Integrand size = 29, antiderivative size = 408 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {a \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right ) x}{16 b^8}-\frac {2 a^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^8 d}+\frac {\left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{105 b^7 d}-\frac {a \left (8 a^4-18 a^2 b^2+11 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^6 d}+\frac {\left (35 a^4-77 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{105 b^5 d}+\frac {\cos (c+d x) \sin ^3(c+d x)}{3 a d}-\frac {\left (6 a^4-13 a^2 b^2+8 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a b^4 d}-\frac {b \cos (c+d x) \sin ^4(c+d x)}{4 a^2 d}+\frac {\left (28 a^4-60 a^2 b^2+35 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{140 a^2 b^3 d}-\frac {a \cos (c+d x) \sin ^5(c+d x)}{6 b^2 d}+\frac {\cos (c+d x) \sin ^6(c+d x)}{7 b d} \]
1/16*a*(16*a^6-40*a^4*b^2+30*a^2*b^4-5*b^6)*x/b^8-2*a^2*(a^2-b^2)^(5/2)*ar ctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^8/d+1/105*(105*a^6-245*a^ 4*b^2+161*a^2*b^4-15*b^6)*cos(d*x+c)/b^7/d-1/16*a*(8*a^4-18*a^2*b^2+11*b^4 )*cos(d*x+c)*sin(d*x+c)/b^6/d+1/105*(35*a^4-77*a^2*b^2+45*b^4)*cos(d*x+c)* sin(d*x+c)^2/b^5/d+1/3*cos(d*x+c)*sin(d*x+c)^3/a/d-1/24*(6*a^4-13*a^2*b^2+ 8*b^4)*cos(d*x+c)*sin(d*x+c)^3/a/b^4/d-1/4*b*cos(d*x+c)*sin(d*x+c)^4/a^2/d +1/140*(28*a^4-60*a^2*b^2+35*b^4)*cos(d*x+c)*sin(d*x+c)^4/a^2/b^3/d-1/6*a* cos(d*x+c)*sin(d*x+c)^5/b^2/d+1/7*cos(d*x+c)*sin(d*x+c)^6/b/d
Time = 2.33 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {-6720 a^7 c+16800 a^5 b^2 c-12600 a^3 b^4 c+2100 a b^6 c-6720 a^7 d x+16800 a^5 b^2 d x-12600 a^3 b^4 d x+2100 a b^6 d x+13440 a^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+105 b \left (-64 a^6+144 a^4 b^2-88 a^2 b^4+5 b^6\right ) \cos (c+d x)+35 \left (16 a^4 b^3-28 a^2 b^5+9 b^7\right ) \cos (3 (c+d x))-84 a^2 b^5 \cos (5 (c+d x))+105 b^7 \cos (5 (c+d x))+15 b^7 \cos (7 (c+d x))+1680 a^5 b^2 \sin (2 (c+d x))-3360 a^3 b^4 \sin (2 (c+d x))+1575 a b^6 \sin (2 (c+d x))-210 a^3 b^4 \sin (4 (c+d x))+315 a b^6 \sin (4 (c+d x))+35 a b^6 \sin (6 (c+d x))}{6720 b^8 d} \]
-1/6720*(-6720*a^7*c + 16800*a^5*b^2*c - 12600*a^3*b^4*c + 2100*a*b^6*c - 6720*a^7*d*x + 16800*a^5*b^2*d*x - 12600*a^3*b^4*d*x + 2100*a*b^6*d*x + 13 440*a^2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 105*b*(-64*a^6 + 144*a^4*b^2 - 88*a^2*b^4 + 5*b^6)*Cos[c + d*x] + 35*(1 6*a^4*b^3 - 28*a^2*b^5 + 9*b^7)*Cos[3*(c + d*x)] - 84*a^2*b^5*Cos[5*(c + d *x)] + 105*b^7*Cos[5*(c + d*x)] + 15*b^7*Cos[7*(c + d*x)] + 1680*a^5*b^2*S in[2*(c + d*x)] - 3360*a^3*b^4*Sin[2*(c + d*x)] + 1575*a*b^6*Sin[2*(c + d* x)] - 210*a^3*b^4*Sin[4*(c + d*x)] + 315*a*b^6*Sin[4*(c + d*x)] + 35*a*b^6 *Sin[6*(c + d*x)])/(b^8*d)
Time = 2.84 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.13, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.828, Rules used = {3042, 3375, 27, 3042, 3528, 27, 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.
| \(\displaystyle \int \frac {\sin ^2(c+d x) \cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^2 \cos (c+d x)^6}{a+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3375 |
| \(\displaystyle \frac {\int \frac {6 \sin ^4(c+d x) \left (-3 \left (28 a^4-60 b^2 a^2+35 b^4\right ) \sin ^2(c+d x)-a b \left (2 a^2-7 b^2\right ) \sin (c+d x)+14 \left (5 a^4-10 b^2 a^2+6 b^4\right )\right )}{a+b \sin (c+d x)}dx}{504 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sin ^4(c+d x) \left (-3 \left (28 a^4-60 b^2 a^2+35 b^4\right ) \sin ^2(c+d x)-a b \left (2 a^2-7 b^2\right ) \sin (c+d x)+14 \left (5 a^4-10 b^2 a^2+6 b^4\right )\right )}{a+b \sin (c+d x)}dx}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^4 \left (-3 \left (28 a^4-60 b^2 a^2+35 b^4\right ) \sin (c+d x)^2-a b \left (2 a^2-7 b^2\right ) \sin (c+d x)+14 \left (5 a^4-10 b^2 a^2+6 b^4\right )\right )}{a+b \sin (c+d x)}dx}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\int -\frac {2 \sin ^3(c+d x) \left (-b \left (7 a^2+10 b^2\right ) \sin (c+d x) a^2-35 \left (6 a^4-13 b^2 a^2+8 b^4\right ) \sin ^2(c+d x) a+6 \left (28 a^4-60 b^2 a^2+35 b^4\right ) a\right )}{a+b \sin (c+d x)}dx}{5 b}+\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \int \frac {\sin ^3(c+d x) \left (-b \left (7 a^2+10 b^2\right ) \sin (c+d x) a^2-35 \left (6 a^4-13 b^2 a^2+8 b^4\right ) \sin ^2(c+d x) a+6 \left (28 a^4-60 b^2 a^2+35 b^4\right ) a\right )}{a+b \sin (c+d x)}dx}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \int \frac {\sin (c+d x)^3 \left (-b \left (7 a^2+10 b^2\right ) \sin (c+d x) a^2-35 \left (6 a^4-13 b^2 a^2+8 b^4\right ) \sin (c+d x)^2 a+6 \left (28 a^4-60 b^2 a^2+35 b^4\right ) a\right )}{a+b \sin (c+d x)}dx}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {\int -\frac {3 \sin ^2(c+d x) \left (-b \left (14 a^2-25 b^2\right ) \sin (c+d x) a^3-8 \left (35 a^4-77 b^2 a^2+45 b^4\right ) \sin ^2(c+d x) a^2+35 \left (6 a^4-13 b^2 a^2+8 b^4\right ) a^2\right )}{a+b \sin (c+d x)}dx}{4 b}+\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \int \frac {\sin ^2(c+d x) \left (-b \left (14 a^2-25 b^2\right ) \sin (c+d x) a^3-8 \left (35 a^4-77 b^2 a^2+45 b^4\right ) \sin ^2(c+d x) a^2+35 \left (6 a^4-13 b^2 a^2+8 b^4\right ) a^2\right )}{a+b \sin (c+d x)}dx}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \int \frac {\sin (c+d x)^2 \left (-b \left (14 a^2-25 b^2\right ) \sin (c+d x) a^3-8 \left (35 a^4-77 b^2 a^2+45 b^4\right ) \sin (c+d x)^2 a^2+35 \left (6 a^4-13 b^2 a^2+8 b^4\right ) a^2\right )}{a+b \sin (c+d x)}dx}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {\int -\frac {\sin (c+d x) \left (-105 \left (8 a^4-18 b^2 a^2+11 b^4\right ) \sin ^2(c+d x) a^3+16 \left (35 a^4-77 b^2 a^2+45 b^4\right ) a^3-b \left (70 a^4-133 b^2 a^2+120 b^4\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}+\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-105 \left (8 a^4-18 b^2 a^2+11 b^4\right ) \sin ^2(c+d x) a^3+16 \left (35 a^4-77 b^2 a^2+45 b^4\right ) a^3-b \left (70 a^4-133 b^2 a^2+120 b^4\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-105 \left (8 a^4-18 b^2 a^2+11 b^4\right ) \sin (c+d x)^2 a^3+16 \left (35 a^4-77 b^2 a^2+45 b^4\right ) a^3-b \left (70 a^4-133 b^2 a^2+120 b^4\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {\int -\frac {105 \left (8 a^4-18 b^2 a^2+11 b^4\right ) a^4-b \left (280 a^4-574 b^2 a^2+285 b^4\right ) \sin (c+d x) a^3-16 \left (105 a^6-245 b^2 a^4+161 b^4 a^2-15 b^6\right ) \sin ^2(c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}+\frac {105 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {105 \left (8 a^4-18 b^2 a^2+11 b^4\right ) a^4-b \left (280 a^4-574 b^2 a^2+285 b^4\right ) \sin (c+d x) a^3-16 \left (105 a^6-245 b^2 a^4+161 b^4 a^2-15 b^6\right ) \sin ^2(c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {105 \left (8 a^4-18 b^2 a^2+11 b^4\right ) a^4-b \left (280 a^4-574 b^2 a^2+285 b^4\right ) \sin (c+d x) a^3-16 \left (105 a^6-245 b^2 a^4+161 b^4 a^2-15 b^6\right ) \sin (c+d x)^2 a^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {\int \frac {105 \left (b \left (8 a^4-18 b^2 a^2+11 b^4\right ) a^4+\left (16 a^6-40 b^2 a^4+30 b^4 a^2-5 b^6\right ) \sin (c+d x) a^3\right )}{a+b \sin (c+d x)}dx}{b}+\frac {16 a^2 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \int \frac {b \left (8 a^4-18 b^2 a^2+11 b^4\right ) a^4+\left (16 a^6-40 b^2 a^4+30 b^4 a^2-5 b^6\right ) \sin (c+d x) a^3}{a+b \sin (c+d x)}dx}{b}+\frac {16 a^2 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \int \frac {b \left (8 a^4-18 b^2 a^2+11 b^4\right ) a^4+\left (16 a^6-40 b^2 a^4+30 b^4 a^2-5 b^6\right ) \sin (c+d x) a^3}{a+b \sin (c+d x)}dx}{b}+\frac {16 a^2 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \left (\frac {a^3 x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{b}-\frac {16 a^4 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {16 a^2 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \left (\frac {a^3 x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{b}-\frac {16 a^4 \left (a^2-b^2\right )^3 \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {16 a^2 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \left (\frac {a^3 x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{b}-\frac {32 a^4 \left (a^2-b^2\right )^3 \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}\right )}{b}+\frac {16 a^2 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {105 \left (\frac {64 a^4 \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {a^3 x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{b}\right )}{b}+\frac {16 a^2 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {b \sin ^4(c+d x) \cos (c+d x)}{4 a^2 d}+\frac {\frac {3 \left (28 a^4-60 a^2 b^2+35 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{5 b d}-\frac {2 \left (\frac {35 a \left (6 a^4-13 a^2 b^2+8 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {8 a^2 \left (35 a^4-77 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {105 a^3 \left (8 a^4-18 a^2 b^2+11 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {16 a^2 \left (105 a^6-245 a^4 b^2+161 a^2 b^4-15 b^6\right ) \cos (c+d x)}{b d}+\frac {105 \left (\frac {a^3 x \left (16 a^6-40 a^4 b^2+30 a^2 b^4-5 b^6\right )}{b}-\frac {32 a^4 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d}\right )}{b}}{2 b}}{3 b}\right )}{4 b}\right )}{5 b}}{84 a^2 b^2}-\frac {a \sin ^5(c+d x) \cos (c+d x)}{6 b^2 d}+\frac {\sin ^3(c+d x) \cos (c+d x)}{3 a d}+\frac {\sin ^6(c+d x) \cos (c+d x)}{7 b d}\) |
(Cos[c + d*x]*Sin[c + d*x]^3)/(3*a*d) - (b*Cos[c + d*x]*Sin[c + d*x]^4)/(4 *a^2*d) - (a*Cos[c + d*x]*Sin[c + d*x]^5)/(6*b^2*d) + (Cos[c + d*x]*Sin[c + d*x]^6)/(7*b*d) + ((3*(28*a^4 - 60*a^2*b^2 + 35*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(5*b*d) - (2*((35*a*(6*a^4 - 13*a^2*b^2 + 8*b^4)*Cos[c + d*x]*Si n[c + d*x]^3)/(4*b*d) - (3*((8*a^2*(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^3*(16*a^6 - 40*a^4*b^2 + 30 *a^2*b^4 - 5*b^6)*x)/b - (32*a^4*(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 + (16*a^2*(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^3*(8*a^4 - 18* a^2*b^2 + 11*b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*b*d))/(3*b)))/(4*b)))/(5*b ))/(84*a^2*b^2)
3.14.21.3.1 Defintions of rubi rules used
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 = 1.31 (sec) , antiderivative size = 601, normalized size of antiderivative = 1.47
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {2 \left (\left (\frac {1}{2} a^{5} b^{2}-\frac {9}{8} a^{3} b^{4}+\frac {11}{16} a \,b^{6}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{6} b -3 a^{4} b^{3}+3 a^{2} b^{5}-b^{7}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a^{5} b^{2}-\frac {7}{2} a^{3} b^{4}+\frac {7}{12} a \,b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{6} b -16 a^{4} b^{3}+12 a^{2} b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{5} b^{2}-\frac {29}{8} a^{3} b^{4}+\frac {85}{48} a \,b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{6} b -\frac {109}{3} a^{4} b^{3}+\frac {73}{3} a^{2} b^{5}-5 b^{7}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (20 a^{6} b -\frac {136}{3} a^{4} b^{3}+\frac {88}{3} a^{2} b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {5}{2} a^{5} b^{2}+\frac {29}{8} a^{3} b^{4}-\frac {85}{48} a \,b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{6} b -33 a^{4} b^{3}+\frac {101}{5} a^{2} b^{5}-3 b^{7}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{5} b^{2}+\frac {7}{2} a^{3} b^{4}-\frac {7}{12} a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{6} b -\frac {40}{3} a^{4} b^{3}+\frac {116}{15} a^{2} b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{5} b^{2}+\frac {9}{8} a^{3} b^{4}-\frac {11}{16} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{6} b -\frac {7 a^{4} b^{3}}{3}+\frac {23 a^{2} b^{5}}{15}-\frac {b^{7}}{7}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {a \left (16 a^{6}-40 a^{4} b^{2}+30 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{b^{8}}-\frac {2 a^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \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^{8} \sqrt {a^{2}-b^{2}}}}{d}\) | \(601\) |
| default | \(\frac {\frac {\frac {2 \left (\left (\frac {1}{2} a^{5} b^{2}-\frac {9}{8} a^{3} b^{4}+\frac {11}{16} a \,b^{6}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (a^{6} b -3 a^{4} b^{3}+3 a^{2} b^{5}-b^{7}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a^{5} b^{2}-\frac {7}{2} a^{3} b^{4}+\frac {7}{12} a \,b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{6} b -16 a^{4} b^{3}+12 a^{2} b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {5}{2} a^{5} b^{2}-\frac {29}{8} a^{3} b^{4}+\frac {85}{48} a \,b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{6} b -\frac {109}{3} a^{4} b^{3}+\frac {73}{3} a^{2} b^{5}-5 b^{7}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (20 a^{6} b -\frac {136}{3} a^{4} b^{3}+\frac {88}{3} a^{2} b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {5}{2} a^{5} b^{2}+\frac {29}{8} a^{3} b^{4}-\frac {85}{48} a \,b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{6} b -33 a^{4} b^{3}+\frac {101}{5} a^{2} b^{5}-3 b^{7}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{5} b^{2}+\frac {7}{2} a^{3} b^{4}-\frac {7}{12} a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (6 a^{6} b -\frac {40}{3} a^{4} b^{3}+\frac {116}{15} a^{2} b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {1}{2} a^{5} b^{2}+\frac {9}{8} a^{3} b^{4}-\frac {11}{16} a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a^{6} b -\frac {7 a^{4} b^{3}}{3}+\frac {23 a^{2} b^{5}}{15}-\frac {b^{7}}{7}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {a \left (16 a^{6}-40 a^{4} b^{2}+30 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{b^{8}}-\frac {2 a^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \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^{8} \sqrt {a^{2}-b^{2}}}}{d}\) | \(601\) |
| risch | \(\frac {\cos \left (5 d x +5 c \right ) a^{2}}{80 b^{3} d}+\frac {a^{3} \sin \left (4 d x +4 c \right )}{32 b^{4} d}-\frac {\cos \left (7 d x +7 c \right )}{448 b d}-\frac {5 a x}{16 b^{2}}-\frac {\cos \left (3 d x +3 c \right ) a^{4}}{12 b^{5} d}-\frac {a^{5} \sin \left (2 d x +2 c \right )}{4 b^{6} d}-\frac {15 a \sin \left (2 d x +2 c \right )}{64 b^{2} d}-\frac {5 a^{5} x}{2 b^{6}}+\frac {15 a^{3} x}{8 b^{4}}-\frac {3 \cos \left (3 d x +3 c \right )}{64 b d}+\frac {7 \cos \left (3 d x +3 c \right ) a^{2}}{48 b^{3} d}-\frac {a \sin \left (6 d x +6 c \right )}{192 b^{2} d}-\frac {3 a \sin \left (4 d x +4 c \right )}{64 b^{2} d}+\frac {a^{3} \sin \left (2 d x +2 c \right )}{2 d \,b^{4}}-\frac {\cos \left (5 d x +5 c \right )}{64 b d}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}}{128 b d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )}}{128 b d}-\frac {\sqrt {-a^{2}+b^{2}}\, a^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{8}}+\frac {{\mathrm e}^{i \left (d x +c \right )} a^{6}}{2 b^{7} d}+\frac {{\mathrm e}^{-i \left (d x +c \right )} a^{6}}{2 b^{7} d}-\frac {9 \,{\mathrm e}^{i \left (d x +c \right )} a^{4}}{8 b^{5} d}+\frac {11 \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{16 b^{3} d}-\frac {9 \,{\mathrm e}^{-i \left (d x +c \right )} a^{4}}{8 b^{5} d}+\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{16 b^{3} d}+\frac {a^{7} x}{b^{8}}+\frac {\sqrt {-a^{2}+b^{2}}\, a^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{8}}+\frac {\sqrt {-a^{2}+b^{2}}\, a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}-\frac {2 \sqrt {-a^{2}+b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{6}}+\frac {2 \sqrt {-a^{2}+b^{2}}\, a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{6}}-\frac {\sqrt {-a^{2}+b^{2}}\, a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}\) | \(736\) |
1/d*(2/b^8*(((1/2*a^5*b^2-9/8*a^3*b^4+11/16*a*b^6)*tan(1/2*d*x+1/2*c)^13+( a^6*b-3*a^4*b^3+3*a^2*b^5-b^7)*tan(1/2*d*x+1/2*c)^12+(2*a^5*b^2-7/2*a^3*b^ 4+7/12*a*b^6)*tan(1/2*d*x+1/2*c)^11+(6*a^6*b-16*a^4*b^3+12*a^2*b^5)*tan(1/ 2*d*x+1/2*c)^10+(5/2*a^5*b^2-29/8*a^3*b^4+85/48*a*b^6)*tan(1/2*d*x+1/2*c)^ 9+(15*a^6*b-109/3*a^4*b^3+73/3*a^2*b^5-5*b^7)*tan(1/2*d*x+1/2*c)^8+(20*a^6 *b-136/3*a^4*b^3+88/3*a^2*b^5)*tan(1/2*d*x+1/2*c)^6+(-5/2*a^5*b^2+29/8*a^3 *b^4-85/48*a*b^6)*tan(1/2*d*x+1/2*c)^5+(15*a^6*b-33*a^4*b^3+101/5*a^2*b^5- 3*b^7)*tan(1/2*d*x+1/2*c)^4+(-2*a^5*b^2+7/2*a^3*b^4-7/12*a*b^6)*tan(1/2*d* x+1/2*c)^3+(6*a^6*b-40/3*a^4*b^3+116/15*a^2*b^5)*tan(1/2*d*x+1/2*c)^2+(-1/ 2*a^5*b^2+9/8*a^3*b^4-11/16*a*b^6)*tan(1/2*d*x+1/2*c)+a^6*b-7/3*a^4*b^3+23 /15*a^2*b^5-1/7*b^7)/(1+tan(1/2*d*x+1/2*c)^2)^7+1/16*a*(16*a^6-40*a^4*b^2+ 30*a^2*b^4-5*b^6)*arctan(tan(1/2*d*x+1/2*c)))-2*a^2*(a^6-3*a^4*b^2+3*a^2*b ^4-b^6)/b^8/(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 = 0.49 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.52 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\left [-\frac {240 \, b^{7} \cos \left (d x + c\right )^{7} - 336 \, a^{2} b^{5} \cos \left (d x + c\right )^{5} + 560 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (16 \, a^{7} - 40 \, a^{5} b^{2} + 30 \, a^{3} b^{4} - 5 \, a b^{6}\right )} d x - 840 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 1680 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right ) + 35 \, {\left (8 \, a b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{5} b^{2} - 14 \, a^{3} b^{4} + 5 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, b^{8} d}, -\frac {240 \, b^{7} \cos \left (d x + c\right )^{7} - 336 \, a^{2} b^{5} \cos \left (d x + c\right )^{5} + 560 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \cos \left (d x + c\right )^{3} - 105 \, {\left (16 \, a^{7} - 40 \, a^{5} b^{2} + 30 \, a^{3} b^{4} - 5 \, a b^{6}\right )} d x - 1680 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - 1680 \, {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right ) + 35 \, {\left (8 \, a b^{6} \cos \left (d x + c\right )^{5} - 2 \, {\left (6 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{5} b^{2} - 14 \, a^{3} b^{4} + 5 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, b^{8} d}\right ] \]
[-1/1680*(240*b^7*cos(d*x + c)^7 - 336*a^2*b^5*cos(d*x + c)^5 + 560*(a^4*b ^3 - a^2*b^5)*cos(d*x + c)^3 - 105*(16*a^7 - 40*a^5*b^2 + 30*a^3*b^4 - 5*a *b^6)*d*x - 840*(a^6 - 2*a^4*b^2 + a^2*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)) - 1680*(a^6*b - 2*a^4*b^3 + a^2*b^5)*cos(d*x + c) + 35*(8*a*b^6*cos(d*x + c)^5 - 2*(6*a^3*b^4 - 5*a*b^6)*cos(d*x + c)^ 3 + 3*(8*a^5*b^2 - 14*a^3*b^4 + 5*a*b^6)*cos(d*x + c))*sin(d*x + c))/(b^8* d), -1/1680*(240*b^7*cos(d*x + c)^7 - 336*a^2*b^5*cos(d*x + c)^5 + 560*(a^ 4*b^3 - a^2*b^5)*cos(d*x + c)^3 - 105*(16*a^7 - 40*a^5*b^2 + 30*a^3*b^4 - 5*a*b^6)*d*x - 1680*(a^6 - 2*a^4*b^2 + a^2*b^4)*sqrt(a^2 - b^2)*arctan(-(a *sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 1680*(a^6*b - 2*a^4*b ^3 + a^2*b^5)*cos(d*x + c) + 35*(8*a*b^6*cos(d*x + c)^5 - 2*(6*a^3*b^4 - 5 *a*b^6)*cos(d*x + c)^3 + 3*(8*a^5*b^2 - 14*a^3*b^4 + 5*a*b^6)*cos(d*x + c) )*sin(d*x + c))/(b^8*d)]
Timed out. \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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. 863 vs. \(2 (383) = 766\).
Time = 0.35 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.12 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
1/1680*(105*(16*a^7 - 40*a^5*b^2 + 30*a^3*b^4 - 5*a*b^6)*(d*x + c)/b^8 - 3 360*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*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^8) + 2*(840*a^5*b*tan(1/2*d*x + 1/2*c)^13 - 1890*a^3*b^3*tan(1 /2*d*x + 1/2*c)^13 + 1155*a*b^5*tan(1/2*d*x + 1/2*c)^13 + 1680*a^6*tan(1/2 *d*x + 1/2*c)^12 - 5040*a^4*b^2*tan(1/2*d*x + 1/2*c)^12 + 5040*a^2*b^4*tan (1/2*d*x + 1/2*c)^12 - 1680*b^6*tan(1/2*d*x + 1/2*c)^12 + 3360*a^5*b*tan(1 /2*d*x + 1/2*c)^11 - 5880*a^3*b^3*tan(1/2*d*x + 1/2*c)^11 + 980*a*b^5*tan( 1/2*d*x + 1/2*c)^11 + 10080*a^6*tan(1/2*d*x + 1/2*c)^10 - 26880*a^4*b^2*ta n(1/2*d*x + 1/2*c)^10 + 20160*a^2*b^4*tan(1/2*d*x + 1/2*c)^10 + 4200*a^5*b *tan(1/2*d*x + 1/2*c)^9 - 6090*a^3*b^3*tan(1/2*d*x + 1/2*c)^9 + 2975*a*b^5 *tan(1/2*d*x + 1/2*c)^9 + 25200*a^6*tan(1/2*d*x + 1/2*c)^8 - 61040*a^4*b^2 *tan(1/2*d*x + 1/2*c)^8 + 40880*a^2*b^4*tan(1/2*d*x + 1/2*c)^8 - 8400*b^6* tan(1/2*d*x + 1/2*c)^8 + 33600*a^6*tan(1/2*d*x + 1/2*c)^6 - 76160*a^4*b^2* tan(1/2*d*x + 1/2*c)^6 + 49280*a^2*b^4*tan(1/2*d*x + 1/2*c)^6 - 4200*a^5*b *tan(1/2*d*x + 1/2*c)^5 + 6090*a^3*b^3*tan(1/2*d*x + 1/2*c)^5 - 2975*a*b^5 *tan(1/2*d*x + 1/2*c)^5 + 25200*a^6*tan(1/2*d*x + 1/2*c)^4 - 55440*a^4*b^2 *tan(1/2*d*x + 1/2*c)^4 + 33936*a^2*b^4*tan(1/2*d*x + 1/2*c)^4 - 5040*b^6* tan(1/2*d*x + 1/2*c)^4 - 3360*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 5880*a^3*b^3* tan(1/2*d*x + 1/2*c)^3 - 980*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 10080*a^6*t...
Time = 14.56 (sec) , antiderivative size = 3797, normalized size of antiderivative = 9.31 \[ \int \frac {\cos ^6(c+d x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
((2*(105*a^6 - 15*b^6 + 161*a^2*b^4 - 245*a^4*b^2))/(105*b^7) + (tan(c/2 + (d*x)/2)^13*(11*a*b^4 + 8*a^5 - 18*a^3*b^2))/(8*b^6) - (tan(c/2 + (d*x)/2 )^3*(7*a*b^4 + 24*a^5 - 42*a^3*b^2))/(6*b^6) + (tan(c/2 + (d*x)/2)^11*(7*a *b^4 + 24*a^5 - 42*a^3*b^2))/(6*b^6) - (tan(c/2 + (d*x)/2)^5*(85*a*b^4 + 1 20*a^5 - 174*a^3*b^2))/(24*b^6) + (tan(c/2 + (d*x)/2)^9*(85*a*b^4 + 120*a^ 5 - 174*a^3*b^2))/(24*b^6) + (2*tan(c/2 + (d*x)/2)^12*(a^6 - b^6 + 3*a^2*b ^4 - 3*a^4*b^2))/b^7 + (4*tan(c/2 + (d*x)/2)^10*(3*a^6 + 6*a^2*b^4 - 8*a^4 *b^2))/b^7 + (8*tan(c/2 + (d*x)/2)^6*(15*a^6 + 22*a^2*b^4 - 34*a^4*b^2))/( 3*b^7) + (4*tan(c/2 + (d*x)/2)^2*(45*a^6 + 58*a^2*b^4 - 100*a^4*b^2))/(15* b^7) + (2*tan(c/2 + (d*x)/2)^8*(45*a^6 - 15*b^6 + 73*a^2*b^4 - 109*a^4*b^2 ))/(3*b^7) + (2*tan(c/2 + (d*x)/2)^4*(75*a^6 - 15*b^6 + 101*a^2*b^4 - 165* a^4*b^2))/(5*b^7) - (a*tan(c/2 + (d*x)/2)*(8*a^4 + 11*b^4 - 18*a^2*b^2))/( 8*b^6))/(d*(7*tan(c/2 + (d*x)/2)^2 + 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 + 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 + 7*tan( c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 + 1)) + (a^2*atan(((a^2*(-(a + b )^5*(a - b)^5)^(1/2)*(((25*a^4*b^19)/8 - (75*a^6*b^17)/2 + (325*a^8*b^15)/ 2 - 320*a^10*b^13 + 320*a^12*b^11 - 160*a^14*b^9 + 32*a^16*b^7)/b^20 + (ta n(c/2 + (d*x)/2)*(50*a^3*b^21 - 881*a^5*b^19 + 4436*a^7*b^17 - 10260*a^9*b ^15 + 12800*a^11*b^13 - 8960*a^13*b^11 + 3328*a^15*b^9 - 512*a^17*b^7))/(8 *b^21) + (a^2*(-(a + b)^5*(a - b)^5)^(1/2)*((10*a^2*b^22 - 38*a^4*b^20 ...