Integrand size = 16, antiderivative size = 161 \[ \int (c+d x)^4 \sin ^2(a+b x) \, dx=\frac {3 d^4 x}{4 b^4}-\frac {d (c+d x)^3}{2 b^2}+\frac {(c+d x)^5}{10 d}-\frac {3 d^4 \cos (a+b x) \sin (a+b x)}{4 b^5}+\frac {3 d^2 (c+d x)^2 \cos (a+b x) \sin (a+b x)}{2 b^3}-\frac {(c+d x)^4 \cos (a+b x) \sin (a+b x)}{2 b}-\frac {3 d^3 (c+d x) \sin ^2(a+b x)}{2 b^4}+\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2} \] Output:
3/4*d^4*x/b^4-1/2*d*(d*x+c)^3/b^2+1/10*(d*x+c)^5/d-3/4*d^4*cos(b*x+a)*sin( b*x+a)/b^5+3/2*d^2*(d*x+c)^2*cos(b*x+a)*sin(b*x+a)/b^3-1/2*(d*x+c)^4*cos(b *x+a)*sin(b*x+a)/b-3/2*d^3*(d*x+c)*sin(b*x+a)^2/b^4+d*(d*x+c)^3*sin(b*x+a) ^2/b^2
Time = 0.62 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.82 \[ \int (c+d x)^4 \sin ^2(a+b x) \, dx=\frac {8 b^5 x \left (5 c^4+10 c^3 d x+10 c^2 d^2 x^2+5 c d^3 x^3+d^4 x^4\right )-20 b d (c+d x) \left (-3 d^2+2 b^2 (c+d x)^2\right ) \cos (2 (a+b x))-10 \left (3 d^4-6 b^2 d^2 (c+d x)^2+2 b^4 (c+d x)^4\right ) \sin (2 (a+b x))}{80 b^5} \] Input:
Integrate[(c + d*x)^4*Sin[a + b*x]^2,x]
Output:
(8*b^5*x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4) - 2 0*b*d*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] - 10*(3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Sin[2*(a + b*x)])/(80*b^5)
Time = 0.50 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {3042, 3792, 17, 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.
| \(\displaystyle \int (c+d x)^4 \sin ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^4 \sin (a+b x)^2dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -\frac {3 d^2 \int (c+d x)^2 \sin ^2(a+b x)dx}{b^2}+\frac {1}{2} \int (c+d x)^4dx+\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}-\frac {(c+d x)^4 \sin (a+b x) \cos (a+b x)}{2 b}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {3 d^2 \int (c+d x)^2 \sin ^2(a+b x)dx}{b^2}+\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}-\frac {(c+d x)^4 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^5}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 d^2 \int (c+d x)^2 \sin (a+b x)^2dx}{b^2}+\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}-\frac {(c+d x)^4 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^5}{10 d}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -\frac {3 d^2 \left (-\frac {d^2 \int \sin ^2(a+b x)dx}{2 b^2}+\frac {1}{2} \int (c+d x)^2dx+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}\right )}{b^2}+\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}-\frac {(c+d x)^4 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^5}{10 d}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {3 d^2 \left (-\frac {d^2 \int \sin ^2(a+b x)dx}{2 b^2}+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{b^2}+\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}-\frac {(c+d x)^4 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^5}{10 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 d^2 \left (-\frac {d^2 \int \sin (a+b x)^2dx}{2 b^2}+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{b^2}+\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}-\frac {(c+d x)^4 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^5}{10 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {3 d^2 \left (-\frac {d^2 \left (\frac {\int 1dx}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{b^2}+\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}-\frac {(c+d x)^4 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^5}{10 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {3 d^2 \left (\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {d^2 \left (\frac {x}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{b^2}+\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}-\frac {(c+d x)^4 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^5}{10 d}\) |
Input:
Int[(c + d*x)^4*Sin[a + b*x]^2,x]
Output:
(c + d*x)^5/(10*d) - ((c + d*x)^4*Cos[a + b*x]*Sin[a + b*x])/(2*b) + (d*(c + d*x)^3*Sin[a + b*x]^2)/b^2 - (3*d^2*((c + d*x)^3/(6*d) - ((c + d*x)^2*C os[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)))/b^2
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]
Time = 1.59 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.90
| method | result | size |
| parallelrisch | \(\frac {\left (-2 \left (d x +c \right )^{4} b^{4}+6 d^{2} \left (d x +c \right )^{2} b^{2}-3 d^{4}\right ) \sin \left (2 b x +2 a \right )+4 b \left (-\left (d x +c \right ) d \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{2}\right ) \cos \left (2 b x +2 a \right )+x \left (\frac {1}{5} d^{4} x^{4}+c \,d^{3} x^{3}+2 c^{2} d^{2} x^{2}+2 c^{3} d x +c^{4}\right ) b^{4}+b^{2} c^{3} d -\frac {3 d^{3} c}{2}\right )}{8 b^{5}}\) | \(145\) |
| risch | \(\frac {d^{4} x^{5}}{10}+\frac {d^{3} c \,x^{4}}{2}+d^{2} c^{2} x^{3}+d \,c^{3} x^{2}+\frac {c^{4} x}{2}+\frac {c^{5}}{10 d}-\frac {d \left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \cos \left (2 b x +2 a \right )}{4 b^{4}}-\frac {\left (2 d^{4} x^{4} b^{4}+8 b^{4} c \,d^{3} x^{3}+12 b^{4} c^{2} d^{2} x^{2}+8 b^{4} c^{3} d x +2 b^{4} c^{4}-6 b^{2} d^{4} x^{2}-12 b^{2} c \,d^{3} x -6 b^{2} c^{2} d^{2}+3 d^{4}\right ) \sin \left (2 b x +2 a \right )}{8 b^{5}}\) | \(227\) |
| orering | \(\frac {\left (2 b^{6} d^{6} x^{7}+14 b^{6} c \,d^{5} x^{6}+42 b^{6} c^{2} d^{4} x^{5}+70 b^{6} c^{3} d^{3} x^{4}+70 b^{6} c^{4} d^{2} x^{3}+20 b^{4} d^{6} x^{5}+40 b^{6} c^{5} d \,x^{2}+100 b^{4} c \,d^{5} x^{4}+10 b^{6} c^{6} x +200 b^{4} c^{2} d^{4} x^{3}+210 b^{4} c^{3} d^{3} x^{2}+120 b^{4} c^{4} d^{2} x -80 b^{2} d^{6} x^{3}+20 b^{4} c^{5} d -255 b^{2} c \,d^{5} x^{2}-270 b^{2} c^{2} d^{4} x -45 b^{2} c^{3} d^{3}+90 d^{6} x +15 c \,d^{5}\right ) \sin \left (b x +a \right )^{2}}{10 b^{6} \left (d x +c \right )^{2}}-\frac {\left (26 b^{4} d^{5} x^{5}+130 b^{4} c \,d^{4} x^{4}+260 b^{4} c^{2} d^{3} x^{3}+260 b^{4} c^{3} d^{2} x^{2}+130 b^{4} c^{4} d x -110 b^{2} d^{5} x^{3}+10 b^{4} c^{5}-330 b^{2} c \,d^{4} x^{2}-330 b^{2} c^{2} d^{3} x -30 b^{2} c^{3} d^{2}+135 d^{5} x +15 c \,d^{4}\right ) \left (4 \left (d x +c \right )^{3} \sin \left (b x +a \right )^{2} d +2 \left (d x +c \right )^{4} \sin \left (b x +a \right ) b \cos \left (b x +a \right )\right )}{40 b^{6} \left (d x +c \right )^{5}}+\frac {x \left (2 d^{4} x^{4} b^{4}+10 b^{4} c \,d^{3} x^{3}+20 b^{4} c^{2} d^{2} x^{2}+20 b^{4} c^{3} d x +10 b^{4} c^{4}-10 b^{2} d^{4} x^{2}-30 b^{2} c \,d^{3} x -30 b^{2} c^{2} d^{2}+15 d^{4}\right ) \left (12 \left (d x +c \right )^{2} \sin \left (b x +a \right )^{2} d^{2}+16 \left (d x +c \right )^{3} \sin \left (b x +a \right ) d b \cos \left (b x +a \right )+2 \left (d x +c \right )^{4} b^{2} \cos \left (b x +a \right )^{2}-2 \left (d x +c \right )^{4} \sin \left (b x +a \right )^{2} b^{2}\right )}{40 b^{6} \left (d x +c \right )^{4}}\) | \(610\) |
| norman | \(\frac {-\frac {d^{4} x^{4} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+d^{3} c \,x^{4} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+\frac {d^{4} x^{4} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{b}+\frac {d^{3} c \,x^{4} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{2}+\frac {\left (2 b^{4} c^{4}+18 b^{2} c^{2} d^{2}-9 d^{4}\right ) x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{2 b^{4}}+\frac {\left (2 b^{4} c^{4}-6 b^{2} c^{2} d^{2}+3 d^{4}\right ) x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{4 b^{4}}+\frac {d^{2} \left (2 b^{2} c^{2}-d^{2}\right ) x^{3}}{2 b^{2}}-\frac {2 c d \left (2 b^{2} c^{2}-3 d^{2}\right ) x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {2 c d \left (2 b^{2} c^{2}-3 d^{2}\right ) x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{b^{3}}+\frac {c d \left (2 b^{2} c^{2}+9 d^{2}\right ) x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b^{2}}+\frac {c d \left (2 b^{2} c^{2}-3 d^{2}\right ) x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{2 b^{2}}+\frac {c d \left (2 b^{2} c^{2}-3 d^{2}\right ) x^{2}}{2 b^{2}}-\frac {3 d^{2} \left (2 b^{2} c^{2}-d^{2}\right ) x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {3 d^{2} \left (2 b^{2} c^{2}-d^{2}\right ) x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{b^{3}}-\frac {4 d^{3} c \,x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {d^{2} \left (2 b^{2} c^{2}+3 d^{2}\right ) x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b^{2}}+\frac {4 d^{3} c \,x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{b}+\frac {d^{2} \left (2 b^{2} c^{2}-d^{2}\right ) x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{2 b^{2}}+\frac {d^{4} x^{5} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}{10}+\frac {2 \left (2 b^{2} c^{3} d -3 d^{3} c \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b^{4}}+\frac {d^{3} c \,x^{4}}{2}+\frac {\left (2 b^{4} c^{4}-6 b^{2} c^{2} d^{2}+3 d^{4}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}{2 b^{5}}+\frac {\left (2 b^{4} c^{4}-6 b^{2} c^{2} d^{2}+3 d^{4}\right ) x}{4 b^{4}}+\frac {d^{4} x^{5} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{5}-\frac {\left (2 b^{4} c^{4}-6 b^{2} c^{2} d^{2}+3 d^{4}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{2 b^{5}}+\frac {d^{4} x^{5}}{10}}{\left (1+\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right )^{2}}\) | \(747\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1030\) |
| default | \(\text {Expression too large to display}\) | \(1030\) |
Input:
int((d*x+c)^4*sin(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/8*((-2*(d*x+c)^4*b^4+6*d^2*(d*x+c)^2*b^2-3*d^4)*sin(2*b*x+2*a)+4*b*(-(d* x+c)*d*((d*x+c)^2*b^2-3/2*d^2)*cos(2*b*x+2*a)+x*(1/5*d^4*x^4+c*d^3*x^3+2*c ^2*d^2*x^2+2*c^3*d*x+c^4)*b^4+b^2*c^3*d-3/2*d^3*c))/b^5
Time = 0.09 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.78 \[ \int (c+d x)^4 \sin ^2(a+b x) \, dx=\frac {2 \, b^{5} d^{4} x^{5} + 10 \, b^{5} c d^{3} x^{4} + 10 \, {\left (2 \, b^{5} c^{2} d^{2} + b^{3} d^{4}\right )} x^{3} + 10 \, {\left (2 \, b^{5} c^{3} d + 3 \, b^{3} c d^{3}\right )} x^{2} - 10 \, {\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 2 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \, {\left (2 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{2} - 5 \, {\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 2 \, b^{4} c^{4} - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 6 \, {\left (2 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 4 \, {\left (2 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 5 \, {\left (2 \, b^{5} c^{4} + 6 \, b^{3} c^{2} d^{2} - 3 \, b d^{4}\right )} x}{20 \, b^{5}} \] Input:
integrate((d*x+c)^4*sin(b*x+a)^2,x, algorithm="fricas")
Output:
1/20*(2*b^5*d^4*x^5 + 10*b^5*c*d^3*x^4 + 10*(2*b^5*c^2*d^2 + b^3*d^4)*x^3 + 10*(2*b^5*c^3*d + 3*b^3*c*d^3)*x^2 - 10*(2*b^3*d^4*x^3 + 6*b^3*c*d^3*x^2 + 2*b^3*c^3*d - 3*b*c*d^3 + 3*(2*b^3*c^2*d^2 - b*d^4)*x)*cos(b*x + a)^2 - 5*(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 2*b^4*c^4 - 6*b^2*c^2*d^2 + 3*d^4 + 6*(2*b^4*c^2*d^2 - b^2*d^4)*x^2 + 4*(2*b^4*c^3*d - 3*b^2*c*d^3)*x)*cos(b*x + a)*sin(b*x + a) + 5*(2*b^5*c^4 + 6*b^3*c^2*d^2 - 3*b*d^4)*x)/b^5
Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (156) = 312\).
Time = 0.70 (sec) , antiderivative size = 660, normalized size of antiderivative = 4.10 \[ \int (c+d x)^4 \sin ^2(a+b x) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)**4*sin(b*x+a)**2,x)
Output:
Piecewise((c**4*x*sin(a + b*x)**2/2 + c**4*x*cos(a + b*x)**2/2 + c**3*d*x* *2*sin(a + b*x)**2 + c**3*d*x**2*cos(a + b*x)**2 + c**2*d**2*x**3*sin(a + b*x)**2 + c**2*d**2*x**3*cos(a + b*x)**2 + c*d**3*x**4*sin(a + b*x)**2/2 + c*d**3*x**4*cos(a + b*x)**2/2 + d**4*x**5*sin(a + b*x)**2/10 + d**4*x**5* cos(a + b*x)**2/10 - c**4*sin(a + b*x)*cos(a + b*x)/(2*b) - 2*c**3*d*x*sin (a + b*x)*cos(a + b*x)/b - 3*c**2*d**2*x**2*sin(a + b*x)*cos(a + b*x)/b - 2*c*d**3*x**3*sin(a + b*x)*cos(a + b*x)/b - d**4*x**4*sin(a + b*x)*cos(a + b*x)/(2*b) + c**3*d*sin(a + b*x)**2/b**2 + 3*c**2*d**2*x*sin(a + b*x)**2/ (2*b**2) - 3*c**2*d**2*x*cos(a + b*x)**2/(2*b**2) + 3*c*d**3*x**2*sin(a + b*x)**2/(2*b**2) - 3*c*d**3*x**2*cos(a + b*x)**2/(2*b**2) + d**4*x**3*sin( a + b*x)**2/(2*b**2) - d**4*x**3*cos(a + b*x)**2/(2*b**2) + 3*c**2*d**2*si n(a + b*x)*cos(a + b*x)/(2*b**3) + 3*c*d**3*x*sin(a + b*x)*cos(a + b*x)/b* *3 + 3*d**4*x**2*sin(a + b*x)*cos(a + b*x)/(2*b**3) - 3*c*d**3*sin(a + b*x )**2/(2*b**4) - 3*d**4*x*sin(a + b*x)**2/(4*b**4) + 3*d**4*x*cos(a + b*x)* *2/(4*b**4) - 3*d**4*sin(a + b*x)*cos(a + b*x)/(4*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, True))
Leaf count of result is larger than twice the leaf count of optimal. 735 vs. \(2 (147) = 294\).
Time = 0.06 (sec) , antiderivative size = 735, normalized size of antiderivative = 4.57 \[ \int (c+d x)^4 \sin ^2(a+b x) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^4*sin(b*x+a)^2,x, algorithm="maxima")
Output:
1/40*(10*(2*b*x + 2*a - sin(2*b*x + 2*a))*c^4 - 40*(2*b*x + 2*a - sin(2*b* x + 2*a))*a*c^3*d/b + 60*(2*b*x + 2*a - sin(2*b*x + 2*a))*a^2*c^2*d^2/b^2 - 40*(2*b*x + 2*a - sin(2*b*x + 2*a))*a^3*c*d^3/b^3 + 10*(2*b*x + 2*a - si n(2*b*x + 2*a))*a^4*d^4/b^4 + 20*(2*(b*x + a)^2 - 2*(b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a))*c^3*d/b - 60*(2*(b*x + a)^2 - 2*(b*x + a)*sin(2*b *x + 2*a) - cos(2*b*x + 2*a))*a*c^2*d^2/b^2 + 60*(2*(b*x + a)^2 - 2*(b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a))*a^2*c*d^3/b^3 - 20*(2*(b*x + a)^2 - 2*(b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a))*a^3*d^4/b^4 + 10*(4*(b *x + a)^3 - 6*(b*x + a)*cos(2*b*x + 2*a) - 3*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*a))*c^2*d^2/b^2 - 20*(4*(b*x + a)^3 - 6*(b*x + a)*cos(2*b*x + 2*a) - 3*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*a))*a*c*d^3/b^3 + 10*(4*(b*x + a)^3 - 6*(b*x + a)*cos(2*b*x + 2*a) - 3*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*a))*a^2 *d^4/b^4 + 10*(2*(b*x + a)^4 - 3*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 2* (2*(b*x + a)^3 - 3*b*x - 3*a)*sin(2*b*x + 2*a))*c*d^3/b^3 - 10*(2*(b*x + a )^4 - 3*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 2*(2*(b*x + a)^3 - 3*b*x - 3*a)*sin(2*b*x + 2*a))*a*d^4/b^4 + (4*(b*x + a)^5 - 10*(2*(b*x + a)^3 - 3* b*x - 3*a)*cos(2*b*x + 2*a) - 5*(2*(b*x + a)^4 - 6*(b*x + a)^2 + 3)*sin(2* b*x + 2*a))*d^4/b^4)/b
Time = 0.38 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.38 \[ \int (c+d x)^4 \sin ^2(a+b x) \, dx=\frac {1}{10} \, d^{4} x^{5} + \frac {1}{2} \, c d^{3} x^{4} + c^{2} d^{2} x^{3} + c^{3} d x^{2} + \frac {1}{2} \, c^{4} x - \frac {{\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{2} d^{2} x + 2 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \cos \left (2 \, b x + 2 \, a\right )}{4 \, b^{5}} - \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 8 \, b^{4} c^{3} d x + 2 \, b^{4} c^{4} - 6 \, b^{2} d^{4} x^{2} - 12 \, b^{2} c d^{3} x - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{5}} \] Input:
integrate((d*x+c)^4*sin(b*x+a)^2,x, algorithm="giac")
Output:
1/10*d^4*x^5 + 1/2*c*d^3*x^4 + c^2*d^2*x^3 + c^3*d*x^2 + 1/2*c^4*x - 1/4*( 2*b^3*d^4*x^3 + 6*b^3*c*d^3*x^2 + 6*b^3*c^2*d^2*x + 2*b^3*c^3*d - 3*b*d^4* x - 3*b*c*d^3)*cos(2*b*x + 2*a)/b^5 - 1/8*(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 12*b^4*c^2*d^2*x^2 + 8*b^4*c^3*d*x + 2*b^4*c^4 - 6*b^2*d^4*x^2 - 12*b^2 *c*d^3*x - 6*b^2*c^2*d^2 + 3*d^4)*sin(2*b*x + 2*a)/b^5
Time = 0.94 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.17 \[ \int (c+d x)^4 \sin ^2(a+b x) \, dx=-\frac {\frac {15\,d^4\,\sin \left (2\,a+2\,b\,x\right )}{2}-10\,b^5\,c^4\,x+5\,b^4\,c^4\,\sin \left (2\,a+2\,b\,x\right )-2\,b^5\,d^4\,x^5+10\,b^3\,c^3\,d\,\cos \left (2\,a+2\,b\,x\right )-20\,b^5\,c^3\,d\,x^2-10\,b^5\,c\,d^3\,x^4-15\,b^2\,c^2\,d^2\,\sin \left (2\,a+2\,b\,x\right )+10\,b^3\,d^4\,x^3\,\cos \left (2\,a+2\,b\,x\right )-20\,b^5\,c^2\,d^2\,x^3-15\,b^2\,d^4\,x^2\,\sin \left (2\,a+2\,b\,x\right )+5\,b^4\,d^4\,x^4\,\sin \left (2\,a+2\,b\,x\right )-15\,b\,c\,d^3\,\cos \left (2\,a+2\,b\,x\right )-15\,b\,d^4\,x\,\cos \left (2\,a+2\,b\,x\right )+30\,b^4\,c^2\,d^2\,x^2\,\sin \left (2\,a+2\,b\,x\right )-30\,b^2\,c\,d^3\,x\,\sin \left (2\,a+2\,b\,x\right )+20\,b^4\,c^3\,d\,x\,\sin \left (2\,a+2\,b\,x\right )+30\,b^3\,c^2\,d^2\,x\,\cos \left (2\,a+2\,b\,x\right )+30\,b^3\,c\,d^3\,x^2\,\cos \left (2\,a+2\,b\,x\right )+20\,b^4\,c\,d^3\,x^3\,\sin \left (2\,a+2\,b\,x\right )}{20\,b^5} \] Input:
int(sin(a + b*x)^2*(c + d*x)^4,x)
Output:
-((15*d^4*sin(2*a + 2*b*x))/2 - 10*b^5*c^4*x + 5*b^4*c^4*sin(2*a + 2*b*x) - 2*b^5*d^4*x^5 + 10*b^3*c^3*d*cos(2*a + 2*b*x) - 20*b^5*c^3*d*x^2 - 10*b^ 5*c*d^3*x^4 - 15*b^2*c^2*d^2*sin(2*a + 2*b*x) + 10*b^3*d^4*x^3*cos(2*a + 2 *b*x) - 20*b^5*c^2*d^2*x^3 - 15*b^2*d^4*x^2*sin(2*a + 2*b*x) + 5*b^4*d^4*x ^4*sin(2*a + 2*b*x) - 15*b*c*d^3*cos(2*a + 2*b*x) - 15*b*d^4*x*cos(2*a + 2 *b*x) + 30*b^4*c^2*d^2*x^2*sin(2*a + 2*b*x) - 30*b^2*c*d^3*x*sin(2*a + 2*b *x) + 20*b^4*c^3*d*x*sin(2*a + 2*b*x) + 30*b^3*c^2*d^2*x*cos(2*a + 2*b*x) + 30*b^3*c*d^3*x^2*cos(2*a + 2*b*x) + 20*b^4*c*d^3*x^3*sin(2*a + 2*b*x))/( 20*b^5)
Time = 0.17 (sec) , antiderivative size = 455, normalized size of antiderivative = 2.83 \[ \int (c+d x)^4 \sin ^2(a+b x) \, dx=\frac {-10 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{4} d^{4} x^{4}+30 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{2} c^{2} d^{2}+30 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{2} d^{4} x^{2}+60 \sin \left (b x +a \right )^{2} b^{3} c^{2} d^{2} x +60 \sin \left (b x +a \right )^{2} b^{3} c \,d^{3} x^{2}-10 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{4} c^{4}+20 \sin \left (b x +a \right )^{2} b^{3} c^{3} d +20 \sin \left (b x +a \right )^{2} b^{3} d^{4} x^{3}-30 \sin \left (b x +a \right )^{2} b c \,d^{3}-30 \sin \left (b x +a \right )^{2} b \,d^{4} x +90 a \,b^{2} c^{2} d^{2}+20 b^{5} c^{3} d \,x^{2}+20 b^{5} c^{2} d^{2} x^{3}+10 b^{5} c \,d^{3} x^{4}-30 b^{3} c^{2} d^{2} x -30 b^{3} c \,d^{3} x^{2}-40 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{4} c^{3} d x -60 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{4} c^{2} d^{2} x^{2}-40 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{4} c \,d^{3} x^{3}+60 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{2} c \,d^{3} x -45 a \,d^{4}-15 \cos \left (b x +a \right ) \sin \left (b x +a \right ) d^{4}+10 a \,b^{4} c^{4}+10 b^{5} c^{4} x +2 b^{5} d^{4} x^{5}-40 b^{3} c^{3} d -10 b^{3} d^{4} x^{3}+60 b c \,d^{3}+15 b \,d^{4} x}{20 b^{5}} \] Input:
int((d*x+c)^4*sin(b*x+a)^2,x)
Output:
( - 10*cos(a + b*x)*sin(a + b*x)*b**4*c**4 - 40*cos(a + b*x)*sin(a + b*x)* b**4*c**3*d*x - 60*cos(a + b*x)*sin(a + b*x)*b**4*c**2*d**2*x**2 - 40*cos( a + b*x)*sin(a + b*x)*b**4*c*d**3*x**3 - 10*cos(a + b*x)*sin(a + b*x)*b**4 *d**4*x**4 + 30*cos(a + b*x)*sin(a + b*x)*b**2*c**2*d**2 + 60*cos(a + b*x) *sin(a + b*x)*b**2*c*d**3*x + 30*cos(a + b*x)*sin(a + b*x)*b**2*d**4*x**2 - 15*cos(a + b*x)*sin(a + b*x)*d**4 + 20*sin(a + b*x)**2*b**3*c**3*d + 60* sin(a + b*x)**2*b**3*c**2*d**2*x + 60*sin(a + b*x)**2*b**3*c*d**3*x**2 + 2 0*sin(a + b*x)**2*b**3*d**4*x**3 - 30*sin(a + b*x)**2*b*c*d**3 - 30*sin(a + b*x)**2*b*d**4*x + 10*a*b**4*c**4 + 90*a*b**2*c**2*d**2 - 45*a*d**4 + 10 *b**5*c**4*x + 20*b**5*c**3*d*x**2 + 20*b**5*c**2*d**2*x**3 + 10*b**5*c*d* *3*x**4 + 2*b**5*d**4*x**5 - 40*b**3*c**3*d - 30*b**3*c**2*d**2*x - 30*b** 3*c*d**3*x**2 - 10*b**3*d**4*x**3 + 60*b*c*d**3 + 15*b*d**4*x)/(20*b**5)