Integrand size = 20, antiderivative size = 120 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=\frac {3 d^3 x}{8 b^3}-\frac {(c+d x)^3}{4 b}-\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}+\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b} \] Output:
3/8*d^3*x/b^3-1/4*(d*x+c)^3/b-3/8*d^3*cos(b*x+a)*sin(b*x+a)/b^4+3/4*d*(d*x +c)^2*cos(b*x+a)*sin(b*x+a)/b^2-3/4*d^2*(d*x+c)*sin(b*x+a)^2/b^3+1/2*(d*x+ c)^3*sin(b*x+a)^2/b
Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.59 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=\frac {-2 b (c+d x) \left (-3 d^2+2 b^2 (c+d x)^2\right ) \cos (2 (a+b x))+3 d \left (-d^2+2 b^2 (c+d x)^2\right ) \sin (2 (a+b x))}{16 b^4} \] Input:
Integrate[(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x],x]
Output:
(-2*b*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + 3*d*(-d^2 + 2*b^2*(c + d*x)^2)*Sin[2*(a + b*x)])/(16*b^4)
Time = 0.39 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {4904, 3042, 3792, 17, 3042, 3115, 24}
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.
| \(\displaystyle \int (c+d x)^3 \sin (a+b x) \cos (a+b x) \, dx\) |
| \(\Big \downarrow \) 4904 |
| \(\displaystyle \frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {3 d \int (c+d x)^2 \sin ^2(a+b x)dx}{2 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {3 d \int (c+d x)^2 \sin (a+b x)^2dx}{2 b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {3 d \left (-\frac {d^2 \int \sin ^2(a+b x)dx}{2 b^2}+\frac {1}{2} \int (c+d x)^2dx+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {3 d \left (-\frac {d^2 \int \sin ^2(a+b x)dx}{2 b^2}+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {3 d \left (-\frac {d^2 \int \sin (a+b x)^2dx}{2 b^2}+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {3 d \left (-\frac {d^2 \left (\frac {\int 1dx}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}-\frac {3 d \left (\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {d^2 \left (\frac {x}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}\) |
Input:
Int[(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x],x]
Output:
((c + d*x)^3*Sin[a + b*x]^2)/(2*b) - (3*d*((c + d*x)^3/(6*d) - ((c + d*x)^ 2*Cos[a + b*x]*Sin[a + b*x])/(2*b) + (d*(c + d*x)*Sin[a + b*x]^2)/(2*b^2) - (d^2*(x/2 - (Cos[a + b*x]*Sin[a + b*x])/(2*b)))/(2*b^2)))/(2*b)
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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 = 0.58 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.71
| method | result | size |
| parallelrisch | \(\frac {-2 \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{2}\right ) \left (d x +c \right ) b \cos \left (2 b x +2 a \right )+3 d \left (\left (d x +c \right )^{2} b^{2}-\frac {d^{2}}{2}\right ) \sin \left (2 b x +2 a \right )+2 b^{3} c^{3}-3 c \,d^{2} b}{8 b^{4}}\) | \(85\) |
| risch | \(-\frac {\left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \cos \left (2 b x +2 a \right )}{8 b^{3}}+\frac {3 d \left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}-d^{2}\right ) \sin \left (2 b x +2 a \right )}{16 b^{4}}\) | \(118\) |
| orering | \(\frac {3 d \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-d^{2}\right ) \cos \left (b x +a \right ) \sin \left (b x +a \right )}{2 b^{4}}-\frac {\left (2 x^{2} d^{2} b^{2}+4 b^{2} c d x +2 b^{2} c^{2}-3 d^{2}\right ) \left (3 \left (d x +c \right )^{2} \cos \left (b x +a \right ) \sin \left (b x +a \right ) d -\left (d x +c \right )^{3} b \sin \left (b x +a \right )^{2}+\left (d x +c \right )^{3} \cos \left (b x +a \right )^{2} b \right )}{8 \left (d x +c \right )^{2} b^{4}}\) | \(154\) |
| norman | \(\frac {\frac {\left (2 b^{2} c^{3}-3 c \,d^{2}\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}}{b^{3}}-\frac {d^{3} x^{3}}{4 b}-\frac {3 c \,d^{2} x^{2}}{4 b}+\frac {3 d \left (2 b^{2} c^{2}-d^{2}\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )}{4 b^{4}}-\frac {3 d \left (2 b^{2} c^{2}-d^{2}\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{3}}{4 b^{4}}-\frac {3 d \left (2 b^{2} c^{2}-d^{2}\right ) x}{8 b^{3}}+\frac {3 d^{3} x^{2} \tan \left (\frac {a}{2}+\frac {b x}{2}\right )}{2 b^{2}}-\frac {3 d^{3} x^{2} \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{3}}{2 b^{2}}+\frac {3 d^{3} x^{3} \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}}{2 b}-\frac {d^{3} x^{3} \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{4}}{4 b}+\frac {3 c \,d^{2} x \tan \left (\frac {a}{2}+\frac {b x}{2}\right )}{b^{2}}-\frac {3 c \,d^{2} x \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{3}}{b^{2}}+\frac {9 c \,d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}}{2 b}-\frac {3 c \,d^{2} x^{2} \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{4}}{4 b}+\frac {9 d \left (2 b^{2} c^{2}-d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}}{4 b^{3}}-\frac {3 d \left (2 b^{2} c^{2}-d^{2}\right ) x \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{4}}{8 b^{3}}}{\left (1+\tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}\right )^{2}}\) | \(388\) |
| derivativedivides | \(\frac {\frac {a^{3} d^{3} \cos \left (b x +a \right )^{2}}{2 b^{3}}-\frac {3 a^{2} c \,d^{2} \cos \left (b x +a \right )^{2}}{2 b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{2}}{2}+\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{3}}+\frac {3 a \,c^{2} d \cos \left (b x +a \right )^{2}}{2 b}-\frac {6 a c \,d^{2} \left (-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{2}}{2}+\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (-\frac {\left (b x +a \right )^{2} \cos \left (b x +a \right )^{2}}{2}+\left (b x +a \right ) \left (\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\sin \left (b x +a \right )^{2}}{4}\right )}{b^{3}}-\frac {c^{3} \cos \left (b x +a \right )^{2}}{2}+\frac {3 c^{2} d \left (-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{2}}{2}+\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b}+\frac {3 c \,d^{2} \left (-\frac {\left (b x +a \right )^{2} \cos \left (b x +a \right )^{2}}{2}+\left (b x +a \right ) \left (\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\sin \left (b x +a \right )^{2}}{4}\right )}{b^{2}}+\frac {d^{3} \left (-\frac {\left (b x +a \right )^{3} \cos \left (b x +a \right )^{2}}{2}+\frac {3 \left (b x +a \right )^{2} \left (\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{2}+\frac {3 \left (b x +a \right ) \cos \left (b x +a \right )^{2}}{4}-\frac {3 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{8}-\frac {3 b x}{8}-\frac {3 a}{8}-\frac {\left (b x +a \right )^{3}}{2}\right )}{b^{3}}}{b}\) | \(466\) |
| default | \(\frac {\frac {a^{3} d^{3} \cos \left (b x +a \right )^{2}}{2 b^{3}}-\frac {3 a^{2} c \,d^{2} \cos \left (b x +a \right )^{2}}{2 b^{2}}+\frac {3 a^{2} d^{3} \left (-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{2}}{2}+\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{3}}+\frac {3 a \,c^{2} d \cos \left (b x +a \right )^{2}}{2 b}-\frac {6 a c \,d^{2} \left (-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{2}}{2}+\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b^{2}}-\frac {3 a \,d^{3} \left (-\frac {\left (b x +a \right )^{2} \cos \left (b x +a \right )^{2}}{2}+\left (b x +a \right ) \left (\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\sin \left (b x +a \right )^{2}}{4}\right )}{b^{3}}-\frac {c^{3} \cos \left (b x +a \right )^{2}}{2}+\frac {3 c^{2} d \left (-\frac {\left (b x +a \right ) \cos \left (b x +a \right )^{2}}{2}+\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{4}+\frac {b x}{4}+\frac {a}{4}\right )}{b}+\frac {3 c \,d^{2} \left (-\frac {\left (b x +a \right )^{2} \cos \left (b x +a \right )^{2}}{2}+\left (b x +a \right ) \left (\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\sin \left (b x +a \right )^{2}}{4}\right )}{b^{2}}+\frac {d^{3} \left (-\frac {\left (b x +a \right )^{3} \cos \left (b x +a \right )^{2}}{2}+\frac {3 \left (b x +a \right )^{2} \left (\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{2}+\frac {3 \left (b x +a \right ) \cos \left (b x +a \right )^{2}}{4}-\frac {3 \sin \left (b x +a \right ) \cos \left (b x +a \right )}{8}-\frac {3 b x}{8}-\frac {3 a}{8}-\frac {\left (b x +a \right )^{3}}{2}\right )}{b^{3}}}{b}\) | \(466\) |
Input:
int((d*x+c)^3*cos(b*x+a)*sin(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/8*(-2*((d*x+c)^2*b^2-3/2*d^2)*(d*x+c)*b*cos(2*b*x+2*a)+3*d*((d*x+c)^2*b^ 2-1/2*d^2)*sin(2*b*x+2*a)+2*b^3*c^3-3*c*d^2*b)/b^4
Time = 0.07 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.38 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=\frac {2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} - 2 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 2 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (2 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 3 \, {\left (2 \, b^{3} c^{2} d - b d^{3}\right )} x}{8 \, b^{4}} \] Input:
integrate((d*x+c)^3*cos(b*x+a)*sin(b*x+a),x, algorithm="fricas")
Output:
1/8*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 - 2*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 2*b^3*c^3 - 3*b*c*d^2 + 3*(2*b^3*c^2*d - b*d^3)*x)*cos(b*x + a)^2 + 3*(2 *b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d - d^3)*cos(b*x + a)*sin(b*x + a ) + 3*(2*b^3*c^2*d - b*d^3)*x)/b^4
Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (116) = 232\).
Time = 0.36 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.85 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=\begin {cases} \frac {c^{3} \sin ^{2}{\left (a + b x \right )}}{2 b} + \frac {3 c^{2} d x \sin ^{2}{\left (a + b x \right )}}{4 b} - \frac {3 c^{2} d x \cos ^{2}{\left (a + b x \right )}}{4 b} + \frac {3 c d^{2} x^{2} \sin ^{2}{\left (a + b x \right )}}{4 b} - \frac {3 c d^{2} x^{2} \cos ^{2}{\left (a + b x \right )}}{4 b} + \frac {d^{3} x^{3} \sin ^{2}{\left (a + b x \right )}}{4 b} - \frac {d^{3} x^{3} \cos ^{2}{\left (a + b x \right )}}{4 b} + \frac {3 c^{2} d \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b^{2}} + \frac {3 c d^{2} x \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{2 b^{2}} + \frac {3 d^{3} x^{2} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{4 b^{2}} - \frac {3 c d^{2} \sin ^{2}{\left (a + b x \right )}}{4 b^{3}} - \frac {3 d^{3} x \sin ^{2}{\left (a + b x \right )}}{8 b^{3}} + \frac {3 d^{3} x \cos ^{2}{\left (a + b x \right )}}{8 b^{3}} - \frac {3 d^{3} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sin {\left (a \right )} \cos {\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**3*cos(b*x+a)*sin(b*x+a),x)
Output:
Piecewise((c**3*sin(a + b*x)**2/(2*b) + 3*c**2*d*x*sin(a + b*x)**2/(4*b) - 3*c**2*d*x*cos(a + b*x)**2/(4*b) + 3*c*d**2*x**2*sin(a + b*x)**2/(4*b) - 3*c*d**2*x**2*cos(a + b*x)**2/(4*b) + d**3*x**3*sin(a + b*x)**2/(4*b) - d* *3*x**3*cos(a + b*x)**2/(4*b) + 3*c**2*d*sin(a + b*x)*cos(a + b*x)/(4*b**2 ) + 3*c*d**2*x*sin(a + b*x)*cos(a + b*x)/(2*b**2) + 3*d**3*x**2*sin(a + b* x)*cos(a + b*x)/(4*b**2) - 3*c*d**2*sin(a + b*x)**2/(4*b**3) - 3*d**3*x*si n(a + b*x)**2/(8*b**3) + 3*d**3*x*cos(a + b*x)**2/(8*b**3) - 3*d**3*sin(a + b*x)*cos(a + b*x)/(8*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)*cos(a), True))
Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (108) = 216\).
Time = 0.05 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.85 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=-\frac {8 \, c^{3} \cos \left (b x + a\right )^{2} - \frac {24 \, a c^{2} d \cos \left (b x + a\right )^{2}}{b} + \frac {24 \, a^{2} c d^{2} \cos \left (b x + a\right )^{2}}{b^{2}} - \frac {8 \, a^{3} d^{3} \cos \left (b x + a\right )^{2}}{b^{3}} + \frac {6 \, {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} c^{2} d}{b} - \frac {12 \, {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} a c d^{2}}{b^{2}} + \frac {6 \, {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} a^{2} d^{3}}{b^{3}} + \frac {6 \, {\left ({\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c d^{2}}{b^{2}} - \frac {6 \, {\left ({\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) - 2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} a d^{3}}{b^{3}} + \frac {{\left (2 \, {\left (2 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, {\left (2 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{3}}{b^{3}}}{16 \, b} \] Input:
integrate((d*x+c)^3*cos(b*x+a)*sin(b*x+a),x, algorithm="maxima")
Output:
-1/16*(8*c^3*cos(b*x + a)^2 - 24*a*c^2*d*cos(b*x + a)^2/b + 24*a^2*c*d^2*c os(b*x + a)^2/b^2 - 8*a^3*d^3*cos(b*x + a)^2/b^3 + 6*(2*(b*x + a)*cos(2*b* x + 2*a) - sin(2*b*x + 2*a))*c^2*d/b - 12*(2*(b*x + a)*cos(2*b*x + 2*a) - sin(2*b*x + 2*a))*a*c*d^2/b^2 + 6*(2*(b*x + a)*cos(2*b*x + 2*a) - sin(2*b* x + 2*a))*a^2*d^3/b^3 + 6*((2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 2*(b*x + a)*sin(2*b*x + 2*a))*c*d^2/b^2 - 6*((2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 2*(b*x + a)*sin(2*b*x + 2*a))*a*d^3/b^3 + (2*(2*(b*x + a)^3 - 3*b*x - 3* a)*cos(2*b*x + 2*a) - 3*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*a))*d^3/b^3)/b
Time = 0.15 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.01 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=-\frac {{\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 6 \, b^{3} c^{2} d x + 2 \, b^{3} c^{3} - 3 \, b d^{3} x - 3 \, b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )}{8 \, b^{4}} + \frac {3 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{16 \, b^{4}} \] Input:
integrate((d*x+c)^3*cos(b*x+a)*sin(b*x+a),x, algorithm="giac")
Output:
-1/8*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 6*b^3*c^2*d*x + 2*b^3*c^3 - 3*b*d^ 3*x - 3*b*c*d^2)*cos(2*b*x + 2*a)/b^4 + 3/16*(2*b^2*d^3*x^2 + 4*b^2*c*d^2* x + 2*b^2*c^2*d - d^3)*sin(2*b*x + 2*a)/b^4
Time = 19.23 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.38 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=\frac {\cos \left (2\,a+2\,b\,x\right )\,\left (\frac {3\,c\,d^2}{4}-\frac {b^2\,c^3}{2}\right )}{2\,b^3}-\frac {3\,\sin \left (2\,a+2\,b\,x\right )\,\left (d^3-2\,b^2\,c^2\,d\right )}{16\,b^4}-\frac {d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )}{4\,b}+\frac {3\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )}{8\,b^2}+\frac {3\,x\,\cos \left (2\,a+2\,b\,x\right )\,\left (d^3-2\,b^2\,c^2\,d\right )}{8\,b^3}+\frac {3\,c\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )}{4\,b^2}-\frac {3\,c\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )}{4\,b} \] Input:
int(cos(a + b*x)*sin(a + b*x)*(c + d*x)^3,x)
Output:
(cos(2*a + 2*b*x)*((3*c*d^2)/4 - (b^2*c^3)/2))/(2*b^3) - (3*sin(2*a + 2*b* x)*(d^3 - 2*b^2*c^2*d))/(16*b^4) - (d^3*x^3*cos(2*a + 2*b*x))/(4*b) + (3*d ^3*x^2*sin(2*a + 2*b*x))/(8*b^2) + (3*x*cos(2*a + 2*b*x)*(d^3 - 2*b^2*c^2* d))/(8*b^3) + (3*c*d^2*x*sin(2*a + 2*b*x))/(4*b^2) - (3*c*d^2*x^2*cos(2*a + 2*b*x))/(4*b)
Time = 0.16 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.20 \[ \int (c+d x)^3 \cos (a+b x) \sin (a+b x) \, dx=\frac {-4 \cos \left (b x +a \right )^{2} b^{3} c^{3}-6 \cos \left (b x +a \right )^{2} b^{3} c^{2} d x -6 \cos \left (b x +a \right )^{2} b^{3} c \,d^{2} x^{2}-2 \cos \left (b x +a \right )^{2} b^{3} d^{3} x^{3}+6 \cos \left (b x +a \right )^{2} b c \,d^{2}+3 \cos \left (b x +a \right )^{2} b \,d^{3} x +6 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{2} c^{2} d +12 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{2} c \,d^{2} x +6 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{2} d^{3} x^{2}-3 \cos \left (b x +a \right ) \sin \left (b x +a \right ) d^{3}+6 \sin \left (b x +a \right )^{2} b^{3} c^{2} d x +6 \sin \left (b x +a \right )^{2} b^{3} c \,d^{2} x^{2}+2 \sin \left (b x +a \right )^{2} b^{3} d^{3} x^{3}-3 \sin \left (b x +a \right )^{2} b \,d^{3} x}{8 b^{4}} \] Input:
int((d*x+c)^3*cos(b*x+a)*sin(b*x+a),x)
Output:
( - 4*cos(a + b*x)**2*b**3*c**3 - 6*cos(a + b*x)**2*b**3*c**2*d*x - 6*cos( a + b*x)**2*b**3*c*d**2*x**2 - 2*cos(a + b*x)**2*b**3*d**3*x**3 + 6*cos(a + b*x)**2*b*c*d**2 + 3*cos(a + b*x)**2*b*d**3*x + 6*cos(a + b*x)*sin(a + b *x)*b**2*c**2*d + 12*cos(a + b*x)*sin(a + b*x)*b**2*c*d**2*x + 6*cos(a + b *x)*sin(a + b*x)*b**2*d**3*x**2 - 3*cos(a + b*x)*sin(a + b*x)*d**3 + 6*sin (a + b*x)**2*b**3*c**2*d*x + 6*sin(a + b*x)**2*b**3*c*d**2*x**2 + 2*sin(a + b*x)**2*b**3*d**3*x**3 - 3*sin(a + b*x)**2*b*d**3*x)/(8*b**4)