Integrand size = 22, antiderivative size = 205 \[ \int (c+d x)^4 \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {160 d^3 (c+d x) \cos (a+b x)}{27 b^4}+\frac {8 d (c+d x)^3 \cos (a+b x)}{9 b^2}+\frac {160 d^4 \sin (a+b x)}{27 b^5}-\frac {8 d^2 (c+d x)^2 \sin (a+b x)}{3 b^3}-\frac {8 d^3 (c+d x) \cos (a+b x) \sin ^2(a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \cos (a+b x) \sin ^2(a+b x)}{9 b^2}+\frac {8 d^4 \sin ^3(a+b x)}{81 b^5}-\frac {4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}+\frac {(c+d x)^4 \sin ^3(a+b x)}{3 b} \] Output:
-160/27*d^3*(d*x+c)*cos(b*x+a)/b^4+8/9*d*(d*x+c)^3*cos(b*x+a)/b^2+160/27*d ^4*sin(b*x+a)/b^5-8/3*d^2*(d*x+c)^2*sin(b*x+a)/b^3-8/27*d^3*(d*x+c)*cos(b* x+a)*sin(b*x+a)^2/b^4+4/9*d*(d*x+c)^3*cos(b*x+a)*sin(b*x+a)^2/b^2+8/81*d^4 *sin(b*x+a)^3/b^5-4/9*d^2*(d*x+c)^2*sin(b*x+a)^3/b^3+1/3*(d*x+c)^4*sin(b*x +a)^3/b
Time = 0.99 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.88 \[ \int (c+d x)^4 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {324 b d (c+d x) \left (-6 d^2+b^2 (c+d x)^2\right ) \cos (a+b x)-12 b d (c+d x) \left (-2 d^2+3 b^2 (c+d x)^2\right ) \cos (3 (a+b x))+81 b^4 c^4 \sin (a+b x)-972 b^2 c^2 d^2 \sin (a+b x)+1944 d^4 \sin (a+b x)+324 b^4 c^3 d x \sin (a+b x)-1944 b^2 c d^3 x \sin (a+b x)+486 b^4 c^2 d^2 x^2 \sin (a+b x)-972 b^2 d^4 x^2 \sin (a+b x)+324 b^4 c d^3 x^3 \sin (a+b x)+81 b^4 d^4 x^4 \sin (a+b x)-27 b^4 c^4 \sin (3 (a+b x))+36 b^2 c^2 d^2 \sin (3 (a+b x))-8 d^4 \sin (3 (a+b x))-108 b^4 c^3 d x \sin (3 (a+b x))+72 b^2 c d^3 x \sin (3 (a+b x))-162 b^4 c^2 d^2 x^2 \sin (3 (a+b x))+36 b^2 d^4 x^2 \sin (3 (a+b x))-108 b^4 c d^3 x^3 \sin (3 (a+b x))-27 b^4 d^4 x^4 \sin (3 (a+b x))}{324 b^5} \] Input:
Integrate[(c + d*x)^4*Cos[a + b*x]*Sin[a + b*x]^2,x]
Output:
(324*b*d*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] - 12*b*d*(c + d *x)*(-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] + 81*b^4*c^4*Sin[a + b*x ] - 972*b^2*c^2*d^2*Sin[a + b*x] + 1944*d^4*Sin[a + b*x] + 324*b^4*c^3*d*x *Sin[a + b*x] - 1944*b^2*c*d^3*x*Sin[a + b*x] + 486*b^4*c^2*d^2*x^2*Sin[a + b*x] - 972*b^2*d^4*x^2*Sin[a + b*x] + 324*b^4*c*d^3*x^3*Sin[a + b*x] + 8 1*b^4*d^4*x^4*Sin[a + b*x] - 27*b^4*c^4*Sin[3*(a + b*x)] + 36*b^2*c^2*d^2* Sin[3*(a + b*x)] - 8*d^4*Sin[3*(a + b*x)] - 108*b^4*c^3*d*x*Sin[3*(a + b*x )] + 72*b^2*c*d^3*x*Sin[3*(a + b*x)] - 162*b^4*c^2*d^2*x^2*Sin[3*(a + b*x) ] + 36*b^2*d^4*x^2*Sin[3*(a + b*x)] - 108*b^4*c*d^3*x^3*Sin[3*(a + b*x)] - 27*b^4*d^4*x^4*Sin[3*(a + b*x)])/(324*b^5)
Time = 1.26 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.21, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {4904, 3042, 3792, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3117, 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.
| \(\displaystyle \int (c+d x)^4 \sin ^2(a+b x) \cos (a+b x) \, dx\) |
| \(\Big \downarrow \) 4904 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \int (c+d x)^3 \sin ^3(a+b x)dx}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \int (c+d x)^3 \sin (a+b x)^3dx}{3 b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin ^3(a+b x)dx}{3 b^2}+\frac {2}{3} \int (c+d x)^3 \sin (a+b x)dx+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \int (c+d x)^3 \sin (a+b x)dx+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {3 d \int (c+d x)^2 \cos (a+b x)dx}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {3 d \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {2 d \int -((c+d x) \sin (a+b x))dx}{b}+\frac {(c+d x)^2 \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \int (c+d x) \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \int (c+d x) \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \int \cos (a+b x)dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin (a+b x)^3dx}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \int \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \int (c+d x) \sin (a+b x)^3dx}{3 b^2}+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \left (\frac {2}{3} \int (c+d x) \sin (a+b x)dx+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \left (\frac {2}{3} \int (c+d x) \sin (a+b x)dx+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \left (\frac {2}{3} \left (\frac {d \int \cos (a+b x)dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \left (\frac {2}{3} \left (\frac {d \int \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac {4 d \left (-\frac {2 d^2 \left (\frac {2}{3} \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )+\frac {d \sin ^3(a+b x)}{9 b^2}-\frac {(c+d x) \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b^2}+\frac {d (c+d x)^2 \sin ^3(a+b x)}{3 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{3 b}\right )}{3 b}\) |
Input:
Int[(c + d*x)^4*Cos[a + b*x]*Sin[a + b*x]^2,x]
Output:
((c + d*x)^4*Sin[a + b*x]^3)/(3*b) - (4*d*(-1/3*((c + d*x)^3*Cos[a + b*x]* Sin[a + b*x]^2)/b + (d*(c + d*x)^2*Sin[a + b*x]^3)/(3*b^2) - (2*d^2*(-1/3* ((c + d*x)*Cos[a + b*x]*Sin[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^2) + (2 *(-(((c + d*x)^3*Cos[a + b*x])/b) + (3*d*(((c + d*x)^2*Sin[a + b*x])/b - ( 2*d*(-(((c + d*x)*Cos[a + b*x])/b) + (d*Sin[a + b*x])/b^2))/b))/b))/3))/(3 *b)
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[((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_)]*((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 = 1.53 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.84
| method | result | size |
| parallelrisch | \(\frac {\left (-27 \left (d x +c \right )^{4} b^{4}+36 d^{2} \left (d x +c \right )^{2} b^{2}-8 d^{4}\right ) \sin \left (3 b x +3 a \right )-36 d \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{3}\right ) \left (d x +c \right ) b \cos \left (3 b x +3 a \right )+81 \left (\left (d x +c \right )^{4} b^{4}-12 d^{2} \left (d x +c \right )^{2} b^{2}+24 d^{4}\right ) \sin \left (b x +a \right )+324 d \left (\left (\left (d x +c \right )^{2} b^{2}-6 d^{2}\right ) \left (d x +c \right ) \cos \left (b x +a \right )+\frac {8 b^{2} c^{3}}{9}-\frac {160 c \,d^{2}}{27}\right ) b}{324 b^{5}}\) | \(173\) |
| risch | \(\frac {d \left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \cos \left (b x +a \right )}{b^{4}}+\frac {\left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{4} c^{3} d x +b^{4} c^{4}-12 b^{2} d^{4} x^{2}-24 b^{2} c \,d^{3} x -12 b^{2} c^{2} d^{2}+24 d^{4}\right ) \sin \left (b x +a \right )}{4 b^{5}}-\frac {d \left (3 b^{2} d^{3} x^{3}+9 b^{2} c \,d^{2} x^{2}+9 b^{2} c^{2} d x +3 b^{2} c^{3}-2 d^{3} x -2 c \,d^{2}\right ) \cos \left (3 b x +3 a \right )}{27 b^{4}}-\frac {\left (27 d^{4} x^{4} b^{4}+108 b^{4} c \,d^{3} x^{3}+162 b^{4} c^{2} d^{2} x^{2}+108 b^{4} c^{3} d x +27 b^{4} c^{4}-36 b^{2} d^{4} x^{2}-72 b^{2} c \,d^{3} x -36 b^{2} c^{2} d^{2}+8 d^{4}\right ) \sin \left (3 b x +3 a \right )}{324 b^{5}}\) | \(343\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(835\) |
| default | \(\text {Expression too large to display}\) | \(835\) |
| orering | \(\frac {16 d \left (135 b^{6} d^{6} x^{6}+810 b^{6} c \,d^{5} x^{5}+2025 b^{6} c^{2} d^{4} x^{4}+2700 b^{6} c^{3} d^{3} x^{3}+2025 b^{6} c^{4} d^{2} x^{2}-891 b^{4} d^{6} x^{4}+810 b^{6} c^{5} d x -3564 b^{4} c \,d^{5} x^{3}+135 b^{6} c^{6}-5346 b^{4} c^{2} d^{4} x^{2}-3564 b^{4} c^{3} d^{3} x -891 b^{4} c^{4} d^{2}-1960 b^{2} d^{6} x^{2}-3920 b^{2} c \,d^{5} x -1960 b^{2} c^{2} d^{4}+5460 d^{6}\right ) \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2}}{243 b^{8} \left (d x +c \right )^{3}}-\frac {2 \left (135 b^{6} d^{6} x^{6}+810 b^{6} c \,d^{5} x^{5}+2025 b^{6} c^{2} d^{4} x^{4}+2700 b^{6} c^{3} d^{3} x^{3}+2025 b^{6} c^{4} d^{2} x^{2}-396 b^{4} d^{6} x^{4}+810 b^{6} c^{5} d x -1584 b^{4} c \,d^{5} x^{3}+135 b^{6} c^{6}-2376 b^{4} c^{2} d^{4} x^{2}-1584 b^{4} c^{3} d^{3} x -396 b^{4} c^{4} d^{2}-10400 b^{2} d^{6} x^{2}-20800 b^{2} c \,d^{5} x -10400 b^{2} c^{2} d^{4}+21840 d^{6}\right ) \left (4 \left (d x +c \right )^{3} \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} d -\left (d x +c \right )^{4} b \sin \left (b x +a \right )^{3}+2 \left (d x +c \right )^{4} \cos \left (b x +a \right )^{2} \sin \left (b x +a \right ) b \right )}{243 b^{8} \left (d x +c \right )^{6}}+\frac {16 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}-105 b^{2} d^{4} x^{2}-210 b^{2} c \,d^{3} x -105 b^{2} c^{2} d^{2}+182 d^{4}\right ) \left (12 \left (d x +c \right )^{2} \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} d^{2}-8 \left (d x +c \right )^{3} b \sin \left (b x +a \right )^{3} d +16 \left (d x +c \right )^{3} \cos \left (b x +a \right )^{2} \sin \left (b x +a \right ) d b -7 \left (d x +c \right )^{4} b^{2} \sin \left (b x +a \right )^{2} \cos \left (b x +a \right )+2 \left (d x +c \right )^{4} \cos \left (b x +a \right )^{3} b^{2}\right )}{81 b^{8} \left (d x +c \right )^{5}}-\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}-360 b^{2} d^{4} x^{2}-720 b^{2} c \,d^{3} x -360 b^{2} c^{2} d^{2}+728 d^{4}\right ) \left (24 \left (d x +c \right ) \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} d^{3}-36 \left (d x +c \right )^{2} b \sin \left (b x +a \right )^{3} d^{2}+72 \left (d x +c \right )^{2} \cos \left (b x +a \right )^{2} \sin \left (b x +a \right ) d^{2} b -84 \left (d x +c \right )^{3} b^{2} \sin \left (b x +a \right )^{2} d \cos \left (b x +a \right )+24 \left (d x +c \right )^{3} \cos \left (b x +a \right )^{3} b^{2} d -20 \left (d x +c \right )^{4} b^{3} \sin \left (b x +a \right ) \cos \left (b x +a \right )^{2}+7 \left (d x +c \right )^{4} b^{3} \sin \left (b x +a \right )^{3}\right )}{243 b^{8} \left (d x +c \right )^{4}}\) | \(965\) |
Input:
int((d*x+c)^4*cos(b*x+a)*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/324*((-27*(d*x+c)^4*b^4+36*d^2*(d*x+c)^2*b^2-8*d^4)*sin(3*b*x+3*a)-36*d* ((d*x+c)^2*b^2-2/3*d^2)*(d*x+c)*b*cos(3*b*x+3*a)+81*((d*x+c)^4*b^4-12*d^2* (d*x+c)^2*b^2+24*d^4)*sin(b*x+a)+324*d*(((d*x+c)^2*b^2-6*d^2)*(d*x+c)*cos( b*x+a)+8/9*b^2*c^3-160/27*c*d^2)*b)/b^5
Time = 0.08 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.72 \[ \int (c+d x)^4 \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {12 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 2 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} - 2 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} - 36 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 14 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} - 14 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right ) - {\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} - 252 \, b^{2} c^{2} d^{2} + 488 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 14 \, b^{2} d^{4}\right )} x^{2} - {\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 2 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 2 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 14 \, b^{2} c d^{3}\right )} x\right )} \sin \left (b x + a\right )}{81 \, b^{5}} \] Input:
integrate((d*x+c)^4*cos(b*x+a)*sin(b*x+a)^2,x, algorithm="fricas")
Output:
-1/81*(12*(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)^3 - 36*(3*b^3*d^4*x^3 + 9*b^3*c*d^3 *x^2 + 3*b^3*c^3*d - 14*b*c*d^3 + (9*b^3*c^2*d^2 - 14*b*d^4)*x)*cos(b*x + a) - (27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 27*b^4*c^4 - 252*b^2*c^2*d^2 + 488*d^4 + 18*(9*b^4*c^2*d^2 - 14*b^2*d^4)*x^2 - (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)^2 + 36*(3*b^4* c^3*d - 14*b^2*c*d^3)*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 \[ \int (c+d x)^4 \cos (a+b x) \sin ^2(a+b x) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)**4*cos(b*x+a)*sin(b*x+a)**2,x)
Output:
Piecewise((c**4*sin(a + b*x)**3/(3*b) + 4*c**3*d*x*sin(a + b*x)**3/(3*b) + 2*c**2*d**2*x**2*sin(a + b*x)**3/b + 4*c*d**3*x**3*sin(a + b*x)**3/(3*b) + d**4*x**4*sin(a + b*x)**3/(3*b) + 4*c**3*d*sin(a + b*x)**2*cos(a + b*x)/ (3*b**2) + 8*c**3*d*cos(a + b*x)**3/(9*b**2) + 4*c**2*d**2*x*sin(a + b*x)* *2*cos(a + b*x)/b**2 + 8*c**2*d**2*x*cos(a + b*x)**3/(3*b**2) + 4*c*d**3*x **2*sin(a + b*x)**2*cos(a + b*x)/b**2 + 8*c*d**3*x**2*cos(a + b*x)**3/(3*b **2) + 4*d**4*x**3*sin(a + b*x)**2*cos(a + b*x)/(3*b**2) + 8*d**4*x**3*cos (a + b*x)**3/(9*b**2) - 28*c**2*d**2*sin(a + b*x)**3/(9*b**3) - 8*c**2*d** 2*sin(a + b*x)*cos(a + b*x)**2/(3*b**3) - 56*c*d**3*x*sin(a + b*x)**3/(9*b **3) - 16*c*d**3*x*sin(a + b*x)*cos(a + b*x)**2/(3*b**3) - 28*d**4*x**2*si n(a + b*x)**3/(9*b**3) - 8*d**4*x**2*sin(a + b*x)*cos(a + b*x)**2/(3*b**3) - 56*c*d**3*sin(a + b*x)**2*cos(a + b*x)/(9*b**4) - 160*c*d**3*cos(a + b* x)**3/(27*b**4) - 56*d**4*x*sin(a + b*x)**2*cos(a + b*x)/(9*b**4) - 160*d* *4*x*cos(a + b*x)**3/(27*b**4) + 488*d**4*sin(a + b*x)**3/(81*b**5) + 160* d**4*sin(a + b*x)*cos(a + b*x)**2/(27*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)**2*cos(a), T rue))
Leaf count of result is larger than twice the leaf count of optimal. 880 vs. \(2 (187) = 374\).
Time = 0.07 (sec) , antiderivative size = 880, normalized size of antiderivative = 4.29 \[ \int (c+d x)^4 \cos (a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^4*cos(b*x+a)*sin(b*x+a)^2,x, algorithm="maxima")
Output:
1/324*(108*c^4*sin(b*x + a)^3 - 432*a*c^3*d*sin(b*x + a)^3/b + 648*a^2*c^2 *d^2*sin(b*x + a)^3/b^2 - 432*a^3*c*d^3*sin(b*x + a)^3/b^3 + 108*a^4*d^4*s in(b*x + a)^3/b^4 - 36*(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^3*d/b + 108*(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^2*d^2/b^2 - 108*(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*c*d^3/b^3 + 36*(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^3*d^4/b^4 - 18*(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^2*d^2/b^2 + 36*(6*(b*x + a)*cos(3*b*x + 3*a) - 54*(b*x + a)*c os(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*c*d^3/b^3 - 18*(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^2*d^4/b^4 - 12*((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) - 27*((b*x + a)^3 - 6*b*x - 6*a)*sin(b*x + a))*c*d^3/b^3 + 12*( (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) - 27*((b*x + a)^3 - 6*b* x - 6*a)*sin(b*x + a))*a*d^4/b^4 - (12*(3*(b*x + a)^3 - 2*b*x - 2*a)*co...
Time = 0.14 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.71 \[ \int (c+d x)^4 \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 9 \, b^{3} c^{2} d^{2} x + 3 \, b^{3} c^{3} d - 2 \, b d^{4} x - 2 \, b c d^{3}\right )} \cos \left (3 \, b x + 3 \, a\right )}{27 \, b^{5}} + \frac {{\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \cos \left (b x + a\right )}{b^{5}} - \frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 108 \, b^{4} c^{3} d x + 27 \, b^{4} c^{4} - 36 \, b^{2} d^{4} x^{2} - 72 \, b^{2} c d^{3} x - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4}\right )} \sin \left (3 \, b x + 3 \, a\right )}{324 \, b^{5}} + \frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4} - 12 \, b^{2} d^{4} x^{2} - 24 \, b^{2} c d^{3} x - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \sin \left (b x + a\right )}{4 \, b^{5}} \] Input:
integrate((d*x+c)^4*cos(b*x+a)*sin(b*x+a)^2,x, algorithm="giac")
Output:
-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)*cos(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)*cos(b*x + a)/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)*sin(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)*sin(b*x + a)/b^5
Time = 18.52 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.19 \[ \int (c+d x)^4 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {{\sin \left (a+b\,x\right )}^3\,\left (27\,b^4\,c^4-252\,b^2\,c^2\,d^2+488\,d^4\right )}{81\,b^5}-\frac {8\,{\cos \left (a+b\,x\right )}^3\,\left (20\,c\,d^3-3\,b^2\,c^3\,d\right )}{27\,b^4}+\frac {8\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (20\,d^4-9\,b^2\,c^2\,d^2\right )}{27\,b^5}-\frac {4\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (14\,c\,d^3-3\,b^2\,c^3\,d\right )}{9\,b^4}+\frac {8\,d^4\,x^3\,{\cos \left (a+b\,x\right )}^3}{9\,b^2}-\frac {8\,x\,{\cos \left (a+b\,x\right )}^3\,\left (20\,d^4-9\,b^2\,c^2\,d^2\right )}{27\,b^4}+\frac {d^4\,x^4\,{\sin \left (a+b\,x\right )}^3}{3\,b}-\frac {4\,x\,{\sin \left (a+b\,x\right )}^3\,\left (14\,c\,d^3-3\,b^2\,c^3\,d\right )}{9\,b^3}-\frac {2\,x^2\,{\sin \left (a+b\,x\right )}^3\,\left (14\,d^4-9\,b^2\,c^2\,d^2\right )}{9\,b^3}+\frac {8\,c\,d^3\,x^2\,{\cos \left (a+b\,x\right )}^3}{3\,b^2}+\frac {4\,d^4\,x^3\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{3\,b^2}-\frac {8\,d^4\,x^2\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{3\,b^3}+\frac {4\,c\,d^3\,x^3\,{\sin \left (a+b\,x\right )}^3}{3\,b}-\frac {4\,x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (14\,d^4-9\,b^2\,c^2\,d^2\right )}{9\,b^4}+\frac {4\,c\,d^3\,x^2\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{b^2}-\frac {16\,c\,d^3\,x\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{3\,b^3} \] Input:
int(cos(a + b*x)*sin(a + b*x)^2*(c + d*x)^4,x)
Output:
(sin(a + b*x)^3*(488*d^4 + 27*b^4*c^4 - 252*b^2*c^2*d^2))/(81*b^5) - (8*co s(a + b*x)^3*(20*c*d^3 - 3*b^2*c^3*d))/(27*b^4) + (8*cos(a + b*x)^2*sin(a + b*x)*(20*d^4 - 9*b^2*c^2*d^2))/(27*b^5) - (4*cos(a + b*x)*sin(a + b*x)^2 *(14*c*d^3 - 3*b^2*c^3*d))/(9*b^4) + (8*d^4*x^3*cos(a + b*x)^3)/(9*b^2) - (8*x*cos(a + b*x)^3*(20*d^4 - 9*b^2*c^2*d^2))/(27*b^4) + (d^4*x^4*sin(a + b*x)^3)/(3*b) - (4*x*sin(a + b*x)^3*(14*c*d^3 - 3*b^2*c^3*d))/(9*b^3) - (2 *x^2*sin(a + b*x)^3*(14*d^4 - 9*b^2*c^2*d^2))/(9*b^3) + (8*c*d^3*x^2*cos(a + b*x)^3)/(3*b^2) + (4*d^4*x^3*cos(a + b*x)*sin(a + b*x)^2)/(3*b^2) - (8* d^4*x^2*cos(a + b*x)^2*sin(a + b*x))/(3*b^3) + (4*c*d^3*x^3*sin(a + b*x)^3 )/(3*b) - (4*x*cos(a + b*x)*sin(a + b*x)^2*(14*d^4 - 9*b^2*c^2*d^2))/(9*b^ 4) + (4*c*d^3*x^2*cos(a + b*x)*sin(a + b*x)^2)/b^2 - (16*c*d^3*x*cos(a + b *x)^2*sin(a + b*x))/(3*b^3)
Time = 0.18 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.44 \[ \int (c+d x)^4 \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {-480 \cos \left (b x +a \right ) b \,d^{4} x +27 \sin \left (b x +a \right )^{3} b^{4} d^{4} x^{4}-36 \sin \left (b x +a \right )^{3} b^{2} c^{2} d^{2}-36 \sin \left (b x +a \right )^{3} b^{2} d^{4} x^{2}-216 \sin \left (b x +a \right ) b^{2} c^{2} d^{2}-216 \sin \left (b x +a \right ) b^{2} d^{4} x^{2}-216 a \,b^{2} c^{2} d^{2}+72 \cos \left (b x +a \right ) b^{3} c^{3} d +72 \cos \left (b x +a \right ) b^{3} d^{4} x^{3}-480 \cos \left (b x +a \right ) b c \,d^{3}+27 \sin \left (b x +a \right )^{3} b^{4} c^{4}+72 b^{3} c^{3} d -192 b c \,d^{3}+108 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{3} c^{2} d^{2} x +108 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{3} c \,d^{3} x^{2}+36 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{3} c^{3} d +36 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{3} d^{4} x^{3}-24 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b c \,d^{3}-24 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b \,d^{4} x +216 \cos \left (b x +a \right ) b^{3} c^{2} d^{2} x +216 \cos \left (b x +a \right ) b^{3} c \,d^{3} x^{2}+108 \sin \left (b x +a \right )^{3} b^{4} c^{3} d x +162 \sin \left (b x +a \right )^{3} b^{4} c^{2} d^{2} x^{2}+108 \sin \left (b x +a \right )^{3} b^{4} c \,d^{3} x^{3}-72 \sin \left (b x +a \right )^{3} b^{2} c \,d^{3} x -432 \sin \left (b x +a \right ) b^{2} c \,d^{3} x +8 \sin \left (b x +a \right )^{3} d^{4}+480 \sin \left (b x +a \right ) d^{4}+192 a \,d^{4}}{81 b^{5}} \] Input:
int((d*x+c)^4*cos(b*x+a)*sin(b*x+a)^2,x)
Output:
(36*cos(a + b*x)*sin(a + b*x)**2*b**3*c**3*d + 108*cos(a + b*x)*sin(a + b* x)**2*b**3*c**2*d**2*x + 108*cos(a + b*x)*sin(a + b*x)**2*b**3*c*d**3*x**2 + 36*cos(a + b*x)*sin(a + b*x)**2*b**3*d**4*x**3 - 24*cos(a + b*x)*sin(a + b*x)**2*b*c*d**3 - 24*cos(a + b*x)*sin(a + b*x)**2*b*d**4*x + 72*cos(a + b*x)*b**3*c**3*d + 216*cos(a + b*x)*b**3*c**2*d**2*x + 216*cos(a + b*x)*b **3*c*d**3*x**2 + 72*cos(a + b*x)*b**3*d**4*x**3 - 480*cos(a + b*x)*b*c*d* *3 - 480*cos(a + b*x)*b*d**4*x + 27*sin(a + b*x)**3*b**4*c**4 + 108*sin(a + b*x)**3*b**4*c**3*d*x + 162*sin(a + b*x)**3*b**4*c**2*d**2*x**2 + 108*si n(a + b*x)**3*b**4*c*d**3*x**3 + 27*sin(a + b*x)**3*b**4*d**4*x**4 - 36*si n(a + b*x)**3*b**2*c**2*d**2 - 72*sin(a + b*x)**3*b**2*c*d**3*x - 36*sin(a + b*x)**3*b**2*d**4*x**2 + 8*sin(a + b*x)**3*d**4 - 216*sin(a + b*x)*b**2 *c**2*d**2 - 432*sin(a + b*x)*b**2*c*d**3*x - 216*sin(a + b*x)*b**2*d**4*x **2 + 480*sin(a + b*x)*d**4 - 216*a*b**2*c**2*d**2 + 192*a*d**4 + 72*b**3* c**3*d - 192*b*c*d**3)/(81*b**5)