Integrand size = 22, antiderivative size = 103 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {4 d (c+d x) \cos (a+b x)}{9 b^2}-\frac {4 d^2 \sin (a+b x)}{9 b^3}+\frac {2 d (c+d x) \cos (a+b x) \sin ^2(a+b x)}{9 b^2}-\frac {2 d^2 \sin ^3(a+b x)}{27 b^3}+\frac {(c+d x)^2 \sin ^3(a+b x)}{3 b} \]
4/9*d*(d*x+c)*cos(b*x+a)/b^2-4/9*d^2*sin(b*x+a)/b^3+2/9*d*(d*x+c)*cos(b*x+ a)*sin(b*x+a)^2/b^2-2/27*d^2*sin(b*x+a)^3/b^3+1/3*(d*x+c)^2*sin(b*x+a)^3/b
Time = 0.66 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {54 b d (c+d x) \cos (a+b x)-6 b d (c+d x) \cos (3 (a+b x))-2 \left (26 d^2-9 b^2 (c+d x)^2+\left (-2 d^2+9 b^2 (c+d x)^2\right ) \cos (2 (a+b x))\right ) \sin (a+b x)}{108 b^3} \]
(54*b*d*(c + d*x)*Cos[a + b*x] - 6*b*d*(c + d*x)*Cos[3*(a + b*x)] - 2*(26* d^2 - 9*b^2*(c + d*x)^2 + (-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[2*(a + b*x)])*S in[a + b*x])/(108*b^3)
Time = 0.45 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4904, 3042, 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.
| \(\displaystyle \int (c+d x)^2 \sin ^2(a+b x) \cos (a+b x) \, dx\) |
| \(\Big \downarrow \) 4904 |
| \(\displaystyle \frac {(c+d x)^2 \sin ^3(a+b x)}{3 b}-\frac {2 d \int (c+d x) \sin ^3(a+b x)dx}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^2 \sin ^3(a+b x)}{3 b}-\frac {2 d \int (c+d x) \sin (a+b x)^3dx}{3 b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {(c+d x)^2 \sin ^3(a+b x)}{3 b}-\frac {2 d \left (\frac {2}{3} \int (c+d x) \sin (a+b x)dx+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^2 \sin ^3(a+b x)}{3 b}-\frac {2 d \left (\frac {2}{3} \int (c+d x) \sin (a+b x)dx+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^2 \sin ^3(a+b x)}{3 b}-\frac {2 d \left (\frac {2}{3} \left (\frac {d \int \cos (a+b x)dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^2 \sin ^3(a+b x)}{3 b}-\frac {2 d \left (\frac {2}{3} \left (\frac {d \int \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {(c+d x)^2 \sin ^3(a+b x)}{3 b}-\frac {2 d \left (\frac {2}{3} \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
((c + d*x)^2*Sin[a + b*x]^3)/(3*b) - (2*d*(-1/3*((c + d*x)*Cos[a + b*x]*Si n[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)
3.1.16.3.1 Defintions of rubi rules used
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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_))*((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]
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] - Simp[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]
Time = 1.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.24
| method | result | size |
| risch | \(\frac {d \left (d x +c \right ) \cos \left (x b +a \right )}{2 b^{2}}+\frac {\left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 d^{2}\right ) \sin \left (x b +a \right )}{4 b^{3}}-\frac {d \left (d x +c \right ) \cos \left (3 x b +3 a \right )}{18 b^{2}}-\frac {\left (9 x^{2} d^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}-2 d^{2}\right ) \sin \left (3 x b +3 a \right )}{108 b^{3}}\) | \(128\) |
| parallelrisch | \(\frac {-12 d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6} x b -24 d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}-36 d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4} x b +\left (\left (72 x^{2} b^{2}-64\right ) d^{2}+144 b^{2} c d x +72 b^{2} c^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+72 b d \left (\frac {d x}{2}+c \right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-24 d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+24 b d \left (\frac {d x}{2}+c \right )}{27 b^{3} \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{3}}\) | \(162\) |
| derivativedivides | \(\frac {\frac {a^{2} d^{2} \sin \left (x b +a \right )^{3}}{3 b^{2}}-\frac {2 a c d \sin \left (x b +a \right )^{3}}{3 b}-\frac {2 a \,d^{2} \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b^{2}}+\frac {c^{2} \sin \left (x b +a \right )^{3}}{3}+\frac {2 c d \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b}+\frac {d^{2} \left (\frac {\left (x b +a \right )^{2} \sin \left (x b +a \right )^{3}}{3}+\frac {2 \left (x b +a \right ) \left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}-\frac {2 \sin \left (x b +a \right )^{3}}{27}-\frac {4 \sin \left (x b +a \right )}{9}\right )}{b^{2}}}{b}\) | \(204\) |
| default | \(\frac {\frac {a^{2} d^{2} \sin \left (x b +a \right )^{3}}{3 b^{2}}-\frac {2 a c d \sin \left (x b +a \right )^{3}}{3 b}-\frac {2 a \,d^{2} \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b^{2}}+\frac {c^{2} \sin \left (x b +a \right )^{3}}{3}+\frac {2 c d \left (\frac {\left (x b +a \right ) \sin \left (x b +a \right )^{3}}{3}+\frac {\left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}\right )}{b}+\frac {d^{2} \left (\frac {\left (x b +a \right )^{2} \sin \left (x b +a \right )^{3}}{3}+\frac {2 \left (x b +a \right ) \left (2+\sin \left (x b +a \right )^{2}\right ) \cos \left (x b +a \right )}{9}-\frac {2 \sin \left (x b +a \right )^{3}}{27}-\frac {4 \sin \left (x b +a \right )}{9}\right )}{b^{2}}}{b}\) | \(204\) |
| norman | \(\frac {\frac {8 c d}{9 b^{2}}-\frac {8 d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{9 b^{3}}-\frac {8 d^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}}{9 b^{3}}+\frac {4 d^{2} x}{9 b^{2}}+\frac {8 \left (9 b^{2} c^{2}-8 d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{27 b^{3}}+\frac {8 c d \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{3 b^{2}}+\frac {4 d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{3 b^{2}}-\frac {4 d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{3 b^{2}}-\frac {4 d^{2} x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{9 b^{2}}+\frac {8 d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{3 b}+\frac {16 c d x \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}}{3 b}}{\left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{3}}\) | \(219\) |
1/2*d*(d*x+c)*cos(b*x+a)/b^2+1/4*(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2-2*d^2)/b ^3*sin(b*x+a)-1/18*d*(d*x+c)*cos(3*b*x+3*a)/b^2-1/108*(9*b^2*d^2*x^2+18*b^ 2*c*d*x+9*b^2*c^2-2*d^2)/b^3*sin(3*b*x+3*a)
Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.26 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {6 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )^{3} - 18 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) - {\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - {\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right )^{2} - 14 \, d^{2}\right )} \sin \left (b x + a\right )}{27 \, b^{3}} \]
-1/27*(6*(b*d^2*x + b*c*d)*cos(b*x + a)^3 - 18*(b*d^2*x + b*c*d)*cos(b*x + a) - (9*b^2*d^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2 - (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)^2 - 14*d^2)*sin(b*x + a))/b^3
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (102) = 204\).
Time = 0.32 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.10 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {c^{2} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {2 c d x \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {d^{2} x^{2} \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac {2 c d \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{2}} + \frac {4 c d \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {2 d^{2} x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{2}} + \frac {4 d^{2} x \cos ^{3}{\left (a + b x \right )}}{9 b^{2}} - \frac {14 d^{2} \sin ^{3}{\left (a + b x \right )}}{27 b^{3}} - \frac {4 d^{2} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{9 b^{3}} & \text {for}\: b \neq 0 \\\left (c^{2} x + c d x^{2} + \frac {d^{2} x^{3}}{3}\right ) \sin ^{2}{\left (a \right )} \cos {\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((c**2*sin(a + b*x)**3/(3*b) + 2*c*d*x*sin(a + b*x)**3/(3*b) + d* *2*x**2*sin(a + b*x)**3/(3*b) + 2*c*d*sin(a + b*x)**2*cos(a + b*x)/(3*b**2 ) + 4*c*d*cos(a + b*x)**3/(9*b**2) + 2*d**2*x*sin(a + b*x)**2*cos(a + b*x) /(3*b**2) + 4*d**2*x*cos(a + b*x)**3/(9*b**2) - 14*d**2*sin(a + b*x)**3/(2 7*b**3) - 4*d**2*sin(a + b*x)*cos(a + b*x)**2/(9*b**3), Ne(b, 0)), ((c**2* x + c*d*x**2 + d**2*x**3/3)*sin(a)**2*cos(a), True))
Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (93) = 186\).
Time = 0.21 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.33 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {36 \, c^{2} \sin \left (b x + a\right )^{3} - \frac {72 \, a c d \sin \left (b x + a\right )^{3}}{b} + \frac {36 \, a^{2} d^{2} \sin \left (b x + a\right )^{3}}{b^{2}} - \frac {6 \, {\left (3 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 9 \, {\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) - 9 \, \cos \left (b x + a\right )\right )} c d}{b} + \frac {6 \, {\left (3 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 9 \, {\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (3 \, b x + 3 \, a\right ) - 9 \, \cos \left (b x + a\right )\right )} a d^{2}}{b^{2}} - \frac {{\left (6 \, {\left (b x + a\right )} \cos \left (3 \, b x + 3 \, a\right ) - 54 \, {\left (b x + a\right )} \cos \left (b x + a\right ) + {\left (9 \, {\left (b x + a\right )}^{2} - 2\right )} \sin \left (3 \, b x + 3 \, a\right ) - 27 \, {\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} d^{2}}{b^{2}}}{108 \, b} \]
1/108*(36*c^2*sin(b*x + a)^3 - 72*a*c*d*sin(b*x + a)^3/b + 36*a^2*d^2*sin( b*x + a)^3/b^2 - 6*(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*d/b + 6*(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*d^2 /b^2 - (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))*d^2/b^2 )/b
Time = 0.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.33 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {{\left (b d^{2} x + b c d\right )} \cos \left (3 \, b x + 3 \, a\right )}{18 \, b^{3}} + \frac {{\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right )}{2 \, b^{3}} - \frac {{\left (9 \, b^{2} d^{2} x^{2} + 18 \, b^{2} c d x + 9 \, b^{2} c^{2} - 2 \, d^{2}\right )} \sin \left (3 \, b x + 3 \, a\right )}{108 \, b^{3}} + \frac {{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \sin \left (b x + a\right )}{4 \, b^{3}} \]
-1/18*(b*d^2*x + b*c*d)*cos(3*b*x + 3*a)/b^3 + 1/2*(b*d^2*x + b*c*d)*cos(b *x + a)/b^3 - 1/108*(9*b^2*d^2*x^2 + 18*b^2*c*d*x + 9*b^2*c^2 - 2*d^2)*sin (3*b*x + 3*a)/b^3 + 1/4*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*sin( b*x + a)/b^3
Time = 22.97 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.56 \[ \int (c+d x)^2 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {4\,d^2\,x\,{\cos \left (a+b\,x\right )}^3}{9\,b^2}-\frac {4\,d^2\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{9\,b^3}-\frac {{\sin \left (a+b\,x\right )}^3\,\left (14\,d^2-9\,b^2\,c^2\right )}{27\,b^3}+\frac {d^2\,x^2\,{\sin \left (a+b\,x\right )}^3}{3\,b}+\frac {4\,c\,d\,{\cos \left (a+b\,x\right )}^3}{9\,b^2}+\frac {2\,c\,d\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{3\,b^2}+\frac {2\,c\,d\,x\,{\sin \left (a+b\,x\right )}^3}{3\,b}+\frac {2\,d^2\,x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{3\,b^2} \]
(4*d^2*x*cos(a + b*x)^3)/(9*b^2) - (4*d^2*cos(a + b*x)^2*sin(a + b*x))/(9* b^3) - (sin(a + b*x)^3*(14*d^2 - 9*b^2*c^2))/(27*b^3) + (d^2*x^2*sin(a + b *x)^3)/(3*b) + (4*c*d*cos(a + b*x)^3)/(9*b^2) + (2*c*d*cos(a + b*x)*sin(a + b*x)^2)/(3*b^2) + (2*c*d*x*sin(a + b*x)^3)/(3*b) + (2*d^2*x*cos(a + b*x) *sin(a + b*x)^2)/(3*b^2)