Integrand size = 16, antiderivative size = 123 \[ \int (c+d x)^2 \sin ^3(a+b x) \, dx=\frac {14 d^2 \cos (a+b x)}{9 b^3}-\frac {2 (c+d x)^2 \cos (a+b x)}{3 b}-\frac {2 d^2 \cos ^3(a+b x)}{27 b^3}+\frac {4 d (c+d x) \sin (a+b x)}{3 b^2}-\frac {(c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2} \] Output:
14/9*d^2*cos(b*x+a)/b^3-2/3*(d*x+c)^2*cos(b*x+a)/b-2/27*d^2*cos(b*x+a)^3/b ^3+4/3*d*(d*x+c)*sin(b*x+a)/b^2-1/3*(d*x+c)^2*cos(b*x+a)*sin(b*x+a)^2/b+2/ 9*d*(d*x+c)*sin(b*x+a)^3/b^2
Time = 0.61 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.70 \[ \int (c+d x)^2 \sin ^3(a+b x) \, dx=\frac {-81 \left (-2 d^2+b^2 (c+d x)^2\right ) \cos (a+b x)+\left (-2 d^2+9 b^2 (c+d x)^2\right ) \cos (3 (a+b x))-6 b d (c+d x) (-27 \sin (a+b x)+\sin (3 (a+b x)))}{108 b^3} \] Input:
Integrate[(c + d*x)^2*Sin[a + b*x]^3,x]
Output:
(-81*(-2*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + (-2*d^2 + 9*b^2*(c + d*x)^2 )*Cos[3*(a + b*x)] - 6*b*d*(c + d*x)*(-27*Sin[a + b*x] + Sin[3*(a + b*x)]) )/(108*b^3)
Time = 0.62 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {3042, 3792, 3042, 3113, 2009, 3777, 3042, 3777, 25, 3042, 3118}
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.
| \(\displaystyle \int (c+d x)^2 \sin ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^2 \sin (a+b x)^3dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -\frac {2 d^2 \int \sin ^3(a+b x)dx}{9 b^2}+\frac {2}{3} \int (c+d x)^2 \sin (a+b x)dx+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 d^2 \int \sin (a+b x)^3dx}{9 b^2}+\frac {2}{3} \int (c+d x)^2 \sin (a+b x)dx+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {2 d^2 \int \left (1-\cos ^2(a+b x)\right )d\cos (a+b x)}{9 b^3}+\frac {2}{3} \int (c+d x)^2 \sin (a+b x)dx+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} \int (c+d x)^2 \sin (a+b x)dx+\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {2}{3} \left (\frac {2 d \int (c+d x) \cos (a+b x)dx}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )+\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \left (\frac {2 d \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )+\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {2}{3} \left (\frac {2 d \left (\frac {d \int -\sin (a+b x)dx}{b}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )+\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{3} \left (\frac {2 d \left (\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )+\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{3} \left (\frac {2 d \left (\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )+\frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {2 d^2 \left (\cos (a+b x)-\frac {1}{3} \cos ^3(a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )-\frac {(c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
Input:
Int[(c + d*x)^2*Sin[a + b*x]^3,x]
Output:
(2*d^2*(Cos[a + b*x] - Cos[a + b*x]^3/3))/(9*b^3) - ((c + d*x)^2*Cos[a + b *x]*Sin[a + b*x]^2)/(3*b) + (2*d*(c + d*x)*Sin[a + b*x]^3)/(9*b^2) + (2*(- (((c + d*x)^2*Cos[a + b*x])/b) + (2*d*((d*Cos[a + b*x])/b^2 + ((c + d*x)*S in[a + b*x])/b))/b))/3
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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]
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]
Time = 2.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.86
| method | result | size |
| parallelrisch | \(\frac {\left (9 \left (d x +c \right )^{2} b^{2}-2 d^{2}\right ) \cos \left (3 b x +3 a \right )-6 b d \left (d x +c \right ) \sin \left (3 b x +3 a \right )+\left (-81 \left (d x +c \right )^{2} b^{2}+162 d^{2}\right ) \cos \left (b x +a \right )+162 b d \left (d x +c \right ) \sin \left (b x +a \right )-72 b^{2} c^{2}+160 d^{2}}{108 b^{3}}\) | \(106\) |
| risch | \(-\frac {3 \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 d^{2}\right ) \cos \left (b x +a \right )}{4 b^{3}}+\frac {3 d \left (d x +c \right ) \sin \left (b x +a \right )}{2 b^{2}}+\frac {\left (9 x^{2} d^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}-2 d^{2}\right ) \cos \left (3 b x +3 a \right )}{108 b^{3}}-\frac {d \left (d x +c \right ) \sin \left (3 b x +3 a \right )}{18 b^{2}}\) | \(128\) |
| derivativedivides | \(\frac {-\frac {a^{2} d^{2} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3 b^{2}}+\frac {2 a c d \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3 b}-\frac {2 a \,d^{2} \left (-\frac {\left (b x +a \right ) \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {\sin \left (b x +a \right )^{3}}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b^{2}}-\frac {c^{2} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {2 c d \left (-\frac {\left (b x +a \right ) \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {\sin \left (b x +a \right )^{3}}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b}+\frac {d^{2} \left (-\frac {\left (b x +a \right )^{2} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {4 \cos \left (b x +a \right )}{3}+\frac {4 \left (b x +a \right ) \sin \left (b x +a \right )}{3}+\frac {2 \left (b x +a \right ) \sin \left (b x +a \right )^{3}}{9}+\frac {2 \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{27}\right )}{b^{2}}}{b}\) | \(265\) |
| default | \(\frac {-\frac {a^{2} d^{2} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3 b^{2}}+\frac {2 a c d \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3 b}-\frac {2 a \,d^{2} \left (-\frac {\left (b x +a \right ) \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {\sin \left (b x +a \right )^{3}}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b^{2}}-\frac {c^{2} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {2 c d \left (-\frac {\left (b x +a \right ) \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {\sin \left (b x +a \right )^{3}}{9}+\frac {2 \sin \left (b x +a \right )}{3}\right )}{b}+\frac {d^{2} \left (-\frac {\left (b x +a \right )^{2} \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{3}+\frac {4 \cos \left (b x +a \right )}{3}+\frac {4 \left (b x +a \right ) \sin \left (b x +a \right )}{3}+\frac {2 \left (b x +a \right ) \sin \left (b x +a \right )^{3}}{9}+\frac {2 \left (2+\sin \left (b x +a \right )^{2}\right ) \cos \left (b x +a \right )}{27}\right )}{b^{2}}}{b}\) | \(265\) |
| norman | \(\frac {\frac {-36 b^{2} c^{2}+80 d^{2}}{27 b^{3}}-\frac {2 d^{2} x^{2}}{3 b}+\frac {8 d^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{3 b^{3}}+\frac {\left (-36 b^{2} c^{2}+56 d^{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{9 b^{3}}-\frac {4 c d x}{3 b}+\frac {8 c d \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{3 b^{2}}+\frac {64 c d \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{9 b^{2}}+\frac {8 c d \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}}{3 b^{2}}-\frac {2 d^{2} x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b}+\frac {2 d^{2} x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{b}+\frac {2 d^{2} x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{3 b}+\frac {8 d^{2} x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{3 b^{2}}+\frac {64 d^{2} x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{9 b^{2}}+\frac {8 d^{2} x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}}{3 b^{2}}-\frac {4 c d x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b}+\frac {4 c d x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{b}+\frac {4 c d x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{6}}{3 b}}{\left (1+\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right )^{3}}\) | \(338\) |
| orering | \(\frac {40 d \left (9 d^{4} x^{4} b^{4}+36 b^{4} c \,d^{3} x^{3}+54 b^{4} c^{2} d^{2} x^{2}+36 b^{4} c^{3} d x +9 b^{4} c^{4}-b^{2} d^{4} x^{2}-2 b^{2} c \,d^{3} x -b^{2} c^{2} d^{2}-12 d^{4}\right ) \sin \left (b x +a \right )^{3}}{81 b^{6} \left (d x +c \right )^{3}}-\frac {2 \left (45 d^{4} x^{4} b^{4}+180 b^{4} c \,d^{3} x^{3}+270 b^{4} c^{2} d^{2} x^{2}+180 b^{4} c^{3} d x +45 b^{4} c^{4}+26 b^{2} d^{4} x^{2}+52 b^{2} c \,d^{3} x +26 b^{2} c^{2} d^{2}-180 d^{4}\right ) \left (2 \left (d x +c \right ) \sin \left (b x +a \right )^{3} d +3 \left (d x +c \right )^{2} \sin \left (b x +a \right )^{2} b \cos \left (b x +a \right )\right )}{81 b^{6} \left (d x +c \right )^{4}}+\frac {8 d \left (3 x^{2} d^{2} b^{2}+6 b^{2} c d x +3 b^{2} c^{2}-5 d^{2}\right ) \left (2 d^{2} \sin \left (b x +a \right )^{3}+12 \left (d x +c \right ) \sin \left (b x +a \right )^{2} d b \cos \left (b x +a \right )+6 \left (d x +c \right )^{2} \sin \left (b x +a \right ) b^{2} \cos \left (b x +a \right )^{2}-3 \left (d x +c \right )^{2} \sin \left (b x +a \right )^{3} b^{2}\right )}{27 b^{6} \left (d x +c \right )^{3}}-\frac {\left (9 x^{2} d^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}-20 d^{2}\right ) \left (18 d^{2} \sin \left (b x +a \right )^{2} b \cos \left (b x +a \right )+36 \left (d x +c \right ) \sin \left (b x +a \right ) d \,b^{2} \cos \left (b x +a \right )^{2}-18 \left (d x +c \right ) \sin \left (b x +a \right )^{3} d \,b^{2}+6 \left (d x +c \right )^{2} b^{3} \cos \left (b x +a \right )^{3}-21 \left (d x +c \right )^{2} \sin \left (b x +a \right )^{2} b^{3} \cos \left (b x +a \right )\right )}{81 b^{6} \left (d x +c \right )^{2}}\) | \(547\) |
Input:
int((d*x+c)^2*sin(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/108*((9*(d*x+c)^2*b^2-2*d^2)*cos(3*b*x+3*a)-6*b*d*(d*x+c)*sin(3*b*x+3*a) +(-81*(d*x+c)^2*b^2+162*d^2)*cos(b*x+a)+162*b*d*(d*x+c)*sin(b*x+a)-72*b^2* c^2+160*d^2)/b^3
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.07 \[ \int (c+d x)^2 \sin ^3(a+b x) \, dx=\frac {{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 14 \, d^{2}\right )} \cos \left (b x + a\right ) + 6 \, {\left (7 \, b d^{2} x + 7 \, b c d - {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )}{27 \, b^{3}} \] Input:
integrate((d*x+c)^2*sin(b*x+a)^3,x, algorithm="fricas")
Output:
1/27*((9*b^2*d^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2 - 2*d^2)*cos(b*x + a)^3 - 3*(9*b^2*d^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2 - 14*d^2)*cos(b*x + a) + 6*(7* b*d^2*x + 7*b*c*d - (b*d^2*x + b*c*d)*cos(b*x + a)^2)*sin(b*x + a))/b^3
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (121) = 242\).
Time = 0.34 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.31 \[ \int (c+d x)^2 \sin ^3(a+b x) \, dx=\begin {cases} - \frac {c^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 c^{2} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {2 c d x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {4 c d x \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 d^{2} x^{2} \cos ^{3}{\left (a + b x \right )}}{3 b} + \frac {14 c d \sin ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {4 c d \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{2}} + \frac {14 d^{2} x \sin ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {4 d^{2} x \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{2}} + \frac {14 d^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{9 b^{3}} + \frac {40 d^{2} \cos ^{3}{\left (a + b x \right )}}{27 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sin ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**2*sin(b*x+a)**3,x)
Output:
Piecewise((-c**2*sin(a + b*x)**2*cos(a + b*x)/b - 2*c**2*cos(a + b*x)**3/( 3*b) - 2*c*d*x*sin(a + b*x)**2*cos(a + b*x)/b - 4*c*d*x*cos(a + b*x)**3/(3 *b) - d**2*x**2*sin(a + b*x)**2*cos(a + b*x)/b - 2*d**2*x**2*cos(a + b*x)* *3/(3*b) + 14*c*d*sin(a + b*x)**3/(9*b**2) + 4*c*d*sin(a + b*x)*cos(a + b* x)**2/(3*b**2) + 14*d**2*x*sin(a + b*x)**3/(9*b**2) + 4*d**2*x*sin(a + b*x )*cos(a + b*x)**2/(3*b**2) + 14*d**2*sin(a + b*x)**2*cos(a + b*x)/(9*b**3) + 40*d**2*cos(a + b*x)**3/(27*b**3), Ne(b, 0)), ((c**2*x + c*d*x**2 + d** 2*x**3/3)*sin(a)**3, True))
Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (111) = 222\).
Time = 0.04 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.20 \[ \int (c+d x)^2 \sin ^3(a+b x) \, dx=\frac {36 \, {\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} c^{2} - \frac {72 \, {\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} a c d}{b} + \frac {36 \, {\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} a^{2} d^{2}}{b^{2}} + \frac {6 \, {\left (3 \, {\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) - 27 \, {\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (3 \, b x + 3 \, a\right ) + 27 \, \sin \left (b x + a\right )\right )} c d}{b} - \frac {6 \, {\left (3 \, {\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) - 27 \, {\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (3 \, b x + 3 \, a\right ) + 27 \, \sin \left (b x + a\right )\right )} a d^{2}}{b^{2}} + \frac {{\left ({\left (9 \, {\left (b x + a\right )}^{2} - 2\right )} \cos \left (3 \, b x + 3 \, a\right ) - 81 \, {\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) - 6 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) + 162 \, {\left (b x + a\right )} \sin \left (b x + a\right )\right )} d^{2}}{b^{2}}}{108 \, b} \] Input:
integrate((d*x+c)^2*sin(b*x+a)^3,x, algorithm="maxima")
Output:
1/108*(36*(cos(b*x + a)^3 - 3*cos(b*x + a))*c^2 - 72*(cos(b*x + a)^3 - 3*c os(b*x + a))*a*c*d/b + 36*(cos(b*x + a)^3 - 3*cos(b*x + a))*a^2*d^2/b^2 + 6*(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))*c*d/b - 6*(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*d^2/b^2 + ((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))*d^2/b^2)/b
Time = 0.39 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.11 \[ \int (c+d x)^2 \sin ^3(a+b x) \, dx=\frac {{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (3 \, b x + 3 \, a\right )}{108 \, b^{3}} - \frac {3 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right )}{4 \, b^{3}} - \frac {{\left (b d^{2} x + b c d\right )} \sin \left (3 \, b x + 3 \, a\right )}{18 \, b^{3}} + \frac {3 \, {\left (b d^{2} x + b c d\right )} \sin \left (b x + a\right )}{2 \, b^{3}} \] Input:
integrate((d*x+c)^2*sin(b*x+a)^3,x, algorithm="giac")
Output:
1/108*(9*b^2*d^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2 - 2*d^2)*cos(3*b*x + 3*a)/ b^3 - 3/4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(b*x + a)/b^3 - 1/18*(b*d^2*x + b*c*d)*sin(3*b*x + 3*a)/b^3 + 3/2*(b*d^2*x + b*c*d)*sin(b *x + a)/b^3
Time = 35.40 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.41 \[ \int (c+d x)^2 \sin ^3(a+b x) \, dx=\frac {\frac {3\,d^2\,x\,\sin \left (a+b\,x\right )}{2}-\frac {d^2\,x\,\sin \left (3\,a+3\,b\,x\right )}{18}+\frac {3\,c\,d\,\sin \left (a+b\,x\right )}{2}-\frac {c\,d\,\sin \left (3\,a+3\,b\,x\right )}{18}}{b^2}-\frac {\frac {3\,c^2\,\cos \left (a+b\,x\right )}{4}-\frac {c^2\,\cos \left (3\,a+3\,b\,x\right )}{12}+\frac {3\,d^2\,x^2\,\cos \left (a+b\,x\right )}{4}-\frac {d^2\,x^2\,\cos \left (3\,a+3\,b\,x\right )}{12}-\frac {c\,d\,x\,\cos \left (3\,a+3\,b\,x\right )}{6}+\frac {3\,c\,d\,x\,\cos \left (a+b\,x\right )}{2}}{b}+\frac {3\,d^2\,\cos \left (a+b\,x\right )}{2\,b^3}-\frac {d^2\,\cos \left (3\,a+3\,b\,x\right )}{54\,b^3} \] Input:
int(sin(a + b*x)^3*(c + d*x)^2,x)
Output:
((3*d^2*x*sin(a + b*x))/2 - (d^2*x*sin(3*a + 3*b*x))/18 + (3*c*d*sin(a + b *x))/2 - (c*d*sin(3*a + 3*b*x))/18)/b^2 - ((3*c^2*cos(a + b*x))/4 - (c^2*c os(3*a + 3*b*x))/12 + (3*d^2*x^2*cos(a + b*x))/4 - (d^2*x^2*cos(3*a + 3*b* x))/12 - (c*d*x*cos(3*a + 3*b*x))/6 + (3*c*d*x*cos(a + b*x))/2)/b + (3*d^2 *cos(a + b*x))/(2*b^3) - (d^2*cos(3*a + 3*b*x))/(54*b^3)
Time = 0.26 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.80 \[ \int (c+d x)^2 \sin ^3(a+b x) \, dx=\frac {-9 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{2} c^{2}-18 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{2} c d x -9 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{2} d^{2} x^{2}+2 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} d^{2}-18 \cos \left (b x +a \right ) b^{2} c^{2}-36 \cos \left (b x +a \right ) b^{2} c d x -18 \cos \left (b x +a \right ) b^{2} d^{2} x^{2}+40 \cos \left (b x +a \right ) d^{2}+6 \sin \left (b x +a \right )^{3} b c d +6 \sin \left (b x +a \right )^{3} b \,d^{2} x +36 \sin \left (b x +a \right ) b c d +36 \sin \left (b x +a \right ) b \,d^{2} x +36 a b c d -18 b^{2} c^{2}+16 d^{2}}{27 b^{3}} \] Input:
int((d*x+c)^2*sin(b*x+a)^3,x)
Output:
( - 9*cos(a + b*x)*sin(a + b*x)**2*b**2*c**2 - 18*cos(a + b*x)*sin(a + b*x )**2*b**2*c*d*x - 9*cos(a + b*x)*sin(a + b*x)**2*b**2*d**2*x**2 + 2*cos(a + b*x)*sin(a + b*x)**2*d**2 - 18*cos(a + b*x)*b**2*c**2 - 36*cos(a + b*x)* b**2*c*d*x - 18*cos(a + b*x)*b**2*d**2*x**2 + 40*cos(a + b*x)*d**2 + 6*sin (a + b*x)**3*b*c*d + 6*sin(a + b*x)**3*b*d**2*x + 36*sin(a + b*x)*b*c*d + 36*sin(a + b*x)*b*d**2*x + 36*a*b*c*d - 18*b**2*c**2 + 16*d**2)/(27*b**3)