Integrand size = 29, antiderivative size = 536 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right ) x}{16 b^9}+\frac {a \sqrt {a^2-b^2} \left (56 a^4-47 a^2 b^2+6 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^9 d}-\frac {a \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{30 b^8 d}+\frac {\left (224 a^4-244 a^2 b^2+43 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 b^7 d}-\frac {\left (280 a^4-291 a^2 b^2+45 b^4\right ) \cos (c+d x) \sin ^2(c+d x)}{30 a b^6 d}+\frac {\left (168 a^4-169 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{24 a^2 b^5 d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 a d (a+b \sin (c+d x))^2}-\frac {b \cos (c+d x) \sin ^5(c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \cos (c+d x) \sin ^5(c+d x)}{60 a^2 b^3 d (a+b \sin (c+d x))^2}-\frac {4 a \cos (c+d x) \sin ^6(c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\cos (c+d x) \sin ^7(c+d x)}{6 b d (a+b \sin (c+d x))^2}-\frac {\left (112 a^4-110 a^2 b^2+15 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{20 a^2 b^4 d (a+b \sin (c+d x))} \]
-1/16*(448*a^6-600*a^4*b^2+180*a^2*b^4-5*b^6)*x/b^9-1/30*a*(840*a^4-985*a^ 2*b^2+213*b^4)*cos(d*x+c)/b^8/d+1/16*(224*a^4-244*a^2*b^2+43*b^4)*cos(d*x+ c)*sin(d*x+c)/b^7/d-1/30*(280*a^4-291*a^2*b^2+45*b^4)*cos(d*x+c)*sin(d*x+c )^2/a/b^6/d+1/24*(168*a^4-169*a^2*b^2+24*b^4)*cos(d*x+c)*sin(d*x+c)^3/a^2/ b^5/d+1/4*cos(d*x+c)*sin(d*x+c)^4/a/d/(a+b*sin(d*x+c))^2-1/10*b*cos(d*x+c) *sin(d*x+c)^5/a^2/d/(a+b*sin(d*x+c))^2-1/60*(56*a^4-60*a^2*b^2+9*b^4)*cos( d*x+c)*sin(d*x+c)^5/a^2/b^3/d/(a+b*sin(d*x+c))^2-4/15*a*cos(d*x+c)*sin(d*x +c)^6/b^2/d/(a+b*sin(d*x+c))^2+1/6*cos(d*x+c)*sin(d*x+c)^7/b/d/(a+b*sin(d* x+c))^2-1/20*(112*a^4-110*a^2*b^2+15*b^4)*cos(d*x+c)*sin(d*x+c)^4/a^2/b^4/ d/(a+b*sin(d*x+c))+a*(56*a^4-47*a^2*b^2+6*b^4)*arctan((b+a*tan(1/2*d*x+1/2 *c))/(a^2-b^2)^(1/2))*(a^2-b^2)^(1/2)/b^9/d
Time = 11.02 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.18 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {3840 a \left (a^2-b^2\right )^{5/2} \left (56 a^4-47 a^2 b^2+6 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+\frac {\left (a^2-b^2\right )^2 \left (-107520 a^8 c+90240 a^6 b^2 c+28800 a^4 b^4 c-20400 a^2 b^6 c+600 b^8 c-107520 a^8 d x+90240 a^6 b^2 d x+28800 a^4 b^4 d x-20400 a^2 b^6 d x+600 b^8 d x-80 a b \left (1344 a^6-1464 a^4 b^2+202 a^2 b^4+33 b^6\right ) \cos (c+d x)+120 b^2 \left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right ) (c+d x) \cos (2 (c+d x))+8960 a^5 b^3 \cos (3 (c+d x))-10880 a^3 b^5 \cos (3 (c+d x))+2436 a b^7 \cos (3 (c+d x))-224 a^3 b^5 \cos (5 (c+d x))+188 a b^7 \cos (5 (c+d x))+16 a b^7 \cos (7 (c+d x))-215040 a^7 b c \sin (c+d x)+288000 a^5 b^3 c \sin (c+d x)-86400 a^3 b^5 c \sin (c+d x)+2400 a b^7 c \sin (c+d x)-215040 a^7 b d x \sin (c+d x)+288000 a^5 b^3 d x \sin (c+d x)-86400 a^3 b^5 d x \sin (c+d x)+2400 a b^7 d x \sin (c+d x)-80640 a^6 b^2 \sin (2 (c+d x))+99040 a^4 b^4 \sin (2 (c+d x))-24600 a^2 b^6 \sin (2 (c+d x))+405 b^8 \sin (2 (c+d x))-1120 a^4 b^4 \sin (4 (c+d x))+1164 a^2 b^6 \sin (4 (c+d x))-140 b^8 \sin (4 (c+d x))+56 a^2 b^6 \sin (6 (c+d x))-35 b^8 \sin (6 (c+d x))-5 b^8 \sin (8 (c+d x))\right )}{(a+b \sin (c+d x))^2}}{3840 (a-b)^2 b^9 (a+b)^2 d} \]
(3840*a*(a^2 - b^2)^(5/2)*(56*a^4 - 47*a^2*b^2 + 6*b^4)*ArcTan[(b + a*Tan[ (c + d*x)/2])/Sqrt[a^2 - b^2]] + ((a^2 - b^2)^2*(-107520*a^8*c + 90240*a^6 *b^2*c + 28800*a^4*b^4*c - 20400*a^2*b^6*c + 600*b^8*c - 107520*a^8*d*x + 90240*a^6*b^2*d*x + 28800*a^4*b^4*d*x - 20400*a^2*b^6*d*x + 600*b^8*d*x - 80*a*b*(1344*a^6 - 1464*a^4*b^2 + 202*a^2*b^4 + 33*b^6)*Cos[c + d*x] + 120 *b^2*(448*a^6 - 600*a^4*b^2 + 180*a^2*b^4 - 5*b^6)*(c + d*x)*Cos[2*(c + d* x)] + 8960*a^5*b^3*Cos[3*(c + d*x)] - 10880*a^3*b^5*Cos[3*(c + d*x)] + 243 6*a*b^7*Cos[3*(c + d*x)] - 224*a^3*b^5*Cos[5*(c + d*x)] + 188*a*b^7*Cos[5* (c + d*x)] + 16*a*b^7*Cos[7*(c + d*x)] - 215040*a^7*b*c*Sin[c + d*x] + 288 000*a^5*b^3*c*Sin[c + d*x] - 86400*a^3*b^5*c*Sin[c + d*x] + 2400*a*b^7*c*S in[c + d*x] - 215040*a^7*b*d*x*Sin[c + d*x] + 288000*a^5*b^3*d*x*Sin[c + d *x] - 86400*a^3*b^5*d*x*Sin[c + d*x] + 2400*a*b^7*d*x*Sin[c + d*x] - 80640 *a^6*b^2*Sin[2*(c + d*x)] + 99040*a^4*b^4*Sin[2*(c + d*x)] - 24600*a^2*b^6 *Sin[2*(c + d*x)] + 405*b^8*Sin[2*(c + d*x)] - 1120*a^4*b^4*Sin[4*(c + d*x )] + 1164*a^2*b^6*Sin[4*(c + d*x)] - 140*b^8*Sin[4*(c + d*x)] + 56*a^2*b^6 *Sin[6*(c + d*x)] - 35*b^8*Sin[6*(c + d*x)] - 5*b^8*Sin[8*(c + d*x)]))/(a + b*Sin[c + d*x])^2)/(3840*(a - b)^2*b^9*(a + b)^2*d)
Time = 4.41 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.25, 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, 3526, 27, 3042, 3526, 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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^6}{(a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3375 |
| \(\displaystyle \frac {\int \frac {10 \sin ^5(c+d x) \left (-2 \left (56 a^4-65 b^2 a^2+12 b^4\right ) \sin ^2(c+d x)-3 a b \left (2 a^2-3 b^2\right ) \sin (c+d x)+3 \left (32 a^4-35 b^2 a^2+6 b^4\right )\right )}{(a+b \sin (c+d x))^3}dx}{600 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sin ^5(c+d x) \left (-2 \left (56 a^4-65 b^2 a^2+12 b^4\right ) \sin ^2(c+d x)-3 a b \left (2 a^2-3 b^2\right ) \sin (c+d x)+3 \left (32 a^4-35 b^2 a^2+6 b^4\right )\right )}{(a+b \sin (c+d x))^3}dx}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin (c+d x)^5 \left (-2 \left (56 a^4-65 b^2 a^2+12 b^4\right ) \sin (c+d x)^2-3 a b \left (2 a^2-3 b^2\right ) \sin (c+d x)+3 \left (32 a^4-35 b^2 a^2+6 b^4\right )\right )}{(a+b \sin (c+d x))^3}dx}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 \sin ^4(c+d x) \left (-2 \left (168 a^6-353 b^2 a^4+215 b^4 a^2-30 b^6\right ) \sin ^2(c+d x)-a b \left (16 a^4-31 b^2 a^2+15 b^4\right ) \sin (c+d x)+5 \left (56 a^6-116 b^2 a^4+69 b^4 a^2-9 b^6\right )\right )}{(a+b \sin (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sin ^4(c+d x) \left (-2 \left (168 a^6-353 b^2 a^4+215 b^4 a^2-30 b^6\right ) \sin ^2(c+d x)-a b \left (16 a^4-31 b^2 a^2+15 b^4\right ) \sin (c+d x)+5 \left (56 a^6-116 b^2 a^4+69 b^4 a^2-9 b^6\right )\right )}{(a+b \sin (c+d x))^2}dx}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sin (c+d x)^4 \left (-2 \left (168 a^6-353 b^2 a^4+215 b^4 a^2-30 b^6\right ) \sin (c+d x)^2-a b \left (16 a^4-31 b^2 a^2+15 b^4\right ) \sin (c+d x)+5 \left (56 a^6-116 b^2 a^4+69 b^4 a^2-9 b^6\right )\right )}{(a+b \sin (c+d x))^2}dx}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {2 \sin ^3(c+d x) \left (-5 \left (168 a^4-169 b^2 a^2+24 b^4\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+6 \left (112 a^4-110 b^2 a^2+15 b^4\right ) \left (a^2-b^2\right )^2-a b \left (28 a^2-15 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {\sin ^3(c+d x) \left (-5 \left (168 a^4-169 b^2 a^2+24 b^4\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+6 \left (112 a^4-110 b^2 a^2+15 b^4\right ) \left (a^2-b^2\right )^2-a b \left (28 a^2-15 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {\sin (c+d x)^3 \left (-5 \left (168 a^4-169 b^2 a^2+24 b^4\right ) \sin (c+d x)^2 \left (a^2-b^2\right )^2+6 \left (112 a^4-110 b^2 a^2+15 b^4\right ) \left (a^2-b^2\right )^2-a b \left (28 a^2-15 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {\int -\frac {3 \sin ^2(c+d x) \left (-4 a \left (280 a^4-291 b^2 a^2+45 b^4\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+5 a \left (168 a^4-169 b^2 a^2+24 b^4\right ) \left (a^2-b^2\right )^2-7 a^2 b \left (8 a^2-5 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{4 b}+\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \int \frac {\sin ^2(c+d x) \left (-4 a \left (280 a^4-291 b^2 a^2+45 b^4\right ) \sin ^2(c+d x) \left (a^2-b^2\right )^2+5 a \left (168 a^4-169 b^2 a^2+24 b^4\right ) \left (a^2-b^2\right )^2-7 a^2 b \left (8 a^2-5 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \int \frac {\sin (c+d x)^2 \left (-4 a \left (280 a^4-291 b^2 a^2+45 b^4\right ) \sin (c+d x)^2 \left (a^2-b^2\right )^2+5 a \left (168 a^4-169 b^2 a^2+24 b^4\right ) \left (a^2-b^2\right )^2-7 a^2 b \left (8 a^2-5 b^2\right ) \sin (c+d x) \left (a^2-b^2\right )^2\right )}{a+b \sin (c+d x)}dx}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {\int -\frac {\sin (c+d x) \left (-b \left (280 a^2-207 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) a^3-15 \left (a^2-b^2\right )^2 \left (224 a^4-244 b^2 a^2+43 b^4\right ) \sin ^2(c+d x) a^2+8 \left (a^2-b^2\right )^2 \left (280 a^4-291 b^2 a^2+45 b^4\right ) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}+\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-b \left (280 a^2-207 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) a^3-15 \left (a^2-b^2\right )^2 \left (224 a^4-244 b^2 a^2+43 b^4\right ) \sin ^2(c+d x) a^2+8 \left (a^2-b^2\right )^2 \left (280 a^4-291 b^2 a^2+45 b^4\right ) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\int \frac {\sin (c+d x) \left (-b \left (280 a^2-207 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x) a^3-15 \left (a^2-b^2\right )^2 \left (224 a^4-244 b^2 a^2+43 b^4\right ) \sin (c+d x)^2 a^2+8 \left (a^2-b^2\right )^2 \left (280 a^4-291 b^2 a^2+45 b^4\right ) a^2\right )}{a+b \sin (c+d x)}dx}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {\int -\frac {-8 \left (a^2-b^2\right )^2 \left (840 a^4-985 b^2 a^2+213 b^4\right ) \sin ^2(c+d x) a^3+15 \left (a^2-b^2\right )^2 \left (224 a^4-244 b^2 a^2+43 b^4\right ) a^3-b \left (a^2-b^2\right )^2 \left (1120 a^4-996 b^2 a^2+75 b^4\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}+\frac {15 a^2 \left (224 a^4-244 a^2 b^2+43 b^4\right ) \left (a^2-b^2\right )^2 \sin (c+d x) \cos (c+d x)}{2 b d}}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {-8 \left (a^2-b^2\right )^2 \left (840 a^4-985 b^2 a^2+213 b^4\right ) \sin ^2(c+d x) a^3+15 \left (a^2-b^2\right )^2 \left (224 a^4-244 b^2 a^2+43 b^4\right ) a^3-b \left (a^2-b^2\right )^2 \left (1120 a^4-996 b^2 a^2+75 b^4\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\int \frac {-8 \left (a^2-b^2\right )^2 \left (840 a^4-985 b^2 a^2+213 b^4\right ) \sin (c+d x)^2 a^3+15 \left (a^2-b^2\right )^2 \left (224 a^4-244 b^2 a^2+43 b^4\right ) a^3-b \left (a^2-b^2\right )^2 \left (1120 a^4-996 b^2 a^2+75 b^4\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{2 b}}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {\int \frac {15 \left (b \left (a^2-b^2\right )^2 \left (224 a^4-244 b^2 a^2+43 b^4\right ) a^3+\left (a^2-b^2\right )^2 \left (448 a^6-600 b^2 a^4+180 b^4 a^2-5 b^6\right ) \sin (c+d x) a^2\right )}{a+b \sin (c+d x)}dx}{b}+\frac {8 a^3 \left (a^2-b^2\right )^2 \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \int \frac {b \left (a^2-b^2\right )^2 \left (224 a^4-244 b^2 a^2+43 b^4\right ) a^3+\left (a^2-b^2\right )^2 \left (448 a^6-600 b^2 a^4+180 b^4 a^2-5 b^6\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{b}+\frac {8 a^3 \left (a^2-b^2\right )^2 \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \int \frac {b \left (a^2-b^2\right )^2 \left (224 a^4-244 b^2 a^2+43 b^4\right ) a^3+\left (a^2-b^2\right )^2 \left (448 a^6-600 b^2 a^4+180 b^4 a^2-5 b^6\right ) \sin (c+d x) a^2}{a+b \sin (c+d x)}dx}{b}+\frac {8 a^3 \left (a^2-b^2\right )^2 \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a^2 x \left (a^2-b^2\right )^2 \left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right )}{b}-\frac {8 a^3 \left (a^2-b^2\right )^3 \left (56 a^4-47 a^2 b^2+6 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {8 a^3 \left (a^2-b^2\right )^2 \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a^2 x \left (a^2-b^2\right )^2 \left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right )}{b}-\frac {8 a^3 \left (a^2-b^2\right )^3 \left (56 a^4-47 a^2 b^2+6 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{b}\right )}{b}+\frac {8 a^3 \left (a^2-b^2\right )^2 \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {a^2 x \left (a^2-b^2\right )^2 \left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right )}{b}-\frac {16 a^3 \left (a^2-b^2\right )^3 \left (56 a^4-47 a^2 b^2+6 b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{b d}\right )}{b}+\frac {8 a^3 \left (a^2-b^2\right )^2 \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {15 \left (\frac {32 a^3 \left (56 a^4-47 a^2 b^2+6 b^4\right ) \left (a^2-b^2\right )^3 \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b d}+\frac {a^2 x \left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right ) \left (a^2-b^2\right )^2}{b}\right )}{b}+\frac {8 a^3 \left (a^2-b^2\right )^2 \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{b d}}{2 b}}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {b \sin ^5(c+d x) \cos (c+d x)}{10 a^2 d (a+b \sin (c+d x))^2}+\frac {\frac {\frac {2 \left (\frac {5 \left (a^2-b^2\right )^2 \left (168 a^4-169 a^2 b^2+24 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4 b d}-\frac {3 \left (\frac {4 a \left (a^2-b^2\right )^2 \left (280 a^4-291 a^2 b^2+45 b^4\right ) \sin ^2(c+d x) \cos (c+d x)}{3 b d}-\frac {\frac {15 a^2 \left (a^2-b^2\right )^2 \left (224 a^4-244 a^2 b^2+43 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {\frac {8 a^3 \left (a^2-b^2\right )^2 \left (840 a^4-985 a^2 b^2+213 b^4\right ) \cos (c+d x)}{b d}+\frac {15 \left (\frac {a^2 x \left (a^2-b^2\right )^2 \left (448 a^6-600 a^4 b^2+180 a^2 b^4-5 b^6\right )}{b}-\frac {16 a^3 \left (a^2-b^2\right )^{5/2} \left (56 a^4-47 a^2 b^2+6 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{b d}\right )}{b}}{2 b}}{3 b}\right )}{4 b}\right )}{b \left (a^2-b^2\right )}-\frac {3 \left (a^2-b^2\right ) \left (112 a^4-110 a^2 b^2+15 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))}}{b \left (a^2-b^2\right )}-\frac {\left (56 a^4-60 a^2 b^2+9 b^4\right ) \sin ^5(c+d x) \cos (c+d x)}{b d (a+b \sin (c+d x))^2}}{60 a^2 b^2}-\frac {4 a \sin ^6(c+d x) \cos (c+d x)}{15 b^2 d (a+b \sin (c+d x))^2}+\frac {\sin ^7(c+d x) \cos (c+d x)}{6 b d (a+b \sin (c+d x))^2}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 a d (a+b \sin (c+d x))^2}\) |
(Cos[c + d*x]*Sin[c + d*x]^4)/(4*a*d*(a + b*Sin[c + d*x])^2) - (b*Cos[c + d*x]*Sin[c + d*x]^5)/(10*a^2*d*(a + b*Sin[c + d*x])^2) - (4*a*Cos[c + d*x] *Sin[c + d*x]^6)/(15*b^2*d*(a + b*Sin[c + d*x])^2) + (Cos[c + d*x]*Sin[c + d*x]^7)/(6*b*d*(a + b*Sin[c + d*x])^2) + (-(((56*a^4 - 60*a^2*b^2 + 9*b^4 )*Cos[c + d*x]*Sin[c + d*x]^5)/(b*d*(a + b*Sin[c + d*x])^2)) + ((-3*(a^2 - b^2)*(112*a^4 - 110*a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(b*d*( a + b*Sin[c + d*x])) + (2*((5*(a^2 - b^2)^2*(168*a^4 - 169*a^2*b^2 + 24*b^ 4)*Cos[c + d*x]*Sin[c + d*x]^3)/(4*b*d) - (3*((4*a*(a^2 - b^2)^2*(280*a^4 - 291*a^2*b^2 + 45*b^4)*Cos[c + d*x]*Sin[c + d*x]^2)/(3*b*d) - (-1/2*((15* ((a^2*(a^2 - b^2)^2*(448*a^6 - 600*a^4*b^2 + 180*a^2*b^4 - 5*b^6)*x)/b - ( 16*a^3*(a^2 - b^2)^(5/2)*(56*a^4 - 47*a^2*b^2 + 6*b^4)*ArcTan[(2*b + 2*a*T an[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(b*d)))/b + (8*a^3*(a^2 - b^2)^2*(8 40*a^4 - 985*a^2*b^2 + 213*b^4)*Cos[c + d*x])/(b*d))/b + (15*a^2*(a^2 - b^ 2)^2*(224*a^4 - 244*a^2*b^2 + 43*b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*b*d))/ (3*b)))/(4*b)))/(b*(a^2 - b^2)))/(b*(a^2 - b^2)))/(60*a^2*b^2)
3.13.66.3.1 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^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -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 = 5.79 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.30
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {\left (\frac {15}{2} a^{4} b^{2}-\frac {27}{4} a^{2} b^{4}+\frac {11}{16} b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 a^{5} b -30 a^{3} b^{3}+9 a \,b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {45}{2} a^{4} b^{2}-\frac {57}{4} a^{2} b^{4}-\frac {5}{48} b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (105 a^{5} b -130 a^{3} b^{3}+27 a \,b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{4} b^{2}-\frac {15}{2} a^{2} b^{4}+\frac {15}{8} b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (210 a^{5} b -\frac {700}{3} a^{3} b^{3}+46 a \,b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-15 a^{4} b^{2}+\frac {15}{2} a^{2} b^{4}-\frac {15}{8} b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (210 a^{5} b -220 a^{3} b^{3}+42 a \,b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {45}{2} a^{4} b^{2}+\frac {57}{4} a^{2} b^{4}+\frac {5}{48} b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (105 a^{5} b -110 a^{3} b^{3}+\frac {93}{5} a \,b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {15}{2} a^{4} b^{2}+\frac {27}{4} a^{2} b^{4}-\frac {11}{16} b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+21 a^{5} b -\frac {70 a^{3} b^{3}}{3}+\frac {23 a \,b^{5}}{5}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {\left (448 a^{6}-600 a^{4} b^{2}+180 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}\right )}{b^{9}}+\frac {2 a \left (\frac {-\frac {a \,b^{2} \left (13 a^{4}-17 a^{2} b^{2}+4 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b \left (14 a^{6}+9 a^{4} b^{2}-33 a^{2} b^{4}+10 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b^{2} a \left (43 a^{4}-59 a^{2} b^{2}+16 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-7 a^{6} b +\frac {19 a^{4} b^{3}}{2}-\frac {5 a^{2} b^{5}}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (56 a^{6}-103 a^{4} b^{2}+53 a^{2} b^{4}-6 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 )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{9}}}{d}\) | \(696\) |
| default | \(\frac {-\frac {2 \left (\frac {\left (\frac {15}{2} a^{4} b^{2}-\frac {27}{4} a^{2} b^{4}+\frac {11}{16} b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (21 a^{5} b -30 a^{3} b^{3}+9 a \,b^{5}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {45}{2} a^{4} b^{2}-\frac {57}{4} a^{2} b^{4}-\frac {5}{48} b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (105 a^{5} b -130 a^{3} b^{3}+27 a \,b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 a^{4} b^{2}-\frac {15}{2} a^{2} b^{4}+\frac {15}{8} b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (210 a^{5} b -\frac {700}{3} a^{3} b^{3}+46 a \,b^{5}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-15 a^{4} b^{2}+\frac {15}{2} a^{2} b^{4}-\frac {15}{8} b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (210 a^{5} b -220 a^{3} b^{3}+42 a \,b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {45}{2} a^{4} b^{2}+\frac {57}{4} a^{2} b^{4}+\frac {5}{48} b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (105 a^{5} b -110 a^{3} b^{3}+\frac {93}{5} a \,b^{5}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {15}{2} a^{4} b^{2}+\frac {27}{4} a^{2} b^{4}-\frac {11}{16} b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+21 a^{5} b -\frac {70 a^{3} b^{3}}{3}+\frac {23 a \,b^{5}}{5}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {\left (448 a^{6}-600 a^{4} b^{2}+180 a^{2} b^{4}-5 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}\right )}{b^{9}}+\frac {2 a \left (\frac {-\frac {a \,b^{2} \left (13 a^{4}-17 a^{2} b^{2}+4 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b \left (14 a^{6}+9 a^{4} b^{2}-33 a^{2} b^{4}+10 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {b^{2} a \left (43 a^{4}-59 a^{2} b^{2}+16 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-7 a^{6} b +\frac {19 a^{4} b^{3}}{2}-\frac {5 a^{2} b^{5}}{2}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (56 a^{6}-103 a^{4} b^{2}+53 a^{2} b^{4}-6 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 )}{2 \sqrt {a^{2}-b^{2}}}\right )}{b^{9}}}{d}\) | \(696\) |
| risch | \(\frac {\sin \left (6 d x +6 c \right )}{192 b^{3} d}+\frac {5 x}{16 b^{3}}-\frac {28 x \,a^{6}}{b^{9}}+\frac {75 x \,a^{4}}{2 b^{7}}-\frac {45 x \,a^{2}}{4 b^{5}}-\frac {3 a \cos \left (5 d x +5 c \right )}{80 d \,b^{4}}-\frac {3 \sin \left (4 d x +4 c \right ) a^{2}}{16 b^{5} d}-\frac {15 i {\mathrm e}^{2 i \left (d x +c \right )}}{128 b^{3} d}+\frac {5 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{12 b^{6} d}-\frac {7 a \,{\mathrm e}^{3 i \left (d x +c \right )}}{32 b^{4} d}-\frac {21 a^{5} {\mathrm e}^{i \left (d x +c \right )}}{2 b^{8} d}+\frac {45 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{4 b^{6} d}-\frac {21 a^{5} {\mathrm e}^{-i \left (d x +c \right )}}{2 b^{8} d}+\frac {45 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{4 b^{6} d}-\frac {33 a \,{\mathrm e}^{-i \left (d x +c \right )}}{16 b^{4} d}-\frac {33 a \,{\mathrm e}^{i \left (d x +c \right )}}{16 b^{4} d}+\frac {5 a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{12 b^{6} d}-\frac {7 a \,{\mathrm e}^{-3 i \left (d x +c \right )}}{32 b^{4} d}+\frac {15 i {\mathrm e}^{-2 i \left (d x +c \right )}}{128 b^{3} d}+\frac {3 \sin \left (4 d x +4 c \right )}{64 b^{3} d}+\frac {3 i \sqrt {a^{2}-b^{2}}\, a \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 {28 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^{9}}+\frac {47 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 )}{2 d \,b^{7}}+\frac {28 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^{9}}-\frac {47 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 )}{2 d \,b^{7}}-\frac {3 i \sqrt {a^{2}-b^{2}}\, a \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 a^{2} \left (-16 i a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}+23 i a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-7 i a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+44 i a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-61 i a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}+17 i a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}+30 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-27 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+6 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-15 a^{4} b^{2}+21 a^{2} b^{4}-6 b^{6}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} d \,b^{9}}-\frac {15 i {\mathrm e}^{2 i \left (d x +c \right )} a^{4}}{8 b^{7} d}+\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} a^{2}}{2 b^{5} d}+\frac {15 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4}}{8 b^{7} d}-\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} a^{2}}{2 b^{5} d}\) | \(967\) |
1/d*(-2/b^9*(((15/2*a^4*b^2-27/4*a^2*b^4+11/16*b^6)*tan(1/2*d*x+1/2*c)^11+ (21*a^5*b-30*a^3*b^3+9*a*b^5)*tan(1/2*d*x+1/2*c)^10+(45/2*a^4*b^2-57/4*a^2 *b^4-5/48*b^6)*tan(1/2*d*x+1/2*c)^9+(105*a^5*b-130*a^3*b^3+27*a*b^5)*tan(1 /2*d*x+1/2*c)^8+(15*a^4*b^2-15/2*a^2*b^4+15/8*b^6)*tan(1/2*d*x+1/2*c)^7+(2 10*a^5*b-700/3*a^3*b^3+46*a*b^5)*tan(1/2*d*x+1/2*c)^6+(-15*a^4*b^2+15/2*a^ 2*b^4-15/8*b^6)*tan(1/2*d*x+1/2*c)^5+(210*a^5*b-220*a^3*b^3+42*a*b^5)*tan( 1/2*d*x+1/2*c)^4+(-45/2*a^4*b^2+57/4*a^2*b^4+5/48*b^6)*tan(1/2*d*x+1/2*c)^ 3+(105*a^5*b-110*a^3*b^3+93/5*a*b^5)*tan(1/2*d*x+1/2*c)^2+(-15/2*a^4*b^2+2 7/4*a^2*b^4-11/16*b^6)*tan(1/2*d*x+1/2*c)+21*a^5*b-70/3*a^3*b^3+23/5*a*b^5 )/(1+tan(1/2*d*x+1/2*c)^2)^6+1/16*(448*a^6-600*a^4*b^2+180*a^2*b^4-5*b^6)* arctan(tan(1/2*d*x+1/2*c)))+2*a/b^9*((-1/2*a*b^2*(13*a^4-17*a^2*b^2+4*b^4) *tan(1/2*d*x+1/2*c)^3-1/2*b*(14*a^6+9*a^4*b^2-33*a^2*b^4+10*b^6)*tan(1/2*d *x+1/2*c)^2-1/2*b^2*a*(43*a^4-59*a^2*b^2+16*b^4)*tan(1/2*d*x+1/2*c)-7*a^6* b+19/2*a^4*b^3-5/2*a^2*b^5)/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c) +a)^2+1/2*(56*a^6-103*a^4*b^2+53*a^2*b^4-6*b^6)/(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 = 0.52 (sec) , antiderivative size = 1128, normalized size of antiderivative = 2.10 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
[-1/240*(64*a*b^7*cos(d*x + c)^7 - 4*(56*a^3*b^5 - 19*a*b^7)*cos(d*x + c)^ 5 + 15*(448*a^6*b^2 - 600*a^4*b^4 + 180*a^2*b^6 - 5*b^8)*d*x*cos(d*x + c)^ 2 + 10*(224*a^5*b^3 - 244*a^3*b^5 + 43*a*b^7)*cos(d*x + c)^3 - 15*(448*a^8 - 152*a^6*b^2 - 420*a^4*b^4 + 175*a^2*b^6 - 5*b^8)*d*x + 60*(56*a^7 + 9*a ^5*b^2 - 41*a^3*b^4 + 6*a*b^6 - (56*a^5*b^2 - 47*a^3*b^4 + 6*a*b^6)*cos(d* x + c)^2 + 2*(56*a^6*b - 47*a^4*b^3 + 6*a^2*b^5)*sin(d*x + c))*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)) - 30*(224*a^7*b - 188*a^ 5*b^3 - 32*a^3*b^5 + 19*a*b^7)*cos(d*x + c) - (40*b^8*cos(d*x + c)^7 - 2*( 56*a^2*b^6 - 5*b^8)*cos(d*x + c)^5 + 5*(112*a^4*b^4 - 94*a^2*b^6 + 5*b^8)* cos(d*x + c)^3 + 30*(448*a^7*b - 600*a^5*b^3 + 180*a^3*b^5 - 5*a*b^7)*d*x + 15*(672*a^6*b^2 - 844*a^4*b^4 + 223*a^2*b^6 - 5*b^8)*cos(d*x + c))*sin(d *x + c))/(b^11*d*cos(d*x + c)^2 - 2*a*b^10*d*sin(d*x + c) - (a^2*b^9 + b^1 1)*d), -1/240*(64*a*b^7*cos(d*x + c)^7 - 4*(56*a^3*b^5 - 19*a*b^7)*cos(d*x + c)^5 + 15*(448*a^6*b^2 - 600*a^4*b^4 + 180*a^2*b^6 - 5*b^8)*d*x*cos(d*x + c)^2 + 10*(224*a^5*b^3 - 244*a^3*b^5 + 43*a*b^7)*cos(d*x + c)^3 - 15*(4 48*a^8 - 152*a^6*b^2 - 420*a^4*b^4 + 175*a^2*b^6 - 5*b^8)*d*x - 120*(56*a^ 7 + 9*a^5*b^2 - 41*a^3*b^4 + 6*a*b^6 - (56*a^5*b^2 - 47*a^3*b^4 + 6*a*b^6) *cos(d*x + c)^2 + 2*(56*a^6*b - 47*a^4*b^3 + 6*a^2*b^5)*sin(d*x + c))*s...
Timed out. \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.41 (sec) , antiderivative size = 968, normalized size of antiderivative = 1.81 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
-1/240*(15*(448*a^6 - 600*a^4*b^2 + 180*a^2*b^4 - 5*b^6)*(d*x + c)/b^9 - 2 40*(56*a^7 - 103*a^5*b^2 + 53*a^3*b^4 - 6*a*b^6)*(pi*floor(1/2*(d*x + c)/p i + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(s qrt(a^2 - b^2)*b^9) + 240*(13*a^6*b*tan(1/2*d*x + 1/2*c)^3 - 17*a^4*b^3*ta n(1/2*d*x + 1/2*c)^3 + 4*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 14*a^7*tan(1/2*d *x + 1/2*c)^2 + 9*a^5*b^2*tan(1/2*d*x + 1/2*c)^2 - 33*a^3*b^4*tan(1/2*d*x + 1/2*c)^2 + 10*a*b^6*tan(1/2*d*x + 1/2*c)^2 + 43*a^6*b*tan(1/2*d*x + 1/2* c) - 59*a^4*b^3*tan(1/2*d*x + 1/2*c) + 16*a^2*b^5*tan(1/2*d*x + 1/2*c) + 1 4*a^7 - 19*a^5*b^2 + 5*a^3*b^4)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d *x + 1/2*c) + a)^2*b^8) + 2*(1800*a^4*b*tan(1/2*d*x + 1/2*c)^11 - 1620*a^2 *b^3*tan(1/2*d*x + 1/2*c)^11 + 165*b^5*tan(1/2*d*x + 1/2*c)^11 + 5040*a^5* tan(1/2*d*x + 1/2*c)^10 - 7200*a^3*b^2*tan(1/2*d*x + 1/2*c)^10 + 2160*a*b^ 4*tan(1/2*d*x + 1/2*c)^10 + 5400*a^4*b*tan(1/2*d*x + 1/2*c)^9 - 3420*a^2*b ^3*tan(1/2*d*x + 1/2*c)^9 - 25*b^5*tan(1/2*d*x + 1/2*c)^9 + 25200*a^5*tan( 1/2*d*x + 1/2*c)^8 - 31200*a^3*b^2*tan(1/2*d*x + 1/2*c)^8 + 6480*a*b^4*tan (1/2*d*x + 1/2*c)^8 + 3600*a^4*b*tan(1/2*d*x + 1/2*c)^7 - 1800*a^2*b^3*tan (1/2*d*x + 1/2*c)^7 + 450*b^5*tan(1/2*d*x + 1/2*c)^7 + 50400*a^5*tan(1/2*d *x + 1/2*c)^6 - 56000*a^3*b^2*tan(1/2*d*x + 1/2*c)^6 + 11040*a*b^4*tan(1/2 *d*x + 1/2*c)^6 - 3600*a^4*b*tan(1/2*d*x + 1/2*c)^5 + 1800*a^2*b^3*tan(1/2 *d*x + 1/2*c)^5 - 450*b^5*tan(1/2*d*x + 1/2*c)^5 + 50400*a^5*tan(1/2*d*...
Time = 48.95 (sec) , antiderivative size = 4362, normalized size of antiderivative = 8.14 \[ \int \frac {\cos ^6(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
- ((840*a^7 + 213*a^3*b^4 - 985*a^5*b^2)/(15*b^8) + (tan(c/2 + (d*x)/2)^14 *(31*a*b^6 + 112*a^7 - 138*a^3*b^4 + 18*a^5*b^2))/(2*b^8) + (tan(c/2 + (d* x)/2)^12*(410*a*b^6 + 1176*a^7 - 1533*a^3*b^4 + 189*a^5*b^2))/(3*b^8) + (t an(c/2 + (d*x)/2)^10*(2281*a*b^6 + 7056*a^7 - 8766*a^3*b^4 + 686*a^5*b^2)) /(6*b^8) + (tan(c/2 + (d*x)/2)^2*(1239*a*b^6 + 11760*a^7 - 2402*a^3*b^4 - 9310*a^5*b^2))/(30*b^8) + (tan(c/2 + (d*x)/2)^4*(3062*a*b^6 + 17640*a^7 - 10011*a^3*b^4 - 8365*a^5*b^2))/(15*b^8) + (tan(c/2 + (d*x)/2)^6*(14155*a*b ^6 + 58800*a^7 - 50514*a^3*b^4 - 12950*a^5*b^2))/(30*b^8) + (tan(c/2 + (d* x)/2)*(23520*a^6 + 6171*a^2*b^4 - 27860*a^4*b^2))/(120*b^7) + (tan(c/2 + ( d*x)/2)^15*(224*a^6 + 43*a^2*b^4 - 244*a^4*b^2))/(8*b^7) + (tan(c/2 + (d*x )/2)^13*(8736*a^6 + 132*b^6 + 1453*a^2*b^4 - 9516*a^4*b^2))/(24*b^7) + (ta n(c/2 + (d*x)/2)^11*(38304*a^6 - 20*b^6 + 8033*a^2*b^4 - 43068*a^4*b^2))/( 24*b^7) + (tan(c/2 + (d*x)/2)^9*(84000*a^6 + 360*b^6 + 20341*a^2*b^4 - 973 24*a^4*b^2))/(24*b^7) + (tan(c/2 + (d*x)/2)^7*(104160*a^6 - 360*b^6 + 2737 1*a^2*b^4 - 123316*a^4*b^2))/(24*b^7) + (tan(c/2 + (d*x)/2)^3*(144480*a^6 - 660*b^6 + 40447*a^2*b^4 - 173060*a^4*b^2))/(120*b^7) + (tan(c/2 + (d*x)/ 2)^5*(372960*a^6 + 100*b^6 + 102971*a^2*b^4 - 446580*a^4*b^2))/(120*b^7) + (tan(c/2 + (d*x)/2)^8*(7*a^2 + 8*b^2)*(213*a*b^4 + 840*a^5 - 985*a^3*b^2) )/(3*b^8))/(d*(tan(c/2 + (d*x)/2)^2*(8*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^1 4*(8*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^4*(28*a^2 + 24*b^2) + tan(c/2 + ...