Integrand size = 29, antiderivative size = 354 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {1}{128} a \left (8 a^2+9 b^2\right ) x-\frac {b \left (27 a^2+4 b^2\right ) \cos (c+d x)}{105 d}+\frac {b \left (27 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac {a \left (8 a^2+9 b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}-\frac {a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d} \] Output:
1/128*a*(8*a^2+9*b^2)*x-1/105*b*(27*a^2+4*b^2)*cos(d*x+c)/d+1/315*b*(27*a^ 2+4*b^2)*cos(d*x+c)^3/d-1/128*a*(8*a^2+9*b^2)*cos(d*x+c)*sin(d*x+c)/d-1/40 32*a*(40*a^4-188*a^2*b^2+189*b^4)*cos(d*x+c)*sin(d*x+c)^3/b^2/d-1/2520*(20 *a^4-93*a^2*b^2+24*b^4)*cos(d*x+c)*sin(d*x+c)^4/b/d-1/1008*a*(20*a^2-87*b^ 2)*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^2/b^2/d-5/126*(a^2-4*b^2)*cos( d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^3/b^2/d+5/72*a*cos(d*x+c)*sin(d*x+c)^ 3*(a+b*sin(d*x+c))^4/b^2/d-1/9*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^4/ b/d
Time = 1.21 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.58 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {15120 a b^2 c+10080 a^3 d x+11340 a b^2 d x-3780 b \left (6 a^2+b^2\right ) \cos (c+d x)-840 \left (9 a^2 b+b^3\right ) \cos (3 (c+d x))+1512 a^2 b \cos (5 (c+d x))+504 b^3 \cos (5 (c+d x))+1080 a^2 b \cos (7 (c+d x))+90 b^3 \cos (7 (c+d x))-70 b^3 \cos (9 (c+d x))+2520 a^3 \sin (2 (c+d x))-2520 a^3 \sin (4 (c+d x))-3780 a b^2 \sin (4 (c+d x))-840 a^3 \sin (6 (c+d x))+\frac {945}{2} a b^2 \sin (8 (c+d x))}{161280 d} \] Input:
Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]
Output:
(15120*a*b^2*c + 10080*a^3*d*x + 11340*a*b^2*d*x - 3780*b*(6*a^2 + b^2)*Co s[c + d*x] - 840*(9*a^2*b + b^3)*Cos[3*(c + d*x)] + 1512*a^2*b*Cos[5*(c + d*x)] + 504*b^3*Cos[5*(c + d*x)] + 1080*a^2*b*Cos[7*(c + d*x)] + 90*b^3*Co s[7*(c + d*x)] - 70*b^3*Cos[9*(c + d*x)] + 2520*a^3*Sin[2*(c + d*x)] - 252 0*a^3*Sin[4*(c + d*x)] - 3780*a*b^2*Sin[4*(c + d*x)] - 840*a^3*Sin[6*(c + d*x)] + (945*a*b^2*Sin[8*(c + d*x)])/2)/(161280*d)
Time = 2.24 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.99, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.655, Rules used = {3042, 3374, 3042, 3528, 27, 3042, 3528, 3042, 3512, 3042, 3502, 27, 3042, 3227, 3042, 3113, 2009, 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.
| \(\displaystyle \int \sin ^2(c+d x) \cos ^4(c+d x) (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^2 \cos (c+d x)^4 (a+b \sin (c+d x))^3dx\) |
| \(\Big \downarrow \) 3374 |
| \(\displaystyle -\frac {\int \sin ^2(c+d x) (a+b \sin (c+d x))^3 \left (-20 \left (a^2-4 b^2\right ) \sin ^2(c+d x)+3 a b \sin (c+d x)+3 \left (5 a^2-24 b^2\right )\right )dx}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \sin (c+d x)^2 (a+b \sin (c+d x))^3 \left (-20 \left (a^2-4 b^2\right ) \sin (c+d x)^2+3 a b \sin (c+d x)+3 \left (5 a^2-24 b^2\right )\right )dx}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {1}{7} \int 3 \sin ^2(c+d x) (a+b \sin (c+d x))^2 \left (-a \left (20 a^2-87 b^2\right ) \sin ^2(c+d x)+2 b \left (a^2-4 b^2\right ) \sin (c+d x)+a \left (15 a^2-88 b^2\right )\right )dx+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3}{7} \int \sin ^2(c+d x) (a+b \sin (c+d x))^2 \left (-a \left (20 a^2-87 b^2\right ) \sin ^2(c+d x)+2 b \left (a^2-4 b^2\right ) \sin (c+d x)+a \left (15 a^2-88 b^2\right )\right )dx+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{7} \int \sin (c+d x)^2 (a+b \sin (c+d x))^2 \left (-a \left (20 a^2-87 b^2\right ) \sin (c+d x)^2+2 b \left (a^2-4 b^2\right ) \sin (c+d x)+a \left (15 a^2-88 b^2\right )\right )dx+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {3}{7} \left (\frac {1}{6} \int \sin ^2(c+d x) (a+b \sin (c+d x)) \left (3 \left (10 a^2-89 b^2\right ) a^2+b \left (2 a^2-141 b^2\right ) \sin (c+d x) a-2 \left (20 a^4-93 b^2 a^2+24 b^4\right ) \sin ^2(c+d x)\right )dx+\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}\right )+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{7} \left (\frac {1}{6} \int \sin (c+d x)^2 (a+b \sin (c+d x)) \left (3 \left (10 a^2-89 b^2\right ) a^2+b \left (2 a^2-141 b^2\right ) \sin (c+d x) a-2 \left (20 a^4-93 b^2 a^2+24 b^4\right ) \sin (c+d x)^2\right )dx+\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}\right )+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle -\frac {\frac {3}{7} \left (\frac {1}{6} \left (\frac {1}{5} \int \sin ^2(c+d x) \left (15 \left (10 a^2-89 b^2\right ) a^3-5 \left (40 a^4-188 b^2 a^2+189 b^4\right ) \sin ^2(c+d x) a-48 b^3 \left (27 a^2+4 b^2\right ) \sin (c+d x)\right )dx+\frac {2 b \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}\right )+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{7} \left (\frac {1}{6} \left (\frac {1}{5} \int \sin (c+d x)^2 \left (15 \left (10 a^2-89 b^2\right ) a^3-5 \left (40 a^4-188 b^2 a^2+189 b^4\right ) \sin (c+d x)^2 a-48 b^3 \left (27 a^2+4 b^2\right ) \sin (c+d x)\right )dx+\frac {2 b \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}\right )+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {\frac {3}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int -3 \sin ^2(c+d x) \left (64 \left (27 a^2+4 b^2\right ) \sin (c+d x) b^3+105 a \left (8 a^2+9 b^2\right ) b^2\right )dx+\frac {5 a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}\right )+\frac {2 b \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}\right )+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {3}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {5 a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {3}{4} \int \sin ^2(c+d x) \left (64 \left (27 a^2+4 b^2\right ) \sin (c+d x) b^3+105 a \left (8 a^2+9 b^2\right ) b^2\right )dx\right )+\frac {2 b \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}\right )+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {5 a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {3}{4} \int \sin (c+d x)^2 \left (64 \left (27 a^2+4 b^2\right ) \sin (c+d x) b^3+105 a \left (8 a^2+9 b^2\right ) b^2\right )dx\right )+\frac {2 b \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}\right )+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle -\frac {\frac {3}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {5 a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {3}{4} \left (105 a b^2 \left (8 a^2+9 b^2\right ) \int \sin ^2(c+d x)dx+64 b^3 \left (27 a^2+4 b^2\right ) \int \sin ^3(c+d x)dx\right )\right )+\frac {2 b \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}\right )+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {3}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {5 a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {3}{4} \left (105 a b^2 \left (8 a^2+9 b^2\right ) \int \sin (c+d x)^2dx+64 b^3 \left (27 a^2+4 b^2\right ) \int \sin (c+d x)^3dx\right )\right )+\frac {2 b \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}\right )+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -\frac {\frac {3}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {5 a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {3}{4} \left (105 a b^2 \left (8 a^2+9 b^2\right ) \int \sin (c+d x)^2dx-\frac {64 b^3 \left (27 a^2+4 b^2\right ) \int \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{d}\right )\right )+\frac {2 b \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}\right )+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {3}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {5 a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {3}{4} \left (105 a b^2 \left (8 a^2+9 b^2\right ) \int \sin (c+d x)^2dx-\frac {64 b^3 \left (27 a^2+4 b^2\right ) \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d}\right )\right )+\frac {2 b \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}\right )+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {3}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {5 a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {3}{4} \left (105 a b^2 \left (8 a^2+9 b^2\right ) \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {64 b^3 \left (27 a^2+4 b^2\right ) \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d}\right )\right )+\frac {2 b \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}\right )+\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}\right )+\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\frac {20 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{7 d}+\frac {3}{7} \left (\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{6 d}+\frac {1}{6} \left (\frac {2 b \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 d}+\frac {1}{5} \left (\frac {5 a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {3}{4} \left (105 a b^2 \left (8 a^2+9 b^2\right ) \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {64 b^3 \left (27 a^2+4 b^2\right ) \left (\cos (c+d x)-\frac {1}{3} \cos ^3(c+d x)\right )}{d}\right )\right )\right )\right )}{72 b^2}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d}\) |
Input:
Int[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]
Output:
(5*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(72*b^2*d) - (Cos [c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^4)/(9*b*d) - ((20*(a^2 - 4*b ^2)*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(7*d) + (3*((a*(20 *a^2 - 87*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(6*d) + ((2*b*(20*a^4 - 93*a^2*b^2 + 24*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(5*d) + ((5*a*(40*a^4 - 188*a^2*b^2 + 189*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d) - (3*((-64*b^3*(27*a^2 + 4*b^2)*(Cos[c + d*x] - Cos[c + d*x]^3/3))/d + 10 5*a*b^2*(8*a^2 + 9*b^2)*(x/2 - (Cos[c + d*x]*Sin[c + d*x])/(2*d))))/4)/5)/ 6))/7)/(72*b^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[a*(n + 3)*Cos[e + f* x]*(d*Sin[e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d*f*(m + n + 3)*(m + n + 4))), x] + (-Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*((a + b*Sin[e + f*x])^(m + 1)/(b*d^2*f*(m + n + 4))), x] - Simp[1/(b^2*(m + n + 3 )*(m + n + 4)) Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4) + a*b*m*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 3)*(m + n + 5))*Sin[e + f*x]^2, x], x], x]) /; F reeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || Integ ersQ[2*m, 2*n]) && !m < -1 && !LtQ[n, -1] && NeQ[m + n + 3, 0] && NeQ[m + n + 4, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Time = 0.38 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.62
\[\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{6}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{2} b \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{5}}{7}-\frac {2 \cos \left (d x +c \right )^{5}}{35}\right )+3 a \,b^{2} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{5}}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+b^{3} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{5}}{9}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{5}}{63}-\frac {8 \cos \left (d x +c \right )^{5}}{315}\right )}{d}\]
Input:
int(cos(d*x+c)^4*sin(d*x+c)^2*(a+b*sin(d*x+c))^3,x)
Output:
1/d*(a^3*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1/24*(cos(d*x+c)^3+3/2*cos(d*x+c))* sin(d*x+c)+1/16*d*x+1/16*c)+3*a^2*b*(-1/7*sin(d*x+c)^2*cos(d*x+c)^5-2/35*c os(d*x+c)^5)+3*a*b^2*(-1/8*sin(d*x+c)^3*cos(d*x+c)^5-1/16*sin(d*x+c)*cos(d *x+c)^5+1/64*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/128*d*x+3/128*c)+b ^3*(-1/9*sin(d*x+c)^4*cos(d*x+c)^5-4/63*sin(d*x+c)^2*cos(d*x+c)^5-8/315*co s(d*x+c)^5))
Time = 0.10 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.46 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {4480 \, b^{3} \cos \left (d x + c\right )^{9} - 5760 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{7} + 8064 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{5} - 315 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} d x - 105 \, {\left (144 \, a b^{2} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, a^{3} + 27 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, d} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^2*(a+b*sin(d*x+c))^3,x, algorithm="frica s")
Output:
-1/40320*(4480*b^3*cos(d*x + c)^9 - 5760*(3*a^2*b + 2*b^3)*cos(d*x + c)^7 + 8064*(3*a^2*b + b^3)*cos(d*x + c)^5 - 315*(8*a^3 + 9*a*b^2)*d*x - 105*(1 44*a*b^2*cos(d*x + c)^7 - 8*(8*a^3 + 27*a*b^2)*cos(d*x + c)^5 + 2*(8*a^3 + 9*a*b^2)*cos(d*x + c)^3 + 3*(8*a^3 + 9*a*b^2)*cos(d*x + c))*sin(d*x + c)) /d
Time = 1.01 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.43 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\begin {cases} \frac {a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {3 a^{2} b \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {6 a^{2} b \cos ^{7}{\left (c + d x \right )}}{35 d} + \frac {9 a b^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {9 a b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {27 a b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {9 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {9 a b^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {9 a b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {33 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {33 a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {9 a b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {4 b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {8 b^{3} \cos ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{3} \sin ^{2}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**4*sin(d*x+c)**2*(a+b*sin(d*x+c))**3,x)
Output:
Piecewise((a**3*x*sin(c + d*x)**6/16 + 3*a**3*x*sin(c + d*x)**4*cos(c + d* x)**2/16 + 3*a**3*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + a**3*x*cos(c + d* x)**6/16 + a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) + a**3*sin(c + d*x)**3 *cos(c + d*x)**3/(6*d) - a**3*sin(c + d*x)*cos(c + d*x)**5/(16*d) - 3*a**2 *b*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 6*a**2*b*cos(c + d*x)**7/(35*d) + 9*a*b**2*x*sin(c + d*x)**8/128 + 9*a*b**2*x*sin(c + d*x)**6*cos(c + d*x )**2/32 + 27*a*b**2*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 9*a*b**2*x*sin( c + d*x)**2*cos(c + d*x)**6/32 + 9*a*b**2*x*cos(c + d*x)**8/128 + 9*a*b**2 *sin(c + d*x)**7*cos(c + d*x)/(128*d) + 33*a*b**2*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) - 33*a*b**2*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 9*a* b**2*sin(c + d*x)*cos(c + d*x)**7/(128*d) - b**3*sin(c + d*x)**4*cos(c + d *x)**5/(5*d) - 4*b**3*sin(c + d*x)**2*cos(c + d*x)**7/(35*d) - 8*b**3*cos( c + d*x)**9/(315*d), Ne(d, 0)), (x*(a + b*sin(c))**3*sin(c)**2*cos(c)**4, True))
Time = 0.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.40 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {1680 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 27648 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{2} b + 945 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{2} - 1024 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} b^{3}}{322560 \, d} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^2*(a+b*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/322560*(1680*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c)) *a^3 + 27648*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^2*b + 945*(24*d*x + 2 4*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*a*b^2 - 1024*(35*cos(d*x + c) ^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*b^3)/d
Time = 0.23 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.58 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {b^{3} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {3 \, a b^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {a^{3} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {1}{128} \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} x + \frac {{\left (12 \, a^{2} b + b^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (3 \, a^{2} b + b^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (9 \, a^{2} b + b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {3 \, {\left (6 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )}{128 \, d} - \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} \] Input:
integrate(cos(d*x+c)^4*sin(d*x+c)^2*(a+b*sin(d*x+c))^3,x, algorithm="giac" )
Output:
-1/2304*b^3*cos(9*d*x + 9*c)/d + 3/1024*a*b^2*sin(8*d*x + 8*c)/d - 1/192*a ^3*sin(6*d*x + 6*c)/d + 1/64*a^3*sin(2*d*x + 2*c)/d + 1/128*(8*a^3 + 9*a*b ^2)*x + 1/1792*(12*a^2*b + b^3)*cos(7*d*x + 7*c)/d + 1/320*(3*a^2*b + b^3) *cos(5*d*x + 5*c)/d - 1/192*(9*a^2*b + b^3)*cos(3*d*x + 3*c)/d - 3/128*(6* a^2*b + b^3)*cos(d*x + c)/d - 1/128*(2*a^3 + 3*a*b^2)*sin(4*d*x + 4*c)/d
Time = 35.28 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.63 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx =\text {Too large to display} \] Input:
int(cos(c + d*x)^4*sin(c + d*x)^2*(a + b*sin(c + d*x))^3,x)
Output:
(a*atan((a*tan(c/2 + (d*x)/2)*(8*a^2 + 9*b^2))/(64*((9*a*b^2)/64 + a^3/8)) )*(8*a^2 + 9*b^2))/(64*d) - (tan(c/2 + (d*x)/2)*((9*a*b^2)/64 + a^3/8) + ( 12*a^2*b)/35 - tan(c/2 + (d*x)/2)^17*((9*a*b^2)/64 + a^3/8) + tan(c/2 + (d *x)/2)^3*((39*a*b^2)/32 - (19*a^3)/12) - tan(c/2 + (d*x)/2)^15*((39*a*b^2) /32 - (19*a^3)/12) - tan(c/2 + (d*x)/2)^5*((465*a*b^2)/32 + (9*a^3)/4) + t an(c/2 + (d*x)/2)^13*((465*a*b^2)/32 + (9*a^3)/4) + tan(c/2 + (d*x)/2)^7*( (507*a*b^2)/32 + (3*a^3)/4) - tan(c/2 + (d*x)/2)^11*((507*a*b^2)/32 + (3*a ^3)/4) + tan(c/2 + (d*x)/2)^10*(12*a^2*b - 16*b^3) + tan(c/2 + (d*x)/2)^12 *(12*a^2*b + (32*b^3)/3) + tan(c/2 + (d*x)/2)^6*((84*a^2*b)/5 - (32*b^3)/5 ) + tan(c/2 + (d*x)/2)^4*((12*a^2*b)/35 + (64*b^3)/35) + tan(c/2 + (d*x)/2 )^2*((108*a^2*b)/35 + (16*b^3)/35) + tan(c/2 + (d*x)/2)^8*((156*a^2*b)/5 + (112*b^3)/5) + (16*b^3)/315 + 12*a^2*b*tan(c/2 + (d*x)/2)^14)/(d*(9*tan(c /2 + (d*x)/2)^2 + 36*tan(c/2 + (d*x)/2)^4 + 84*tan(c/2 + (d*x)/2)^6 + 126* tan(c/2 + (d*x)/2)^8 + 126*tan(c/2 + (d*x)/2)^10 + 84*tan(c/2 + (d*x)/2)^1 2 + 36*tan(c/2 + (d*x)/2)^14 + 9*tan(c/2 + (d*x)/2)^16 + tan(c/2 + (d*x)/2 )^18 + 1)) - (a*(8*a^2 + 9*b^2)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(64* d)
Time = 0.18 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.92 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {-4480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} b^{3}-15120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a \,b^{2}-17280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a^{2} b +6400 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b^{3}-6720 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a^{3}+22680 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a \,b^{2}+27648 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2} b -384 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{3}+11760 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{3}-1890 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a \,b^{2}-3456 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2} b -512 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{3}-2520 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3}-2835 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{2}-6912 \cos \left (d x +c \right ) a^{2} b -1024 \cos \left (d x +c \right ) b^{3}+2520 a^{3} d x +6912 a^{2} b +2835 a \,b^{2} d x +1024 b^{3}}{40320 d} \] Input:
int(cos(d*x+c)^4*sin(d*x+c)^2*(a+b*sin(d*x+c))^3,x)
Output:
( - 4480*cos(c + d*x)*sin(c + d*x)**8*b**3 - 15120*cos(c + d*x)*sin(c + d* x)**7*a*b**2 - 17280*cos(c + d*x)*sin(c + d*x)**6*a**2*b + 6400*cos(c + d* x)*sin(c + d*x)**6*b**3 - 6720*cos(c + d*x)*sin(c + d*x)**5*a**3 + 22680*c os(c + d*x)*sin(c + d*x)**5*a*b**2 + 27648*cos(c + d*x)*sin(c + d*x)**4*a* *2*b - 384*cos(c + d*x)*sin(c + d*x)**4*b**3 + 11760*cos(c + d*x)*sin(c + d*x)**3*a**3 - 1890*cos(c + d*x)*sin(c + d*x)**3*a*b**2 - 3456*cos(c + d*x )*sin(c + d*x)**2*a**2*b - 512*cos(c + d*x)*sin(c + d*x)**2*b**3 - 2520*co s(c + d*x)*sin(c + d*x)*a**3 - 2835*cos(c + d*x)*sin(c + d*x)*a*b**2 - 691 2*cos(c + d*x)*a**2*b - 1024*cos(c + d*x)*b**3 + 2520*a**3*d*x + 6912*a**2 *b + 2835*a*b**2*d*x + 1024*b**3)/(40320*d)