^2\,d\right )}{9\,b^4}-\frac {x\,{\sin \left (a+b\,x\right )}^3\,\left (14\,d^3-9\,b^2\,c^2\,d\right )}{9\,b^3}-\frac {2\,{\cos \left (a+b\,x\right )}^3\,\left (20\,d^3-9\,b^2\,c^2\,d\right )}{27\,b^4}+\frac {d^3\,x^3\,{\sin \left (a+b\,x\right )}^3}{3\,b}-\frac {4\,c\,d^2\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{3\,b^3}+\frac {4\,c\,d^2\,x\,{\cos \left (a+b\,x\right )}^3}{3\,b^2}-\frac {4\,d^3\,x\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{3\,b^3}+\frac {d^3\,x^2\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{b^2}+\frac {c\,d^2\,x^2\,{\sin \left (a+b\,x\right )}^3}{b}+\frac {2\,c\,d^2\,x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{b^2} \]

[In]

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

[Out]

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