Integrand size = 21, antiderivative size = 653 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {165 a b^4 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{19/2} d}+\frac {b \sec ^3(c+d x)}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac {17 a b \sec ^3(c+d x)}{42 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{14 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{56 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{168 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac {b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{112 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))}-\frac {\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^8 d}+\frac {\sec (c+d x) \left (3465 a b^3 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )+\left (224 a^{10}-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8-4096 b^{10}\right ) \sin (c+d x)\right )}{336 \left (a^2-b^2\right )^9 d} \]
165/8*a*b^4*(32*a^6+112*a^4*b^2+70*a^2*b^4+7*b^6)*arctan((b+a*tan(1/2*d*x+ 1/2*c))/(a^2-b^2)^(1/2))/(a^2-b^2)^(19/2)/d+1/7*b*sec(d*x+c)^3/(a^2-b^2)/d /(a+b*sin(d*x+c))^7+17/42*a*b*sec(d*x+c)^3/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^ 6+1/14*b*(13*a^2+4*b^2)*sec(d*x+c)^3/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^5+1/56 *a*b*(118*a^2+103*b^2)*sec(d*x+c)^3/(a^2-b^2)^4/d/(a+b*sin(d*x+c))^4+1/168 *b*(882*a^4+1421*a^2*b^2+128*b^4)*sec(d*x+c)^3/(a^2-b^2)^5/d/(a+b*sin(d*x+ c))^3+13/112*a*b*(140*a^4+336*a^2*b^2+85*b^4)*sec(d*x+c)^3/(a^2-b^2)^6/d/( a+b*sin(d*x+c))^2+1/112*b*(9212*a^6+28420*a^4*b^2+12907*a^2*b^4+512*b^6)*s ec(d*x+c)^3/(a^2-b^2)^7/d/(a+b*sin(d*x+c))-1/336*sec(d*x+c)^3*(1155*a*b*(3 2*a^6+112*a^4*b^2+70*a^2*b^4+7*b^6)-(112*a^8+52528*a^6*b^2+142902*a^4*b^4+ 57665*a^2*b^6+2048*b^8)*sin(d*x+c))/(a^2-b^2)^8/d+1/336*sec(d*x+c)*(3465*a *b^3*(32*a^6+112*a^4*b^2+70*a^2*b^4+7*b^6)+(224*a^10-6048*a^8*b^2-207332*a ^6*b^4-413024*a^4*b^6-135489*a^2*b^8-4096*b^10)*sin(d*x+c))/(a^2-b^2)^9/d
Time = 5.35 (sec) , antiderivative size = 597, normalized size of antiderivative = 0.91 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {\frac {6930 a b^4 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{19/2}}+\frac {48 b^5 \cos (c+d x)}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^7}+\frac {328 a b^5 \cos (c+d x)}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))^6}+\frac {8 b^5 \left (167 a^2+24 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^5 (a+b \sin (c+d x))^5}+\frac {2 a b^5 \left (2138 a^2+925 b^2\right ) \cos (c+d x)}{\left (a^2-b^2\right )^6 (a+b \sin (c+d x))^4}+\frac {2 b^5 \left (6058 a^4+5273 a^2 b^2+296 b^4\right ) \cos (c+d x)}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))^3}+\frac {a b^5 \left (33284 a^4+48820 a^2 b^2+8287 b^4\right ) \cos (c+d x)}{\left (a^2-b^2\right )^8 (a+b \sin (c+d x))^2}+\frac {b^5 \left (103844 a^6+234272 a^4 b^2+81057 a^2 b^4+2528 b^6\right ) \cos (c+d x)}{\left (a^2-b^2\right )^9 (a+b \sin (c+d x))}+\frac {112 \sec ^3(c+d x) \left (-8 a b \left (a^6+7 a^4 b^2+7 a^2 b^4+b^6\right )+\left (a^8+28 a^6 b^2+70 a^4 b^4+28 a^2 b^6+b^8\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^8}+\frac {224 \sec (c+d x) \left (12 \left (15 a^7 b^3+63 a^5 b^5+45 a^3 b^7+5 a b^9\right )+\left (a^{10}-27 a^8 b^2-462 a^6 b^4-798 a^4 b^6-243 a^2 b^8-7 b^{10}\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^9}}{336 d} \]
((6930*a*b^4*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6)*ArcTan[(b + a*Tan [(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(19/2) + (48*b^5*Cos[c + d*x] )/((a^2 - b^2)^3*(a + b*Sin[c + d*x])^7) + (328*a*b^5*Cos[c + d*x])/((a^2 - b^2)^4*(a + b*Sin[c + d*x])^6) + (8*b^5*(167*a^2 + 24*b^2)*Cos[c + d*x]) /((a^2 - b^2)^5*(a + b*Sin[c + d*x])^5) + (2*a*b^5*(2138*a^2 + 925*b^2)*Co s[c + d*x])/((a^2 - b^2)^6*(a + b*Sin[c + d*x])^4) + (2*b^5*(6058*a^4 + 52 73*a^2*b^2 + 296*b^4)*Cos[c + d*x])/((a^2 - b^2)^7*(a + b*Sin[c + d*x])^3) + (a*b^5*(33284*a^4 + 48820*a^2*b^2 + 8287*b^4)*Cos[c + d*x])/((a^2 - b^2 )^8*(a + b*Sin[c + d*x])^2) + (b^5*(103844*a^6 + 234272*a^4*b^2 + 81057*a^ 2*b^4 + 2528*b^6)*Cos[c + d*x])/((a^2 - b^2)^9*(a + b*Sin[c + d*x])) + (11 2*Sec[c + d*x]^3*(-8*a*b*(a^6 + 7*a^4*b^2 + 7*a^2*b^4 + b^6) + (a^8 + 28*a ^6*b^2 + 70*a^4*b^4 + 28*a^2*b^6 + b^8)*Sin[c + d*x]))/(a^2 - b^2)^8 + (22 4*Sec[c + d*x]*(12*(15*a^7*b^3 + 63*a^5*b^5 + 45*a^3*b^7 + 5*a*b^9) + (a^1 0 - 27*a^8*b^2 - 462*a^6*b^4 - 798*a^4*b^6 - 243*a^2*b^8 - 7*b^10)*Sin[c + d*x]))/(a^2 - b^2)^9)/(336*d)
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.
| \(\displaystyle \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^4 (a+b \sin (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3173 |
| \(\displaystyle \frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}-\frac {\int -\frac {\sec ^4(c+d x) (7 a-10 b \sin (c+d x))}{(a+b \sin (c+d x))^7}dx}{7 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sec ^4(c+d x) (7 a-10 b \sin (c+d x))}{(a+b \sin (c+d x))^7}dx}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {7 a-10 b \sin (c+d x)}{\cos (c+d x)^4 (a+b \sin (c+d x))^7}dx}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle \frac {\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}-\frac {\int -\frac {3 \sec ^4(c+d x) \left (2 \left (7 a^2+10 b^2\right )-51 a b \sin (c+d x)\right )}{(a+b \sin (c+d x))^6}dx}{6 \left (a^2-b^2\right )}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sec ^4(c+d x) \left (2 \left (7 a^2+10 b^2\right )-51 a b \sin (c+d x)\right )}{(a+b \sin (c+d x))^6}dx}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (7 a^2+10 b^2\right )-51 a b \sin (c+d x)}{\cos (c+d x)^4 (a+b \sin (c+d x))^6}dx}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle \frac {\frac {\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}-\frac {\int -\frac {5 \sec ^4(c+d x) \left (a \left (14 a^2+71 b^2\right )-8 b \left (13 a^2+4 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^5}dx}{5 \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sec ^4(c+d x) \left (a \left (14 a^2+71 b^2\right )-8 b \left (13 a^2+4 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^5}dx}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a \left (14 a^2+71 b^2\right )-8 b \left (13 a^2+4 b^2\right ) \sin (c+d x)}{\cos (c+d x)^4 (a+b \sin (c+d x))^5}dx}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle \frac {\frac {\frac {\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}-\frac {\int -\frac {\sec ^4(c+d x) \left (4 \left (14 a^4+175 b^2 a^2+32 b^4\right )-7 a b \left (118 a^2+103 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {\sec ^4(c+d x) \left (4 \left (14 a^4+175 b^2 a^2+32 b^4\right )-7 a b \left (118 a^2+103 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {4 \left (14 a^4+175 b^2 a^2+32 b^4\right )-7 a b \left (118 a^2+103 b^2\right ) \sin (c+d x)}{\cos (c+d x)^4 (a+b \sin (c+d x))^4}dx}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}-\frac {\int -\frac {3 \sec ^4(c+d x) \left (a \left (56 a^4+1526 b^2 a^2+849 b^4\right )-2 b \left (882 a^4+1421 b^2 a^2+128 b^4\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^3}dx}{3 \left (a^2-b^2\right )}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {\sec ^4(c+d x) \left (a \left (56 a^4+1526 b^2 a^2+849 b^4\right )-2 b \left (882 a^4+1421 b^2 a^2+128 b^4\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^3}dx}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {a \left (56 a^4+1526 b^2 a^2+849 b^4\right )-2 b \left (882 a^4+1421 b^2 a^2+128 b^4\right ) \sin (c+d x)}{\cos (c+d x)^4 (a+b \sin (c+d x))^3}dx}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {\int -\frac {\sec ^4(c+d x) \left (2 \left (56 a^6+3290 b^2 a^4+3691 b^4 a^2+256 b^6\right )-65 a b \left (140 a^4+336 b^2 a^2+85 b^4\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\int \frac {\sec ^4(c+d x) \left (2 \left (56 a^6+3290 b^2 a^4+3691 b^4 a^2+256 b^6\right )-65 a b \left (140 a^4+336 b^2 a^2+85 b^4\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\int \frac {2 \left (56 a^6+3290 b^2 a^4+3691 b^4 a^2+256 b^6\right )-65 a b \left (140 a^4+336 b^2 a^2+85 b^4\right ) \sin (c+d x)}{\cos (c+d x)^4 (a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {\int -\frac {\sec ^4(c+d x) \left (a \left (112 a^6+15680 b^2 a^4+29222 b^4 a^2+6037 b^6\right )-4 b \left (9212 a^6+28420 b^2 a^4+12907 b^4 a^2+512 b^6\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a^2-b^2}}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\int \frac {\sec ^4(c+d x) \left (a \left (112 a^6+15680 b^2 a^4+29222 b^4 a^2+6037 b^6\right )-4 b \left (9212 a^6+28420 b^2 a^4+12907 b^4 a^2+512 b^6\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a^2-b^2}+\frac {b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\int \frac {a \left (112 a^6+15680 b^2 a^4+29222 b^4 a^2+6037 b^6\right )-4 b \left (9212 a^6+28420 b^2 a^4+12907 b^4 a^2+512 b^6\right ) \sin (c+d x)}{\cos (c+d x)^4 (a+b \sin (c+d x))}dx}{a^2-b^2}+\frac {b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {-\frac {\int -\frac {\sec ^2(c+d x) \left (224 a^9-5824 b^2 a^7-102276 b^4 a^5-127220 b^6 a^3-20159 b^8 a+2 b \left (112 a^8+52528 b^2 a^6+142902 b^4 a^4+57665 b^6 a^2+2048 b^8\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {\int \frac {\sec ^2(c+d x) \left (224 a^9-5824 b^2 a^7-102276 b^4 a^5-127220 b^6 a^3-20159 b^8 a+2 b \left (112 a^8+52528 b^2 a^6+142902 b^4 a^4+57665 b^6 a^2+2048 b^8\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {\int \frac {224 a^9-5824 b^2 a^7-102276 b^4 a^5-127220 b^6 a^3-20159 b^8 a+2 b \left (112 a^8+52528 b^2 a^6+142902 b^4 a^4+57665 b^6 a^2+2048 b^8\right ) \sin (c+d x)}{\cos (c+d x)^2 (a+b \sin (c+d x))}dx}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {\sec (c+d x) \left (3465 a b^3 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )+\left (224 a^{10}-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8-4096 b^{10}\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\int -\frac {3465 a b^4 \left (32 a^6+112 b^2 a^4+70 b^4 a^2+7 b^6\right )}{a+b \sin (c+d x)}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {3465 a b^4 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}+\frac {\sec (c+d x) \left (3465 a b^3 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )+\left (224 a^{10}-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8-4096 b^{10}\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {3465 a b^4 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}+\frac {\sec (c+d x) \left (3465 a b^3 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )+\left (224 a^{10}-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8-4096 b^{10}\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {6930 a b^4 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{d \left (a^2-b^2\right )}+\frac {\sec (c+d x) \left (3465 a b^3 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )+\left (224 a^{10}-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8-4096 b^{10}\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {\frac {\sec (c+d x) \left (3465 a b^3 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )+\left (224 a^{10}-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8-4096 b^{10}\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {13860 a b^4 \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{d \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (1155 a b \left (32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right )-\left (112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {b \left (9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {13 a b \left (140 a^4+336 a^2 b^2+85 b^4\right ) \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}}{a^2-b^2}+\frac {b \left (882 a^4+1421 a^2 b^2+128 b^4\right ) \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}}{4 \left (a^2-b^2\right )}+\frac {a b \left (118 a^2+103 b^2\right ) \sec ^3(c+d x)}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}}{a^2-b^2}+\frac {b \left (13 a^2+4 b^2\right ) \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}}{2 \left (a^2-b^2\right )}+\frac {17 a b \sec ^3(c+d x)}{6 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}}{7 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{7 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}\) |
3.5.72.3.1 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_) + (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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b ^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p *(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ [a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -1] && IntegerQ[2*m]
Leaf count of result is larger than twice the leaf count of optimal. \(1379\) vs. \(2(628)=1256\).
Time = 111.82 (sec) , antiderivative size = 1380, normalized size of antiderivative = 2.11
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1380\) |
| default | \(\text {Expression too large to display}\) | \(1380\) |
| risch | \(\text {Expression too large to display}\) | \(3047\) |
1/d*(-1/3/(a-b)^8/(tan(1/2*d*x+1/2*c)+1)^3+1/2/(a-b)^8/(tan(1/2*d*x+1/2*c) +1)^2-(a-5*b)/(a-b)^9/(tan(1/2*d*x+1/2*c)+1)-1/3/(a+b)^8/(tan(1/2*d*x+1/2* c)-1)^3-1/2/(a+b)^8/(tan(1/2*d*x+1/2*c)-1)^2-(a+5*b)/(a+b)^9/(tan(1/2*d*x+ 1/2*c)-1)+2*b^4/(a-b)^9/(a+b)^9*((1/16*b^2*(11088*a^12+6798*a^10*b^2+3091* a^8*b^4-1344*a^6*b^6+576*a^4*b^8-144*a^2*b^10+16*b^12)/a*tan(1/2*d*x+1/2*c )^13+1/16*b*(7392*a^14+132528*a^12*b^2+100518*a^10*b^4+25991*a^8*b^6-8064* a^6*b^8+3456*a^4*b^10-864*a^2*b^12+96*b^14)/a^2*tan(1/2*d*x+1/2*c)^12+1/24 /a^3*b^2*(221760*a^14+1107612*a^12*b^2+885544*a^10*b^4+155169*a^8*b^6-3225 6*a^6*b^8+15264*a^4*b^10-4096*a^2*b^12+480*b^14)*tan(1/2*d*x+1/2*c)^11+1/2 4/a^4*b*(66528*a^16+1574496*a^14*b^2+3884100*a^12*b^4+2736860*a^10*b^6+381 885*a^8*b^8-48960*a^6*b^10+26640*a^4*b^12-7760*a^2*b^14+960*b^16)*tan(1/2* d*x+1/2*c)^10+1/48/a^5*b^2*(1718640*a^16+11886930*a^14*b^2+18491825*a^12*b ^4+9856770*a^10*b^6+1146588*a^8*b^8-59760*a^6*b^10+46960*a^4*b^12-16512*a^ 2*b^14+2304*b^16)*tan(1/2*d*x+1/2*c)^9+1/48/a^6*b*(332640*a^18+8651280*a^1 6*b^2+27807890*a^14*b^4+29152473*a^12*b^6+10622738*a^10*b^8+917592*a^8*b^1 0+48960*a^6*b^12+5440*a^4*b^14-7808*a^2*b^16+1536*b^18)*tan(1/2*d*x+1/2*c) ^8+1/84/a^7*b^2*(5433120*a^18+40230960*a^16*b^2+73645726*a^14*b^4+49633899 *a^12*b^6+11312812*a^10*b^8+549276*a^8*b^10+136320*a^6*b^12-34432*a^4*b^14 +1280*a^2*b^16+768*b^18)*tan(1/2*d*x+1/2*c)^7+1/12/a^6*b*(110880*a^18+2766 720*a^16*b^2+8967200*a^14*b^4+9794970*a^12*b^6+3768737*a^10*b^8+417528*...
Leaf count of result is larger than twice the leaf count of optimal. 2208 vs. \(2 (628) = 1256\).
Time = 0.86 (sec) , antiderivative size = 4500, normalized size of antiderivative = 6.89 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
[1/672*(224*a^18*b - 2016*a^16*b^3 + 8064*a^14*b^5 - 18816*a^12*b^7 + 2822 4*a^10*b^9 - 28224*a^8*b^11 + 18816*a^6*b^13 - 8064*a^4*b^15 + 2016*a^2*b^ 17 - 224*b^19 - 2*(224*a^12*b^7 - 6272*a^10*b^9 - 201284*a^8*b^11 - 205692 *a^6*b^13 + 277535*a^4*b^15 + 131393*a^2*b^17 + 4096*b^19)*cos(d*x + c)^10 + 28*(336*a^14*b^5 - 9352*a^12*b^7 - 252014*a^10*b^9 - 230159*a^8*b^11 + 297312*a^6*b^13 + 165122*a^4*b^15 + 27731*a^2*b^17 + 1024*b^19)*cos(d*x + c)^8 - 70*(224*a^16*b^3 - 5936*a^14*b^5 - 126448*a^12*b^7 - 243082*a^10*b^ 9 - 29747*a^8*b^11 + 284285*a^6*b^13 + 109607*a^4*b^15 + 10585*a^2*b^17 + 512*b^19)*cos(d*x + c)^6 + 28*(112*a^18*b - 2296*a^16*b^3 - 35224*a^14*b^5 - 308392*a^12*b^7 - 337750*a^10*b^9 + 149783*a^8*b^11 + 394751*a^6*b^13 + 130949*a^4*b^15 + 7427*a^2*b^17 + 640*b^19)*cos(d*x + c)^4 - 224*(7*a^18* b - 46*a^16*b^3 + 116*a^14*b^5 - 112*a^12*b^7 - 70*a^10*b^9 + 308*a^8*b^11 - 364*a^6*b^13 + 224*a^4*b^15 - 73*a^2*b^17 + 10*b^19)*cos(d*x + c)^2 + 3 465*(7*(32*a^8*b^10 + 112*a^6*b^12 + 70*a^4*b^14 + 7*a^2*b^16)*cos(d*x + c )^9 - 7*(160*a^10*b^8 + 656*a^8*b^10 + 686*a^6*b^12 + 245*a^4*b^14 + 21*a^ 2*b^16)*cos(d*x + c)^7 + 7*(96*a^12*b^6 + 656*a^10*b^8 + 1426*a^8*b^10 + 1 057*a^6*b^12 + 280*a^4*b^14 + 21*a^2*b^16)*cos(d*x + c)^5 - (32*a^14*b^4 + 784*a^12*b^6 + 3542*a^10*b^8 + 5621*a^8*b^10 + 3381*a^6*b^12 + 735*a^4*b^ 14 + 49*a^2*b^16)*cos(d*x + c)^3 + ((32*a^7*b^11 + 112*a^5*b^13 + 70*a^3*b ^15 + 7*a*b^17)*cos(d*x + c)^9 - 3*(224*a^9*b^9 + 816*a^7*b^11 + 602*a^...
Timed out. \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Exception raised: ValueError} \]
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. 3047 vs. \(2 (628) = 1256\).
Time = 0.72 (sec) , antiderivative size = 3047, normalized size of antiderivative = 4.67 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
1/168*(3465*(32*a^7*b^4 + 112*a^5*b^6 + 70*a^3*b^8 + 7*a*b^10)*(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)))/((a^18 - 9*a^16*b^2 + 36*a^14*b^4 - 84*a^12*b^6 + 126*a^10*b^8 - 126*a^8*b^10 + 84*a^6*b^12 - 36*a^4*b^14 + 9*a^2*b^16 - b^18)*sqrt(a^2 - b^2)) - 112*(3*a^10*tan(1/2*d*x + 1/2*c)^5 - 27*a^8*b^2*tan(1/2*d*x + 1/ 2*c)^5 - 882*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 1638*a^4*b^6*tan(1/2*d*x + 1 /2*c)^5 - 513*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 - 15*b^10*tan(1/2*d*x + 1/2*c )^5 - 24*a^9*b*tan(1/2*d*x + 1/2*c)^4 + 216*a^7*b^3*tan(1/2*d*x + 1/2*c)^4 + 1512*a^5*b^5*tan(1/2*d*x + 1/2*c)^4 + 1224*a^3*b^7*tan(1/2*d*x + 1/2*c) ^4 + 144*a*b^9*tan(1/2*d*x + 1/2*c)^4 - 2*a^10*tan(1/2*d*x + 1/2*c)^3 + 16 2*a^8*b^2*tan(1/2*d*x + 1/2*c)^3 + 1932*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 + 3 108*a^4*b^6*tan(1/2*d*x + 1/2*c)^3 + 918*a^2*b^8*tan(1/2*d*x + 1/2*c)^3 + 26*b^10*tan(1/2*d*x + 1/2*c)^3 - 720*a^7*b^3*tan(1/2*d*x + 1/2*c)^2 - 3024 *a^5*b^5*tan(1/2*d*x + 1/2*c)^2 - 2160*a^3*b^7*tan(1/2*d*x + 1/2*c)^2 - 24 0*a*b^9*tan(1/2*d*x + 1/2*c)^2 + 3*a^10*tan(1/2*d*x + 1/2*c) - 27*a^8*b^2* tan(1/2*d*x + 1/2*c) - 882*a^6*b^4*tan(1/2*d*x + 1/2*c) - 1638*a^4*b^6*tan (1/2*d*x + 1/2*c) - 513*a^2*b^8*tan(1/2*d*x + 1/2*c) - 15*b^10*tan(1/2*d*x + 1/2*c) - 8*a^9*b + 312*a^7*b^3 + 1512*a^5*b^5 + 1128*a^3*b^7 + 128*a*b^ 9)/((a^18 - 9*a^16*b^2 + 36*a^14*b^4 - 84*a^12*b^6 + 126*a^10*b^8 - 126*a^ 8*b^10 + 84*a^6*b^12 - 36*a^4*b^14 + 9*a^2*b^16 - b^18)*(tan(1/2*d*x + ...
Timed out. \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Hanged} \]