Integrand size = 22, antiderivative size = 205 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin (a+b x) \, dx=-\frac {160 d^4 \cos (a+b x)}{27 b^5}+\frac {8 d^2 (c+d x)^2 \cos (a+b x)}{3 b^3}-\frac {8 d^4 \cos ^3(a+b x)}{81 b^5}+\frac {4 d^2 (c+d x)^2 \cos ^3(a+b x)}{9 b^3}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}-\frac {160 d^3 (c+d x) \sin (a+b x)}{27 b^4}+\frac {8 d (c+d x)^3 \sin (a+b x)}{9 b^2}-\frac {8 d^3 (c+d x) \cos ^2(a+b x) \sin (a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \cos ^2(a+b x) \sin (a+b x)}{9 b^2} \]
-160/27*d^4*cos(b*x+a)/b^5+8/3*d^2*(d*x+c)^2*cos(b*x+a)/b^3-8/81*d^4*cos(b *x+a)^3/b^5+4/9*d^2*(d*x+c)^2*cos(b*x+a)^3/b^3-1/3*(d*x+c)^4*cos(b*x+a)^3/ b-160/27*d^3*(d*x+c)*sin(b*x+a)/b^4+8/9*d*(d*x+c)^3*sin(b*x+a)/b^2-8/27*d^ 3*(d*x+c)*cos(b*x+a)^2*sin(b*x+a)/b^4+4/9*d*(d*x+c)^3*cos(b*x+a)^2*sin(b*x +a)/b^2
Time = 1.78 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.73 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin (a+b x) \, dx=-\frac {81 \left (24 d^4-12 b^2 d^2 (c+d x)^2+b^4 (c+d x)^4\right ) \cos (a+b x)+\left (8 d^4-36 b^2 d^2 (c+d x)^2+27 b^4 (c+d x)^4\right ) \cos (3 (a+b x))-24 b d (c+d x) \left (-82 d^2+15 b^2 (c+d x)^2+\left (-2 d^2+3 b^2 (c+d x)^2\right ) \cos (2 (a+b x))\right ) \sin (a+b x)}{324 b^5} \]
-1/324*(81*(24*d^4 - 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Cos[a + b*x ] + (8*d^4 - 36*b^2*d^2*(c + d*x)^2 + 27*b^4*(c + d*x)^4)*Cos[3*(a + b*x)] - 24*b*d*(c + d*x)*(-82*d^2 + 15*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^5
Time = 1.22 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.20, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.864, Rules used = {4905, 3042, 3792, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3118, 3791, 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.
| \(\displaystyle \int (c+d x)^4 \sin (a+b x) \cos ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 4905 |
| \(\displaystyle \frac {4 d \int (c+d x)^3 \cos ^3(a+b x)dx}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d \int (c+d x)^3 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \int (c+d x) \cos ^3(a+b x)dx}{3 b^2}+\frac {2}{3} \int (c+d x)^3 \cos (a+b x)dx+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \int (c+d x)^3 \sin \left (a+b x+\frac {\pi }{2}\right )dx+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {3 d \int -(c+d x)^2 \sin (a+b x)dx}{b}+\frac {(c+d x)^3 \sin (a+b x)}{b}\right )+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \int (c+d x)^2 \sin (a+b x)dx}{b}\right )+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \int (c+d x)^2 \sin (a+b x)dx}{b}\right )+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \int (c+d x) \cos (a+b x)dx}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \int -\sin (a+b x)dx}{b}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \left (\frac {2}{3} \int (c+d x) \cos (a+b x)dx+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \left (\frac {2}{3} \int (c+d x) \sin \left (a+b x+\frac {\pi }{2}\right )dx+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \left (\frac {2}{3} \left (\frac {d \int -\sin (a+b x)dx}{b}+\frac {(c+d x) \sin (a+b x)}{b}\right )+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \left (\frac {2}{3} \left (\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x)dx}{b}\right )+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \left (\frac {2}{3} \left (\frac {(c+d x) \sin (a+b x)}{b}-\frac {d \int \sin (a+b x)dx}{b}\right )+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {4 d \left (-\frac {2 d^2 \left (\frac {2}{3} \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )+\frac {d \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x) \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \cos ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {3 d \left (\frac {2 d \left (\frac {d \cos (a+b x)}{b^2}+\frac {(c+d x) \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^2 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^3 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b}-\frac {(c+d x)^4 \cos ^3(a+b x)}{3 b}\) |
-1/3*((c + d*x)^4*Cos[a + b*x]^3)/b + (4*d*((d*(c + d*x)^2*Cos[a + b*x]^3) /(3*b^2) + ((c + d*x)^3*Cos[a + b*x]^2*Sin[a + b*x])/(3*b) - (2*d^2*((d*Co s[a + b*x]^3)/(9*b^2) + ((c + d*x)*Cos[a + b*x]^2*Sin[a + b*x])/(3*b) + (2 *((d*Cos[a + b*x])/b^2 + ((c + d*x)*Sin[a + b*x])/b))/3))/(3*b^2) + (2*((( c + d*x)^3*Sin[a + b*x])/b - (3*d*(-(((c + d*x)^2*Cos[a + b*x])/b) + (2*d* ((d*Cos[a + b*x])/b^2 + ((c + d*x)*Sin[a + b*x])/b))/b))/b))/3))/(3*b)
3.1.71.3.1 Defintions of rubi rules used
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[((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]
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] + Simp[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]
Time = 1.77 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.88
| method | result | size |
| parallelrisch | \(\frac {\left (-27 b^{4} \left (d x +c \right )^{4}+36 d^{2} \left (d x +c \right )^{2} b^{2}-8 d^{4}\right ) \cos \left (3 x b +3 a \right )+36 b d \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{3}\right ) \left (d x +c \right ) \sin \left (3 x b +3 a \right )+\left (-81 b^{4} \left (d x +c \right )^{4}+972 d^{2} \left (d x +c \right )^{2} b^{2}-1944 d^{4}\right ) \cos \left (x b +a \right )+324 b d \left (\left (d x +c \right )^{2} b^{2}-6 d^{2}\right ) \left (d x +c \right ) \sin \left (x b +a \right )-108 b^{4} c^{4}+1008 b^{2} c^{2} d^{2}-1952 d^{4}}{324 b^{5}}\) | \(181\) |
| risch | \(-\frac {\left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{4} c^{3} d x +b^{4} c^{4}-12 b^{2} d^{4} x^{2}-24 b^{2} c \,d^{3} x -12 b^{2} c^{2} d^{2}+24 d^{4}\right ) \cos \left (x b +a \right )}{4 b^{5}}+\frac {d \left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \sin \left (x b +a \right )}{b^{4}}-\frac {\left (27 d^{4} x^{4} b^{4}+108 b^{4} c \,d^{3} x^{3}+162 b^{4} c^{2} d^{2} x^{2}+108 b^{4} c^{3} d x +27 b^{4} c^{4}-36 b^{2} d^{4} x^{2}-72 b^{2} c \,d^{3} x -36 b^{2} c^{2} d^{2}+8 d^{4}\right ) \cos \left (3 x b +3 a \right )}{324 b^{5}}+\frac {d \left (3 b^{2} d^{3} x^{3}+9 b^{2} c \,d^{2} x^{2}+9 b^{2} c^{2} d x +3 b^{2} c^{3}-2 d^{3} x -2 c \,d^{2}\right ) \sin \left (3 x b +3 a \right )}{27 b^{4}}\) | \(343\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(835\) |
| default | \(\text {Expression too large to display}\) | \(835\) |
1/324*((-27*b^4*(d*x+c)^4+36*d^2*(d*x+c)^2*b^2-8*d^4)*cos(3*b*x+3*a)+36*b* d*((d*x+c)^2*b^2-2/3*d^2)*(d*x+c)*sin(3*b*x+3*a)+(-81*b^4*(d*x+c)^4+972*d^ 2*(d*x+c)^2*b^2-1944*d^4)*cos(b*x+a)+324*b*d*((d*x+c)^2*b^2-6*d^2)*(d*x+c) *sin(b*x+a)-108*b^4*c^4+1008*b^2*c^2*d^2-1952*d^4)/b^5
Time = 0.25 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.43 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin (a+b x) \, dx=-\frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 2 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 2 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} - 24 \, {\left (9 \, b^{2} d^{4} x^{2} + 18 \, b^{2} c d^{3} x + 9 \, b^{2} c^{2} d^{2} - 20 \, d^{4}\right )} \cos \left (b x + a\right ) - 12 \, {\left (6 \, b^{3} d^{4} x^{3} + 18 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{3} d - 40 \, b c d^{3} + {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 2 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} - 2 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (9 \, b^{3} c^{2} d^{2} - 20 \, b d^{4}\right )} x\right )} \sin \left (b x + a\right )}{81 \, b^{5}} \]
-1/81*((27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 27*b^4*c^4 - 36*b^2*c^2*d^2 + 8*d^4 + 18*(9*b^4*c^2*d^2 - 2*b^2*d^4)*x^2 + 36*(3*b^4*c^3*d - 2*b^2*c*d^ 3)*x)*cos(b*x + a)^3 - 24*(9*b^2*d^4*x^2 + 18*b^2*c*d^3*x + 9*b^2*c^2*d^2 - 20*d^4)*cos(b*x + a) - 12*(6*b^3*d^4*x^3 + 18*b^3*c*d^3*x^2 + 6*b^3*c^3* d - 40*b*c*d^3 + (3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 3*b^3*c^3*d - 2*b*c*d^ 3 + (9*b^3*c^2*d^2 - 2*b*d^4)*x)*cos(b*x + a)^2 + 2*(9*b^3*c^2*d^2 - 20*b* d^4)*x)*sin(b*x + a))/b^5
Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (207) = 414\).
Time = 0.64 (sec) , antiderivative size = 646, normalized size of antiderivative = 3.15 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin (a+b x) \, dx=\begin {cases} - \frac {c^{4} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {4 c^{3} d x \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {2 c^{2} d^{2} x^{2} \cos ^{3}{\left (a + b x \right )}}{b} - \frac {4 c d^{3} x^{3} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {d^{4} x^{4} \cos ^{3}{\left (a + b x \right )}}{3 b} + \frac {8 c^{3} d \sin ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {4 c^{3} d \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{2}} + \frac {8 c^{2} d^{2} x \sin ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {4 c^{2} d^{2} x \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b^{2}} + \frac {8 c d^{3} x^{2} \sin ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {4 c d^{3} x^{2} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b^{2}} + \frac {8 d^{4} x^{3} \sin ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {4 d^{4} x^{3} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{2}} + \frac {8 c^{2} d^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{3}} + \frac {28 c^{2} d^{2} \cos ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {16 c d^{3} x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{3}} + \frac {56 c d^{3} x \cos ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {8 d^{4} x^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{3}} + \frac {28 d^{4} x^{2} \cos ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {160 c d^{3} \sin ^{3}{\left (a + b x \right )}}{27 b^{4}} - \frac {56 c d^{3} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{9 b^{4}} - \frac {160 d^{4} x \sin ^{3}{\left (a + b x \right )}}{27 b^{4}} - \frac {56 d^{4} x \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{9 b^{4}} - \frac {160 d^{4} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{27 b^{5}} - \frac {488 d^{4} \cos ^{3}{\left (a + b x \right )}}{81 b^{5}} & \text {for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac {d^{4} x^{5}}{5}\right ) \sin {\left (a \right )} \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((-c**4*cos(a + b*x)**3/(3*b) - 4*c**3*d*x*cos(a + b*x)**3/(3*b) - 2*c**2*d**2*x**2*cos(a + b*x)**3/b - 4*c*d**3*x**3*cos(a + b*x)**3/(3*b) - d**4*x**4*cos(a + b*x)**3/(3*b) + 8*c**3*d*sin(a + b*x)**3/(9*b**2) + 4 *c**3*d*sin(a + b*x)*cos(a + b*x)**2/(3*b**2) + 8*c**2*d**2*x*sin(a + b*x) **3/(3*b**2) + 4*c**2*d**2*x*sin(a + b*x)*cos(a + b*x)**2/b**2 + 8*c*d**3* x**2*sin(a + b*x)**3/(3*b**2) + 4*c*d**3*x**2*sin(a + b*x)*cos(a + b*x)**2 /b**2 + 8*d**4*x**3*sin(a + b*x)**3/(9*b**2) + 4*d**4*x**3*sin(a + b*x)*co s(a + b*x)**2/(3*b**2) + 8*c**2*d**2*sin(a + b*x)**2*cos(a + b*x)/(3*b**3) + 28*c**2*d**2*cos(a + b*x)**3/(9*b**3) + 16*c*d**3*x*sin(a + b*x)**2*cos (a + b*x)/(3*b**3) + 56*c*d**3*x*cos(a + b*x)**3/(9*b**3) + 8*d**4*x**2*si n(a + b*x)**2*cos(a + b*x)/(3*b**3) + 28*d**4*x**2*cos(a + b*x)**3/(9*b**3 ) - 160*c*d**3*sin(a + b*x)**3/(27*b**4) - 56*c*d**3*sin(a + b*x)*cos(a + b*x)**2/(9*b**4) - 160*d**4*x*sin(a + b*x)**3/(27*b**4) - 56*d**4*x*sin(a + b*x)*cos(a + b*x)**2/(9*b**4) - 160*d**4*sin(a + b*x)**2*cos(a + b*x)/(2 7*b**5) - 488*d**4*cos(a + b*x)**3/(81*b**5), Ne(b, 0)), ((c**4*x + 2*c**3 *d*x**2 + 2*c**2*d**2*x**3 + c*d**3*x**4 + d**4*x**5/5)*sin(a)*cos(a)**2, True))
Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (187) = 374\).
Time = 0.30 (sec) , antiderivative size = 889, normalized size of antiderivative = 4.34 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin (a+b x) \, dx=\text {Too large to display} \]
-1/324*(108*c^4*cos(b*x + a)^3 - 432*a*c^3*d*cos(b*x + a)^3/b + 648*a^2*c^ 2*d^2*cos(b*x + a)^3/b^2 - 432*a^3*c*d^3*cos(b*x + a)^3/b^3 + 108*a^4*d^4* cos(b*x + a)^3/b^4 + 36*(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^3*d/b - 108*(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^2*d^2/b^2 + 108*(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*c*d^3/b^3 - 36*(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^3*d^4/b^4 + 18*((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)*si n(b*x + a))*c^2*d^2/b^2 - 36*((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*c*d^3/b^3 + 18*((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^2*d^4/b^4 + 12*(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))*c*d^3/b^3 - 12* (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))*a*d^4/b^4 + ((27*(b*x + a)^4 - 36*(b*x + a)^2 +...
Time = 0.31 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.71 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin (a+b x) \, dx=-\frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 108 \, b^{4} c^{3} d x + 27 \, b^{4} c^{4} - 36 \, b^{2} d^{4} x^{2} - 72 \, b^{2} c d^{3} x - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4}\right )} \cos \left (3 \, b x + 3 \, a\right )}{324 \, b^{5}} - \frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4} - 12 \, b^{2} d^{4} x^{2} - 24 \, b^{2} c d^{3} x - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \cos \left (b x + a\right )}{4 \, b^{5}} + \frac {{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 9 \, b^{3} c^{2} d^{2} x + 3 \, b^{3} c^{3} d - 2 \, b d^{4} x - 2 \, b c d^{3}\right )} \sin \left (3 \, b x + 3 \, a\right )}{27 \, b^{5}} + \frac {{\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \sin \left (b x + a\right )}{b^{5}} \]
-1/324*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 162*b^4*c^2*d^2*x^2 + 108*b^4 *c^3*d*x + 27*b^4*c^4 - 36*b^2*d^4*x^2 - 72*b^2*c*d^3*x - 36*b^2*c^2*d^2 + 8*d^4)*cos(3*b*x + 3*a)/b^5 - 1/4*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4* c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4 - 12*b^2*d^4*x^2 - 24*b^2*c*d^3*x - 12*b^2*c^2*d^2 + 24*d^4)*cos(b*x + a)/b^5 + 1/27*(3*b^3*d^4*x^3 + 9*b^3*c* d^3*x^2 + 9*b^3*c^2*d^2*x + 3*b^3*c^3*d - 2*b*d^4*x - 2*b*c*d^3)*sin(3*b*x + 3*a)/b^5 + (b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + b^3*c^3*d - 6*b*d^4*x - 6*b*c*d^3)*sin(b*x + a)/b^5
Time = 25.72 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.19 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin (a+b x) \, dx=\frac {4\,x\,{\cos \left (a+b\,x\right )}^3\,\left (14\,c\,d^3-3\,b^2\,c^3\,d\right )}{9\,b^3}-\frac {{\cos \left (a+b\,x\right )}^3\,\left (27\,b^4\,c^4-252\,b^2\,c^2\,d^2+488\,d^4\right )}{81\,b^5}-\frac {8\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (20\,d^4-9\,b^2\,c^2\,d^2\right )}{27\,b^5}-\frac {4\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (14\,c\,d^3-3\,b^2\,c^3\,d\right )}{9\,b^4}-\frac {d^4\,x^4\,{\cos \left (a+b\,x\right )}^3}{3\,b}-\frac {8\,{\sin \left (a+b\,x\right )}^3\,\left (20\,c\,d^3-3\,b^2\,c^3\,d\right )}{27\,b^4}+\frac {8\,d^4\,x^3\,{\sin \left (a+b\,x\right )}^3}{9\,b^2}-\frac {8\,x\,{\sin \left (a+b\,x\right )}^3\,\left (20\,d^4-9\,b^2\,c^2\,d^2\right )}{27\,b^4}+\frac {2\,x^2\,{\cos \left (a+b\,x\right )}^3\,\left (14\,d^4-9\,b^2\,c^2\,d^2\right )}{9\,b^3}-\frac {4\,c\,d^3\,x^3\,{\cos \left (a+b\,x\right )}^3}{3\,b}+\frac {4\,d^4\,x^3\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{3\,b^2}+\frac {8\,d^4\,x^2\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{3\,b^3}+\frac {8\,c\,d^3\,x^2\,{\sin \left (a+b\,x\right )}^3}{3\,b^2}-\frac {4\,x\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (14\,d^4-9\,b^2\,c^2\,d^2\right )}{9\,b^4}+\frac {4\,c\,d^3\,x^2\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{b^2}+\frac {16\,c\,d^3\,x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{3\,b^3} \]
(4*x*cos(a + b*x)^3*(14*c*d^3 - 3*b^2*c^3*d))/(9*b^3) - (cos(a + b*x)^3*(4 88*d^4 + 27*b^4*c^4 - 252*b^2*c^2*d^2))/(81*b^5) - (8*cos(a + b*x)*sin(a + b*x)^2*(20*d^4 - 9*b^2*c^2*d^2))/(27*b^5) - (4*cos(a + b*x)^2*sin(a + b*x )*(14*c*d^3 - 3*b^2*c^3*d))/(9*b^4) - (d^4*x^4*cos(a + b*x)^3)/(3*b) - (8* sin(a + b*x)^3*(20*c*d^3 - 3*b^2*c^3*d))/(27*b^4) + (8*d^4*x^3*sin(a + b*x )^3)/(9*b^2) - (8*x*sin(a + b*x)^3*(20*d^4 - 9*b^2*c^2*d^2))/(27*b^4) + (2 *x^2*cos(a + b*x)^3*(14*d^4 - 9*b^2*c^2*d^2))/(9*b^3) - (4*c*d^3*x^3*cos(a + b*x)^3)/(3*b) + (4*d^4*x^3*cos(a + b*x)^2*sin(a + b*x))/(3*b^2) + (8*d^ 4*x^2*cos(a + b*x)*sin(a + b*x)^2)/(3*b^3) + (8*c*d^3*x^2*sin(a + b*x)^3)/ (3*b^2) - (4*x*cos(a + b*x)^2*sin(a + b*x)*(14*d^4 - 9*b^2*c^2*d^2))/(9*b^ 4) + (4*c*d^3*x^2*cos(a + b*x)^2*sin(a + b*x))/b^2 + (16*c*d^3*x*cos(a + b *x)*sin(a + b*x)^2)/(3*b^3)