Integrand size = 29, antiderivative size = 467 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {\left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right ) x}{128 b^9}+\frac {2 a^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^9 d}-\frac {a \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^8 d}+\frac {\left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x) \sin (c+d x)}{128 b^7 d}-\frac {a \left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^6 d}+\frac {\left (48 a^4-104 a^2 b^2+59 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{192 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d}-\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a b^4 d}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{5 a^2 d}+\frac {\left (40 a^4-85 a^2 b^2+48 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{240 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^6(c+d x)}{7 b^2 d}+\frac {\cos (c+d x) \sin ^7(c+d x)}{8 b d} \] Output:
-1/128*(128*a^8-320*a^6*b^2+240*a^4*b^4-40*a^2*b^6-5*b^8)*x/b^9+2*a^3*(a^2 -b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^9/d-1/105*a *(105*a^6-245*a^4*b^2+161*a^2*b^4-15*b^6)*cos(d*x+c)/b^8/d+1/128*(64*a^6-1 44*a^4*b^2+88*a^2*b^4-5*b^6)*cos(d*x+c)*sin(d*x+c)/b^7/d-1/105*a*(35*a^4-7 7*a^2*b^2+45*b^4)*cos(d*x+c)*sin(d*x+c)^2/b^6/d+1/192*(48*a^4-104*a^2*b^2+ 59*b^4)*cos(d*x+c)*sin(d*x+c)^3/b^5/d+1/4*cos(d*x+c)*sin(d*x+c)^4/a/d-1/14 0*(28*a^4-60*a^2*b^2+35*b^4)*cos(d*x+c)*sin(d*x+c)^4/a/b^4/d-1/5*b*cos(d*x +c)*sin(d*x+c)^5/a^2/d+1/240*(40*a^4-85*a^2*b^2+48*b^4)*cos(d*x+c)*sin(d*x +c)^5/a^2/b^3/d-1/7*a*cos(d*x+c)*sin(d*x+c)^6/b^2/d+1/8*cos(d*x+c)*sin(d*x +c)^7/b/d
Time = 3.66 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-107520 a^8 c+268800 a^6 b^2 c-201600 a^4 b^4 c+33600 a^2 b^6 c+4200 b^8 c-107520 a^8 d x+268800 a^6 b^2 d x-201600 a^4 b^4 d x+33600 a^2 b^6 d x+4200 b^8 d x+215040 a^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )-1680 a b \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \cos (c+d x)+560 \left (16 a^5 b^3-28 a^3 b^5+9 a b^7\right ) \cos (3 (c+d x))-1344 a^3 b^5 \cos (5 (c+d x))+1680 a b^7 \cos (5 (c+d x))+240 a b^7 \cos (7 (c+d x))+26880 a^6 b^2 \sin (2 (c+d x))-53760 a^4 b^4 \sin (2 (c+d x))+25200 a^2 b^6 \sin (2 (c+d x))+1680 b^8 \sin (2 (c+d x))-3360 a^4 b^4 \sin (4 (c+d x))+5040 a^2 b^6 \sin (4 (c+d x))-840 b^8 \sin (4 (c+d x))+560 a^2 b^6 \sin (6 (c+d x))-560 b^8 \sin (6 (c+d x))-105 b^8 \sin (8 (c+d x))}{107520 b^9 d} \] Input:
Integrate[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]
Output:
(-107520*a^8*c + 268800*a^6*b^2*c - 201600*a^4*b^4*c + 33600*a^2*b^6*c + 4 200*b^8*c - 107520*a^8*d*x + 268800*a^6*b^2*d*x - 201600*a^4*b^4*d*x + 336 00*a^2*b^6*d*x + 4200*b^8*d*x + 215040*a^3*(a^2 - b^2)^(5/2)*ArcTan[(b + a *Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 1680*a*b*(64*a^6 - 144*a^4*b^2 + 88* a^2*b^4 - 5*b^6)*Cos[c + d*x] + 560*(16*a^5*b^3 - 28*a^3*b^5 + 9*a*b^7)*Co s[3*(c + d*x)] - 1344*a^3*b^5*Cos[5*(c + d*x)] + 1680*a*b^7*Cos[5*(c + d*x )] + 240*a*b^7*Cos[7*(c + d*x)] + 26880*a^6*b^2*Sin[2*(c + d*x)] - 53760*a ^4*b^4*Sin[2*(c + d*x)] + 25200*a^2*b^6*Sin[2*(c + d*x)] + 1680*b^8*Sin[2* (c + d*x)] - 3360*a^4*b^4*Sin[4*(c + d*x)] + 5040*a^2*b^6*Sin[4*(c + d*x)] - 840*b^8*Sin[4*(c + d*x)] + 560*a^2*b^6*Sin[6*(c + d*x)] - 560*b^8*Sin[6 *(c + d*x)] - 105*b^8*Sin[8*(c + d*x)])/(107520*b^9*d)
Time = 3.75 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.14, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.931, Rules used = {3042, 3375, 27, 3042, 3528, 27, 3042, 3528, 25, 3042, 3528, 27, 3042, 3528, 25, 3042, 3528, 25, 3042, 3502, 27, 3042, 3214, 3042, 3139, 1083, 217}
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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^6}{a+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3375 |
| \(\displaystyle \frac {\int \frac {4 \sin ^5(c+d x) \left (-7 \left (40 a^4-85 b^2 a^2+48 b^4\right ) \sin ^2(c+d x)-a b \left (5 a^2-14 b^2\right ) \sin (c+d x)+10 \left (24 a^4-49 b^2 a^2+28 b^4\right )\right )}{a+b \sin (c+d x)}dx}{1120 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sin ^5(c+d x) \left (-7 \left (40 a^4-85 b^2 a^2+48 b^4\right ) \sin ^2(c+d x)-a b \left (5 a^2-14 b^2\right ) \sin (c+d x)+10 \left (24 a^4-49 b^2 a^2+28 b^4\right )\right )}{a+b \sin (c+d x)}dx}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^5 \left (-7 \left (40 a^4-85 b^2 a^2+48 b^4\right ) \sin (c+d x)^2-a b \left (5 a^2-14 b^2\right ) \sin (c+d x)+10 \left (24 a^4-49 b^2 a^2+28 b^4\right )\right )}{a+b \sin (c+d x)}dx}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\int -\frac {5 \sin ^4(c+d x) \left (-b \left (8 a^2+7 b^2\right ) \sin (c+d x) a^2-12 \left (28 a^4-60 b^2 a^2+35 b^4\right ) \sin ^2(c+d x) a+7 \left (40 a^4-85 b^2 a^2+48 b^4\right ) a\right )}{a+b \sin (c+d x)}dx}{6 b}+\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \int \frac {\sin ^4(c+d x) \left (-b \left (8 a^2+7 b^2\right ) \sin (c+d x) a^2-12 \left (28 a^4-60 b^2 a^2+35 b^4\right ) \sin ^2(c+d x) a+7 \left (40 a^4-85 b^2 a^2+48 b^4\right ) a\right )}{a+b \sin (c+d x)}dx}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \int \frac {\sin (c+d x)^4 \left (-b \left (8 a^2+7 b^2\right ) \sin (c+d x) a^2-12 \left (28 a^4-60 b^2 a^2+35 b^4\right ) \sin (c+d x)^2 a+7 \left (40 a^4-85 b^2 a^2+48 b^4\right ) a\right )}{a+b \sin (c+d x)}dx}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {\int -\frac {\sin ^3(c+d x) \left (-b \left (56 a^2-95 b^2\right ) \sin (c+d x) a^3-35 \left (48 a^4-104 b^2 a^2+59 b^4\right ) \sin ^2(c+d x) a^2+48 \left (28 a^4-60 b^2 a^2+35 b^4\right ) a^2\right )}{a+b \sin (c+d x)}dx}{5 b}+\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\int \frac {\sin ^3(c+d x) \left (-b \left (56 a^2-95 b^2\right ) \sin (c+d x) a^3-35 \left (48 a^4-104 b^2 a^2+59 b^4\right ) \sin ^2(c+d x) a^2+48 \left (28 a^4-60 b^2 a^2+35 b^4\right ) a^2\right )}{a+b \sin (c+d x)}dx}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\int \frac {\sin (c+d x)^3 \left (-b \left (56 a^2-95 b^2\right ) \sin (c+d x) a^3-35 \left (48 a^4-104 b^2 a^2+59 b^4\right ) \sin (c+d x)^2 a^2+48 \left (28 a^4-60 b^2 a^2+35 b^4\right ) a^2\right )}{a+b \sin (c+d x)}dx}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {\int -\frac {3 \sin ^2(c+d x) \left (-64 \left (35 a^4-77 b^2 a^2+45 b^4\right ) \sin ^2(c+d x) a^3+35 \left (48 a^4-104 b^2 a^2+59 b^4\right ) a^3-b \left (112 a^4-200 b^2 a^2+175 b^4\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{4 b}+\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \int \frac {\sin ^2(c+d x) \left (-64 \left (35 a^4-77 b^2 a^2+45 b^4\right ) \sin ^2(c+d x) a^3+35 \left (48 a^4-104 b^2 a^2+59 b^4\right ) a^3-b \left (112 a^4-200 b^2 a^2+175 b^4\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \int \frac {\sin (c+d x)^2 \left (-64 \left (35 a^4-77 b^2 a^2+45 b^4\right ) \sin (c+d x)^2 a^3+35 \left (48 a^4-104 b^2 a^2+59 b^4\right ) a^3-b \left (112 a^4-200 b^2 a^2+175 b^4\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {\int -\frac {\sin (c+d x) \left (128 \left (35 a^4-77 b^2 a^2+45 b^4\right ) a^4-b \left (560 a^4-1064 b^2 a^2+435 b^4\right ) \sin (c+d x) a^3-105 \left (64 a^6-144 b^2 a^4+88 b^4 a^2-5 b^6\right ) \sin ^2(c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}+\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (128 \left (35 a^4-77 b^2 a^2+45 b^4\right ) a^4-b \left (560 a^4-1064 b^2 a^2+435 b^4\right ) \sin (c+d x) a^3-105 \left (64 a^6-144 b^2 a^4+88 b^4 a^2-5 b^6\right ) \sin ^2(c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (128 \left (35 a^4-77 b^2 a^2+45 b^4\right ) a^4-b \left (560 a^4-1064 b^2 a^2+435 b^4\right ) \sin (c+d x) a^3-105 \left (64 a^6-144 b^2 a^4+88 b^4 a^2-5 b^6\right ) \sin (c+d x)^2 a^2\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {\int -\frac {-128 \left (105 a^6-245 b^2 a^4+161 b^4 a^2-15 b^6\right ) \sin ^2(c+d x) a^3+105 \left (64 a^6-144 b^2 a^4+88 b^4 a^2-5 b^6\right ) a^3-b \left (2240 a^6-4592 b^2 a^4+2280 b^4 a^2+525 b^6\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}+\frac {105 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {-128 \left (105 a^6-245 b^2 a^4+161 b^4 a^2-15 b^6\right ) \sin ^2(c+d x) a^3+105 \left (64 a^6-144 b^2 a^4+88 b^4 a^2-5 b^6\right ) a^3-b \left (2240 a^6-4592 b^2 a^4+2280 b^4 a^2+525 b^6\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {-128 \left (105 a^6-245 b^2 a^4+161 b^4 a^2-15 b^6\right ) \sin (c+d x)^2 a^3+105 \left (64 a^6-144 b^2 a^4+88 b^4 a^2-5 b^6\right ) a^3-b \left (2240 a^6-4592 b^2 a^4+2280 b^4 a^2+525 b^6\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {\int \frac {105 \left (b \left (64 a^6-144 b^2 a^4+88 b^4 a^2-5 b^6\right ) a^3+\left (128 a^8-320 b^2 a^6+240 b^4 a^4-40 b^6 a^2-5 b^8\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{b}+\frac {128 a^3 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \int \frac {b \left (64 a^6-144 b^2 a^4+88 b^4 a^2-5 b^6\right ) a^3+\left (128 a^8-320 b^2 a^6+240 b^4 a^4-40 b^6 a^2-5 b^8\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{b}+\frac {128 a^3 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \int \frac {b \left (64 a^6-144 b^2 a^4+88 b^4 a^2-5 b^6\right ) a^3+\left (128 a^8-320 b^2 a^6+240 b^4 a^4-40 b^6 a^2-5 b^8\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{b}+\frac {128 a^3 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \left (\frac {a^2 x \left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right )}{b}-\frac {128 a^5 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {128 a^3 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \left (\frac {a^2 x \left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right )}{b}-\frac {128 a^5 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {128 a^3 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \left (\frac {a^2 x \left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right )}{b}-\frac {256 a^5 \left (a^2-b^2\right )^3 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}\right )}{b}+\frac {128 a^3 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \left (\frac {512 a^5 \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {a^2 x \left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right )}{b}\right )}{b}+\frac {128 a^3 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {b \sin ^5(c+d x) \cos (c+d x)}{5 a^2 d}+\frac {\frac {7 \left (40 a^4-85 a^2 b^2+48 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{6 b d}-\frac {5 \left (\frac {12 a \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {\frac {35 a^2 \left (48 a^4-104 a^2 b^2+59 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {64 a^3 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^2 \left (64 a^6-144 a^4 b^2+88 a^2 b^4-5 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {128 a^3 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}+\frac {105 \left (\frac {a^2 x \left (128 a^8-320 a^6 b^2+240 a^4 b^4-40 a^2 b^6-5 b^8\right )}{b}-\frac {256 a^5 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d}\right )}{b}}{2 b}}{3 b}\right )}{4 b}}{5 b}\right )}{6 b}}{280 a^2 b^2}-\frac {a \sin ^6(c+d x) \cos (c+d x)}{7 b^2 d}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d}+\frac {\sin ^7(c+d x) \cos (c+d x)}{8 b d}\) |
Input:
Int[(Cos[c + d*x]^6*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]
Output:
(Cos[c + d*x]*Sin[c + d*x]^4)/(4*a*d) - (b*Cos[c + d*x]*Sin[c + d*x]^5)/(5 *a^2*d) - (a*Cos[c + d*x]*Sin[c + d*x]^6)/(7*b^2*d) + (Cos[c + d*x]*Sin[c + d*x]^7)/(8*b*d) + ((7*(40*a^4 - 85*a^2*b^2 + 48*b^4)*Cos[c + d*x]*Sin[c + d*x]^5)/(6*b*d) - (5*((12*a*(28*a^4 - 60*a^2*b^2 + 35*b^4)*Cos[c + d*x]* Sin[c + d*x]^4)/(5*b*d) - ((35*a^2*(48*a^4 - 104*a^2*b^2 + 59*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(4*b*d) - (3*((64*a^3*(35*a^4 - 77*a^2*b^2 + 45*b^4) *Cos[c + d*x]*Sin[c + d*x]^2)/(3*b*d) - (-1/2*((105*((a^2*(128*a^8 - 320*a ^6*b^2 + 240*a^4*b^4 - 40*a^2*b^6 - 5*b^8)*x)/b - (256*a^5*(a^2 - b^2)^(5/ 2)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(b*d)))/b + ( 128*a^3*(105*a^6 - 245*a^4*b^2 + 161*a^2*b^4 - 15*b^6)*Cos[c + d*x])/(b*d) )/b + (105*a^2*(64*a^6 - 144*a^4*b^2 + 88*a^2*b^4 - 5*b^6)*Cos[c + d*x]*Si n[c + d*x])/(2*b*d))/(3*b)))/(4*b))/(5*b)))/(6*b))/(280*a^2*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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[cos[(e_.) + (f_.)*(x_)]^6*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]*(d*Sin[ e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1))), x] + (-Si mp[b*(m + n + 2)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*((a + b*Sin[e + f*x] )^(m + 1)/(a^2*d^2*f*(n + 1)*(n + 2))), x] - Simp[a*(n + 5)*Cos[e + f*x]*(d *Sin[e + f*x])^(n + 3)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d^3*f*(m + n + 5) *(m + n + 6))), x] + Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 4)*((a + b*Sin [e + f*x])^(m + 1)/(b*d^4*f*(m + n + 6))), x] + Simp[1/(a^2*b^2*d^2*(n + 1) *(n + 2)*(m + n + 5)*(m + n + 6)) Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin [e + f*x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2* n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m + n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e + f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4 )*(m + n + 5)*(m + n + 6) - a^2*b^2*(n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && Ne Q[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0] && NeQ[m + n + 6, 0] && !IGtQ[m, 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_)])^(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 = 2.54 (sec) , antiderivative size = 797, normalized size of antiderivative = 1.71
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {a^{7} b -\frac {7 a^{5} b^{3}}{3}+\frac {23 a^{3} b^{5}}{15}-\frac {a \,b^{7}}{7}+\left (-\frac {1}{2} a^{6} b^{2}+\frac {9}{8} a^{4} b^{4}-\frac {11}{16} a^{2} b^{6}+\frac {5}{128} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (7 a^{7} b -\frac {47}{3} a^{5} b^{3}+\frac {139}{15} a^{3} b^{5}-\frac {1}{7} a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {5}{2} a^{6} b^{2}+\frac {37}{8} a^{4} b^{4}-\frac {61}{48} a^{2} b^{6}-\frac {397}{384} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (21 a^{7} b -\frac {139}{3} a^{5} b^{3}+\frac {419}{15} a^{3} b^{5}-3 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {9}{2} a^{6} b^{2}+\frac {57}{8} a^{4} b^{4}-\frac {113}{48} a^{2} b^{6}+\frac {895}{384} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (35 a^{7} b -\frac {235}{3} a^{5} b^{3}+\frac {743}{15} a^{3} b^{5}-3 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {5}{2} a^{6} b^{2}+\frac {29}{8} a^{4} b^{4}-\frac {85}{48} a^{2} b^{6}-\frac {1765}{384} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (35 a^{7} b -\frac {245}{3} a^{5} b^{3}+\frac {161}{3} a^{3} b^{5}-5 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {5}{2} a^{6} b^{2}-\frac {29}{8} a^{4} b^{4}+\frac {85}{48} a^{2} b^{6}+\frac {1765}{384} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (21 a^{7} b -\frac {157}{3} a^{5} b^{3}+\frac {109}{3} a^{3} b^{5}-5 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {9}{2} a^{6} b^{2}-\frac {57}{8} a^{4} b^{4}+\frac {113}{48} a^{2} b^{6}-\frac {895}{384} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (7 a^{7} b -19 a^{5} b^{3}+15 a^{3} b^{5}-a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {5}{2} a^{6} b^{2}-\frac {37}{8} a^{4} b^{4}+\frac {61}{48} a^{2} b^{6}+\frac {397}{384} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+\left (a^{7} b -3 a^{5} b^{3}+3 a^{3} b^{5}-a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {1}{2} a^{6} b^{2}-\frac {9}{8} a^{4} b^{4}+\frac {11}{16} a^{2} b^{6}-\frac {5}{128} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}+\frac {\left (128 a^{8}-320 a^{6} b^{2}+240 a^{4} b^{4}-40 a^{2} b^{6}-5 b^{8}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}\right )}{b^{9}}+\frac {2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) a^{3} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{9} \sqrt {a^{2}-b^{2}}}}{d}\) | \(797\) |
| default | \(\frac {-\frac {2 \left (\frac {a^{7} b -\frac {7 a^{5} b^{3}}{3}+\frac {23 a^{3} b^{5}}{15}-\frac {a \,b^{7}}{7}+\left (-\frac {1}{2} a^{6} b^{2}+\frac {9}{8} a^{4} b^{4}-\frac {11}{16} a^{2} b^{6}+\frac {5}{128} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (7 a^{7} b -\frac {47}{3} a^{5} b^{3}+\frac {139}{15} a^{3} b^{5}-\frac {1}{7} a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {5}{2} a^{6} b^{2}+\frac {37}{8} a^{4} b^{4}-\frac {61}{48} a^{2} b^{6}-\frac {397}{384} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (21 a^{7} b -\frac {139}{3} a^{5} b^{3}+\frac {419}{15} a^{3} b^{5}-3 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {9}{2} a^{6} b^{2}+\frac {57}{8} a^{4} b^{4}-\frac {113}{48} a^{2} b^{6}+\frac {895}{384} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (35 a^{7} b -\frac {235}{3} a^{5} b^{3}+\frac {743}{15} a^{3} b^{5}-3 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {5}{2} a^{6} b^{2}+\frac {29}{8} a^{4} b^{4}-\frac {85}{48} a^{2} b^{6}-\frac {1765}{384} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (35 a^{7} b -\frac {245}{3} a^{5} b^{3}+\frac {161}{3} a^{3} b^{5}-5 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {5}{2} a^{6} b^{2}-\frac {29}{8} a^{4} b^{4}+\frac {85}{48} a^{2} b^{6}+\frac {1765}{384} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (21 a^{7} b -\frac {157}{3} a^{5} b^{3}+\frac {109}{3} a^{3} b^{5}-5 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {9}{2} a^{6} b^{2}-\frac {57}{8} a^{4} b^{4}+\frac {113}{48} a^{2} b^{6}-\frac {895}{384} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (7 a^{7} b -19 a^{5} b^{3}+15 a^{3} b^{5}-a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {5}{2} a^{6} b^{2}-\frac {37}{8} a^{4} b^{4}+\frac {61}{48} a^{2} b^{6}+\frac {397}{384} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+\left (a^{7} b -3 a^{5} b^{3}+3 a^{3} b^{5}-a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {1}{2} a^{6} b^{2}-\frac {9}{8} a^{4} b^{4}+\frac {11}{16} a^{2} b^{6}-\frac {5}{128} b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}+\frac {\left (128 a^{8}-320 a^{6} b^{2}+240 a^{4} b^{4}-40 a^{2} b^{6}-5 b^{8}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}\right )}{b^{9}}+\frac {2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) a^{3} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{9} \sqrt {a^{2}-b^{2}}}}{d}\) | \(797\) |
| risch | \(-\frac {11 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 b^{4} d}+\frac {5 a \,{\mathrm e}^{i \left (d x +c \right )}}{128 b^{2} d}-\frac {11 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 b^{4} d}+\frac {5 a \,{\mathrm e}^{-i \left (d x +c \right )}}{128 b^{2} d}+\frac {3 a \cos \left (3 d x +3 c \right )}{64 b^{2} d}+\frac {a \cos \left (7 d x +7 c \right )}{448 d \,b^{2}}+\frac {5 x}{128 b}+\frac {\sin \left (6 d x +6 c \right ) a^{2}}{192 b^{3} d}-\frac {a^{3} \cos \left (5 d x +5 c \right )}{80 d \,b^{4}}+\frac {a \cos \left (5 d x +5 c \right )}{64 d \,b^{2}}-\frac {\sin \left (4 d x +4 c \right ) a^{4}}{32 b^{5} d}+\frac {3 \sin \left (4 d x +4 c \right ) a^{2}}{64 b^{3} d}+\frac {a^{5} \cos \left (3 d x +3 c \right )}{12 d \,b^{6}}-\frac {7 a^{3} \cos \left (3 d x +3 c \right )}{48 d \,b^{4}}+\frac {\sin \left (2 d x +2 c \right ) a^{6}}{4 b^{7} d}-\frac {\sin \left (2 d x +2 c \right ) a^{4}}{2 b^{5} d}+\frac {i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{5}}-\frac {i \sqrt {a^{2}-b^{2}}\, a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{5}}-\frac {15 x \,a^{4}}{8 b^{5}}+\frac {5 x \,a^{2}}{16 b^{3}}-\frac {i \sqrt {a^{2}-b^{2}}\, a^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{9}}+\frac {2 i \sqrt {a^{2}-b^{2}}\, a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{7}}+\frac {i \sqrt {a^{2}-b^{2}}\, a^{7} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{9}}-\frac {2 i \sqrt {a^{2}-b^{2}}\, a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{7}}+\frac {15 a^{2} \sin \left (2 d x +2 c \right )}{64 b^{3} d}-\frac {\sin \left (6 d x +6 c \right )}{192 b d}-\frac {a^{7} {\mathrm e}^{i \left (d x +c \right )}}{2 b^{8} d}+\frac {9 a^{5} {\mathrm e}^{i \left (d x +c \right )}}{8 b^{6} d}-\frac {a^{7} {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{8} d}+\frac {9 a^{5} {\mathrm e}^{-i \left (d x +c \right )}}{8 b^{6} d}-\frac {\sin \left (8 d x +8 c \right )}{1024 b d}-\frac {x \,a^{8}}{b^{9}}+\frac {5 x \,a^{6}}{2 b^{7}}+\frac {\sin \left (2 d x +2 c \right )}{64 b d}-\frac {\sin \left (4 d x +4 c \right )}{128 b d}\) | \(826\) |
Input:
int(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(-2/b^9*((a^7*b-7/3*a^5*b^3+23/15*a^3*b^5-1/7*a*b^7+(-1/2*a^6*b^2+9/8* a^4*b^4-11/16*a^2*b^6+5/128*b^8)*tan(1/2*d*x+1/2*c)+(7*a^7*b-47/3*a^5*b^3+ 139/15*a^3*b^5-1/7*a*b^7)*tan(1/2*d*x+1/2*c)^2+(-5/2*a^6*b^2+37/8*a^4*b^4- 61/48*a^2*b^6-397/384*b^8)*tan(1/2*d*x+1/2*c)^3+(21*a^7*b-139/3*a^5*b^3+41 9/15*a^3*b^5-3*a*b^7)*tan(1/2*d*x+1/2*c)^4+(-9/2*a^6*b^2+57/8*a^4*b^4-113/ 48*a^2*b^6+895/384*b^8)*tan(1/2*d*x+1/2*c)^5+(35*a^7*b-235/3*a^5*b^3+743/1 5*a^3*b^5-3*a*b^7)*tan(1/2*d*x+1/2*c)^6+(-5/2*a^6*b^2+29/8*a^4*b^4-85/48*a ^2*b^6-1765/384*b^8)*tan(1/2*d*x+1/2*c)^7+(35*a^7*b-245/3*a^5*b^3+161/3*a^ 3*b^5-5*a*b^7)*tan(1/2*d*x+1/2*c)^8+(5/2*a^6*b^2-29/8*a^4*b^4+85/48*a^2*b^ 6+1765/384*b^8)*tan(1/2*d*x+1/2*c)^9+(21*a^7*b-157/3*a^5*b^3+109/3*a^3*b^5 -5*a*b^7)*tan(1/2*d*x+1/2*c)^10+(9/2*a^6*b^2-57/8*a^4*b^4+113/48*a^2*b^6-8 95/384*b^8)*tan(1/2*d*x+1/2*c)^11+(7*a^7*b-19*a^5*b^3+15*a^3*b^5-a*b^7)*ta n(1/2*d*x+1/2*c)^12+(5/2*a^6*b^2-37/8*a^4*b^4+61/48*a^2*b^6+397/384*b^8)*t an(1/2*d*x+1/2*c)^13+(a^7*b-3*a^5*b^3+3*a^3*b^5-a*b^7)*tan(1/2*d*x+1/2*c)^ 14+(1/2*a^6*b^2-9/8*a^4*b^4+11/16*a^2*b^6-5/128*b^8)*tan(1/2*d*x+1/2*c)^15 )/(1+tan(1/2*d*x+1/2*c)^2)^8+1/128*(128*a^8-320*a^6*b^2+240*a^4*b^4-40*a^2 *b^6-5*b^8)*arctan(tan(1/2*d*x+1/2*c)))+2/b^9*(a^6-3*a^4*b^2+3*a^2*b^4-b^6 )*a^3/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1 /2)))
Time = 0.15 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.51 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="fricas" )
Output:
[1/13440*(1920*a*b^7*cos(d*x + c)^7 - 2688*a^3*b^5*cos(d*x + c)^5 + 4480*( a^5*b^3 - a^3*b^5)*cos(d*x + c)^3 - 105*(128*a^8 - 320*a^6*b^2 + 240*a^4*b ^4 - 40*a^2*b^6 - 5*b^8)*d*x + 6720*(a^7 - 2*a^5*b^2 + a^3*b^4)*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2 *cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 13440*(a^7*b - 2*a^5* b^3 + a^3*b^5)*cos(d*x + c) - 35*(48*b^8*cos(d*x + c)^7 - 8*(8*a^2*b^6 + b ^8)*cos(d*x + c)^5 + 2*(48*a^4*b^4 - 40*a^2*b^6 - 5*b^8)*cos(d*x + c)^3 - 3*(64*a^6*b^2 - 112*a^4*b^4 + 40*a^2*b^6 + 5*b^8)*cos(d*x + c))*sin(d*x + c))/(b^9*d), 1/13440*(1920*a*b^7*cos(d*x + c)^7 - 2688*a^3*b^5*cos(d*x + c )^5 + 4480*(a^5*b^3 - a^3*b^5)*cos(d*x + c)^3 - 105*(128*a^8 - 320*a^6*b^2 + 240*a^4*b^4 - 40*a^2*b^6 - 5*b^8)*d*x - 13440*(a^7 - 2*a^5*b^2 + a^3*b^ 4)*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 13440*(a^7*b - 2*a^5*b^3 + a^3*b^5)*cos(d*x + c) - 35*(48*b^8*cos( d*x + c)^7 - 8*(8*a^2*b^6 + b^8)*cos(d*x + c)^5 + 2*(48*a^4*b^4 - 40*a^2*b ^6 - 5*b^8)*cos(d*x + c)^3 - 3*(64*a^6*b^2 - 112*a^4*b^4 + 40*a^2*b^6 + 5* b^8)*cos(d*x + c))*sin(d*x + c))/(b^9*d)]
Timed out. \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**6*sin(d*x+c)**3/(a+b*sin(d*x+c)),x)
Output:
Timed out
Exception generated. \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="maxima" )
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 1244 vs. \(2 (440) = 880\).
Time = 0.25 (sec) , antiderivative size = 1244, normalized size of antiderivative = 2.66 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
-1/13440*(105*(128*a^8 - 320*a^6*b^2 + 240*a^4*b^4 - 40*a^2*b^6 - 5*b^8)*( d*x + c)/b^9 - 26880*(a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*(pi*floor(1/2 *(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^9) + 2*(6720*a^6*b*tan(1/2*d*x + 1/2*c)^15 - 15120*a^4*b^3*tan(1/2*d*x + 1/2*c)^15 + 9240*a^2*b^5*tan(1/2*d*x + 1/2*c)^ 15 - 525*b^7*tan(1/2*d*x + 1/2*c)^15 + 13440*a^7*tan(1/2*d*x + 1/2*c)^14 - 40320*a^5*b^2*tan(1/2*d*x + 1/2*c)^14 + 40320*a^3*b^4*tan(1/2*d*x + 1/2*c )^14 - 13440*a*b^6*tan(1/2*d*x + 1/2*c)^14 + 33600*a^6*b*tan(1/2*d*x + 1/2 *c)^13 - 62160*a^4*b^3*tan(1/2*d*x + 1/2*c)^13 + 17080*a^2*b^5*tan(1/2*d*x + 1/2*c)^13 + 13895*b^7*tan(1/2*d*x + 1/2*c)^13 + 94080*a^7*tan(1/2*d*x + 1/2*c)^12 - 255360*a^5*b^2*tan(1/2*d*x + 1/2*c)^12 + 201600*a^3*b^4*tan(1 /2*d*x + 1/2*c)^12 - 13440*a*b^6*tan(1/2*d*x + 1/2*c)^12 + 60480*a^6*b*tan (1/2*d*x + 1/2*c)^11 - 95760*a^4*b^3*tan(1/2*d*x + 1/2*c)^11 + 31640*a^2*b ^5*tan(1/2*d*x + 1/2*c)^11 - 31325*b^7*tan(1/2*d*x + 1/2*c)^11 + 282240*a^ 7*tan(1/2*d*x + 1/2*c)^10 - 703360*a^5*b^2*tan(1/2*d*x + 1/2*c)^10 + 48832 0*a^3*b^4*tan(1/2*d*x + 1/2*c)^10 - 67200*a*b^6*tan(1/2*d*x + 1/2*c)^10 + 33600*a^6*b*tan(1/2*d*x + 1/2*c)^9 - 48720*a^4*b^3*tan(1/2*d*x + 1/2*c)^9 + 23800*a^2*b^5*tan(1/2*d*x + 1/2*c)^9 + 61775*b^7*tan(1/2*d*x + 1/2*c)^9 + 470400*a^7*tan(1/2*d*x + 1/2*c)^8 - 1097600*a^5*b^2*tan(1/2*d*x + 1/2*c) ^8 + 721280*a^3*b^4*tan(1/2*d*x + 1/2*c)^8 - 67200*a*b^6*tan(1/2*d*x + ...
Time = 21.88 (sec) , antiderivative size = 4505, normalized size of antiderivative = 9.65 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
int((cos(c + d*x)^6*sin(c + d*x)^3)/(a + b*sin(c + d*x)),x)
Output:
((2*(15*a*b^6 - 105*a^7 - 161*a^3*b^4 + 245*a^5*b^2))/(105*b^8) + (2*tan(c /2 + (d*x)/2)^14*(a*b^6 - a^7 - 3*a^3*b^4 + 3*a^5*b^2))/b^8 + (2*tan(c/2 + (d*x)/2)^12*(a*b^6 - 7*a^7 - 15*a^3*b^4 + 19*a^5*b^2))/b^8 + (2*tan(c/2 + (d*x)/2)^10*(15*a*b^6 - 63*a^7 - 109*a^3*b^4 + 157*a^5*b^2))/(3*b^8) + (2 *tan(c/2 + (d*x)/2)^8*(15*a*b^6 - 105*a^7 - 161*a^3*b^4 + 245*a^5*b^2))/(3 *b^8) + (2*tan(c/2 + (d*x)/2)^4*(45*a*b^6 - 315*a^7 - 419*a^3*b^4 + 695*a^ 5*b^2))/(15*b^8) + (2*tan(c/2 + (d*x)/2)^6*(45*a*b^6 - 525*a^7 - 743*a^3*b ^4 + 1175*a^5*b^2))/(15*b^8) + (2*tan(c/2 + (d*x)/2)^2*(15*a*b^6 - 735*a^7 - 973*a^3*b^4 + 1645*a^5*b^2))/(105*b^8) + (tan(c/2 + (d*x)/2)*(64*a^6 - 5*b^6 + 88*a^2*b^4 - 144*a^4*b^2))/(64*b^7) - (tan(c/2 + (d*x)/2)^15*(64*a ^6 - 5*b^6 + 88*a^2*b^4 - 144*a^4*b^2))/(64*b^7) + (tan(c/2 + (d*x)/2)^3*( 960*a^6 + 397*b^6 + 488*a^2*b^4 - 1776*a^4*b^2))/(192*b^7) - (tan(c/2 + (d *x)/2)^13*(960*a^6 + 397*b^6 + 488*a^2*b^4 - 1776*a^4*b^2))/(192*b^7) + (t an(c/2 + (d*x)/2)^7*(960*a^6 + 1765*b^6 + 680*a^2*b^4 - 1392*a^4*b^2))/(19 2*b^7) - (tan(c/2 + (d*x)/2)^9*(960*a^6 + 1765*b^6 + 680*a^2*b^4 - 1392*a^ 4*b^2))/(192*b^7) + (tan(c/2 + (d*x)/2)^5*(1728*a^6 - 895*b^6 + 904*a^2*b^ 4 - 2736*a^4*b^2))/(192*b^7) - (tan(c/2 + (d*x)/2)^11*(1728*a^6 - 895*b^6 + 904*a^2*b^4 - 2736*a^4*b^2))/(192*b^7))/(d*(8*tan(c/2 + (d*x)/2)^2 + 28* tan(c/2 + (d*x)/2)^4 + 56*tan(c/2 + (d*x)/2)^6 + 70*tan(c/2 + (d*x)/2)^8 + 56*tan(c/2 + (d*x)/2)^10 + 28*tan(c/2 + (d*x)/2)^12 + 8*tan(c/2 + (d*x...
Time = 0.40 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-53760 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{5} b^{2}+26880 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{3} b^{4}+3360 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{4} b^{4}-7280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b^{6}-4480 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{5} b^{3}+9856 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3} b^{5}-5760 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a \,b^{7}+6720 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{6} b^{2}-15120 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4} b^{4}+9240 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{6}-13440 a^{8} c +525 b^{8} c -1920 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a \,b^{7}+2240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a^{2} b^{6}+26880 \sqrt {a^{2}-b^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b}{\sqrt {a^{2}-b^{2}}}\right ) a^{7}+4130 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{8}-13440 \cos \left (d x +c \right ) a^{7} b +31360 \cos \left (d x +c \right ) a^{5} b^{3}-10080 a^{7} b +21280 a^{5} b^{3}-10528 a^{3} b^{5}-1440 a \,b^{7}+33600 a^{6} b^{2} d x -25200 a^{4} b^{4} d x +4200 a^{2} b^{6} d x -2688 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{3} b^{5}+5760 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a \,b^{7}+1680 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} b^{8}-4760 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{8}-525 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{8}+1920 \cos \left (d x +c \right ) a \,b^{7}-20608 \cos \left (d x +c \right ) a^{3} b^{5}-13440 a^{8} d x +33600 a^{6} b^{2} c -25200 a^{4} b^{4} c +4200 a^{2} b^{6} c +525 b^{8} d x}{13440 b^{9} d} \] Input:
int(cos(d*x+c)^6*sin(d*x+c)^3/(a+b*sin(d*x+c)),x)
Output:
(26880*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))* a**7 - 53760*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b **2))*a**5*b**2 + 26880*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sq rt(a**2 - b**2))*a**3*b**4 + 1680*cos(c + d*x)*sin(c + d*x)**7*b**8 - 1920 *cos(c + d*x)*sin(c + d*x)**6*a*b**7 + 2240*cos(c + d*x)*sin(c + d*x)**5*a **2*b**6 - 4760*cos(c + d*x)*sin(c + d*x)**5*b**8 - 2688*cos(c + d*x)*sin( c + d*x)**4*a**3*b**5 + 5760*cos(c + d*x)*sin(c + d*x)**4*a*b**7 + 3360*co s(c + d*x)*sin(c + d*x)**3*a**4*b**4 - 7280*cos(c + d*x)*sin(c + d*x)**3*a **2*b**6 + 4130*cos(c + d*x)*sin(c + d*x)**3*b**8 - 4480*cos(c + d*x)*sin( c + d*x)**2*a**5*b**3 + 9856*cos(c + d*x)*sin(c + d*x)**2*a**3*b**5 - 5760 *cos(c + d*x)*sin(c + d*x)**2*a*b**7 + 6720*cos(c + d*x)*sin(c + d*x)*a**6 *b**2 - 15120*cos(c + d*x)*sin(c + d*x)*a**4*b**4 + 9240*cos(c + d*x)*sin( c + d*x)*a**2*b**6 - 525*cos(c + d*x)*sin(c + d*x)*b**8 - 13440*cos(c + d* x)*a**7*b + 31360*cos(c + d*x)*a**5*b**3 - 20608*cos(c + d*x)*a**3*b**5 + 1920*cos(c + d*x)*a*b**7 - 13440*a**8*c - 13440*a**8*d*x - 10080*a**7*b + 33600*a**6*b**2*c + 33600*a**6*b**2*d*x + 21280*a**5*b**3 - 25200*a**4*b** 4*c - 25200*a**4*b**4*d*x - 10528*a**3*b**5 + 4200*a**2*b**6*c + 4200*a**2 *b**6*d*x - 1440*a*b**7 + 525*b**8*c + 525*b**8*d*x)/(13440*b**9*d)