Integrand size = 21, antiderivative size = 539 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) x}{16 a^9}-\frac {\sqrt {a-b} b \sqrt {a+b} \left (6 a^4-47 a^2 b^2+56 b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^9 d}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))} \]
1/16*(5*a^6-180*a^4*b^2+600*a^2*b^4-448*b^6)*x/a^9+1/30*b*(213*a^4-985*a^2 *b^2+840*b^4)*sin(d*x+c)/a^8/d-1/16*(43*a^4-244*a^2*b^2+224*b^4)*cos(d*x+c )*sin(d*x+c)/a^7/d+1/30*(45*a^4-291*a^2*b^2+280*b^4)*cos(d*x+c)^2*sin(d*x+ c)/a^6/b/d-1/24*(24*a^4-169*a^2*b^2+168*b^4)*cos(d*x+c)^3*sin(d*x+c)/a^5/b ^2/d-1/4*cos(d*x+c)^4*sin(d*x+c)/b/d/(b+a*cos(d*x+c))^2+1/10*a*cos(d*x+c)^ 5*sin(d*x+c)/b^2/d/(b+a*cos(d*x+c))^2+1/60*(9*a^4-60*a^2*b^2+56*b^4)*cos(d *x+c)^5*sin(d*x+c)/a^3/b^2/d/(b+a*cos(d*x+c))^2+4/15*b*cos(d*x+c)^6*sin(d* x+c)/a^2/d/(b+a*cos(d*x+c))^2-1/6*cos(d*x+c)^7*sin(d*x+c)/a/d/(b+a*cos(d*x +c))^2+1/20*(15*a^4-110*a^2*b^2+112*b^4)*cos(d*x+c)^4*sin(d*x+c)/a^4/b^2/d /(b+a*cos(d*x+c))-b*(6*a^4-47*a^2*b^2+56*b^4)*arctanh((a-b)^(1/2)*tan(1/2* d*x+1/2*c)/(a+b)^(1/2))*(a-b)^(1/2)*(a+b)^(1/2)/a^9/d
Time = 9.15 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.11 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {-7680 b \left (-a^2+b^2\right )^3 \left (6 a^4-47 a^2 b^2+56 b^4\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^2+2 \left (a^2-b^2\right )^{5/2} \left (600 a^8 c-20400 a^6 b^2 c+28800 a^4 b^4 c+90240 a^2 b^6 c-107520 b^8 c+600 a^8 d x-20400 a^6 b^2 d x+28800 a^4 b^4 d x+90240 a^2 b^6 d x-107520 b^8 d x+480 a b \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) (c+d x) \cos (c+d x)+120 a^2 \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) (c+d x) \cos (2 (c+d x))+2640 a^7 b \sin (c+d x)+16160 a^5 b^3 \sin (c+d x)-117120 a^3 b^5 \sin (c+d x)+107520 a b^7 \sin (c+d x)-405 a^8 \sin (2 (c+d x))+24600 a^6 b^2 \sin (2 (c+d x))-99040 a^4 b^4 \sin (2 (c+d x))+80640 a^2 b^6 \sin (2 (c+d x))+2436 a^7 b \sin (3 (c+d x))-10880 a^5 b^3 \sin (3 (c+d x))+8960 a^3 b^5 \sin (3 (c+d x))-140 a^8 \sin (4 (c+d x))+1164 a^6 b^2 \sin (4 (c+d x))-1120 a^4 b^4 \sin (4 (c+d x))-188 a^7 b \sin (5 (c+d x))+224 a^5 b^3 \sin (5 (c+d x))+35 a^8 \sin (6 (c+d x))-56 a^6 b^2 \sin (6 (c+d x))+16 a^7 b \sin (7 (c+d x))-5 a^8 \sin (8 (c+d x))\right )}{7680 a^9 (a-b)^2 (a+b)^2 \sqrt {a^2-b^2} d (b+a \cos (c+d x))^2} \]
(-7680*b*(-a^2 + b^2)^3*(6*a^4 - 47*a^2*b^2 + 56*b^4)*ArcTanh[((-a + b)*Ta n[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2 + 2*(a^2 - b^2)^(5 /2)*(600*a^8*c - 20400*a^6*b^2*c + 28800*a^4*b^4*c + 90240*a^2*b^6*c - 107 520*b^8*c + 600*a^8*d*x - 20400*a^6*b^2*d*x + 28800*a^4*b^4*d*x + 90240*a^ 2*b^6*d*x - 107520*b^8*d*x + 480*a*b*(5*a^6 - 180*a^4*b^2 + 600*a^2*b^4 - 448*b^6)*(c + d*x)*Cos[c + d*x] + 120*a^2*(5*a^6 - 180*a^4*b^2 + 600*a^2*b ^4 - 448*b^6)*(c + d*x)*Cos[2*(c + d*x)] + 2640*a^7*b*Sin[c + d*x] + 16160 *a^5*b^3*Sin[c + d*x] - 117120*a^3*b^5*Sin[c + d*x] + 107520*a*b^7*Sin[c + d*x] - 405*a^8*Sin[2*(c + d*x)] + 24600*a^6*b^2*Sin[2*(c + d*x)] - 99040* a^4*b^4*Sin[2*(c + d*x)] + 80640*a^2*b^6*Sin[2*(c + d*x)] + 2436*a^7*b*Sin [3*(c + d*x)] - 10880*a^5*b^3*Sin[3*(c + d*x)] + 8960*a^3*b^5*Sin[3*(c + d *x)] - 140*a^8*Sin[4*(c + d*x)] + 1164*a^6*b^2*Sin[4*(c + d*x)] - 1120*a^4 *b^4*Sin[4*(c + d*x)] - 188*a^7*b*Sin[5*(c + d*x)] + 224*a^5*b^3*Sin[5*(c + d*x)] + 35*a^8*Sin[6*(c + d*x)] - 56*a^6*b^2*Sin[6*(c + d*x)] + 16*a^7*b *Sin[7*(c + d*x)] - 5*a^8*Sin[8*(c + d*x)]))/(7680*a^9*(a - b)^2*(a + b)^2 *Sqrt[a^2 - b^2]*d*(b + a*Cos[c + d*x])^2)
Time = 4.10 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.26, number of steps used = 31, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.429, Rules used = {3042, 4360, 25, 25, 3042, 3375, 27, 3042, 3526, 27, 3042, 3526, 27, 3042, 3528, 27, 3042, 3528, 25, 3042, 3528, 25, 3042, 3502, 27, 3042, 3214, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^6}{\left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\sin ^6(c+d x) \cos ^3(c+d x)}{(-a \cos (c+d x)-b)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos ^3(c+d x) \sin ^6(c+d x)}{(b+a \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\sin ^6(c+d x) \cos ^3(c+d x)}{(a \cos (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \cos \left (c+d x+\frac {\pi }{2}\right )^6}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right )^3}dx\) |
| \(\Big \downarrow \) 3375 |
| \(\displaystyle \frac {\int \frac {10 \cos ^5(c+d x) \left (-2 \left (12 a^4-65 b^2 a^2+56 b^4\right ) \cos ^2(c+d x)+3 a b \left (3 a^2-2 b^2\right ) \cos (c+d x)+3 \left (6 a^4-35 b^2 a^2+32 b^4\right )\right )}{(b+a \cos (c+d x))^3}dx}{600 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos ^5(c+d x) \left (-2 \left (12 a^4-65 b^2 a^2+56 b^4\right ) \cos ^2(c+d x)+3 a b \left (3 a^2-2 b^2\right ) \cos (c+d x)+3 \left (6 a^4-35 b^2 a^2+32 b^4\right )\right )}{(b+a \cos (c+d x))^3}dx}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^5 \left (-2 \left (12 a^4-65 b^2 a^2+56 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+3 a b \left (3 a^2-2 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (6 a^4-35 b^2 a^2+32 b^4\right )\right )}{\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\frac {\int \frac {2 \cos ^4(c+d x) \left (-2 \left (30 a^6-215 b^2 a^4+353 b^4 a^2-168 b^6\right ) \cos ^2(c+d x)+a b \left (15 a^4-31 b^2 a^2+16 b^4\right ) \cos (c+d x)+5 \left (9 a^6-69 b^2 a^4+116 b^4 a^2-56 b^6\right )\right )}{(b+a \cos (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\cos ^4(c+d x) \left (-2 \left (30 a^6-215 b^2 a^4+353 b^4 a^2-168 b^6\right ) \cos ^2(c+d x)+a b \left (15 a^4-31 b^2 a^2+16 b^4\right ) \cos (c+d x)+5 \left (9 a^6-69 b^2 a^4+116 b^4 a^2-56 b^6\right )\right )}{(b+a \cos (c+d x))^2}dx}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (-2 \left (30 a^6-215 b^2 a^4+353 b^4 a^2-168 b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a b \left (15 a^4-31 b^2 a^2+16 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 \left (9 a^6-69 b^2 a^4+116 b^4 a^2-56 b^6\right )\right )}{\left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 \cos ^3(c+d x) \left (-5 \left (24 a^4-169 b^2 a^2+168 b^4\right ) \cos ^2(c+d x) \left (a^2-b^2\right )^2+6 \left (15 a^4-110 b^2 a^2+112 b^4\right ) \left (a^2-b^2\right )^2+a b \left (15 a^2-28 b^2\right ) \cos (c+d x) \left (a^2-b^2\right )^2\right )}{b+a \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {\cos ^3(c+d x) \left (-5 \left (24 a^4-169 b^2 a^2+168 b^4\right ) \cos ^2(c+d x) \left (a^2-b^2\right )^2+6 \left (15 a^4-110 b^2 a^2+112 b^4\right ) \left (a^2-b^2\right )^2+a b \left (15 a^2-28 b^2\right ) \cos (c+d x) \left (a^2-b^2\right )^2\right )}{b+a \cos (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (-5 \left (24 a^4-169 b^2 a^2+168 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a^2-b^2\right )^2+6 \left (15 a^4-110 b^2 a^2+112 b^4\right ) \left (a^2-b^2\right )^2+a b \left (15 a^2-28 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )^2\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {\int -\frac {3 \cos ^2(c+d x) \left (-4 b \left (45 a^4-291 b^2 a^2+280 b^4\right ) \cos ^2(c+d x) \left (a^2-b^2\right )^2+5 b \left (24 a^4-169 b^2 a^2+168 b^4\right ) \left (a^2-b^2\right )^2+7 a b^2 \left (5 a^2-8 b^2\right ) \cos (c+d x) \left (a^2-b^2\right )^2\right )}{b+a \cos (c+d x)}dx}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \int \frac {\cos ^2(c+d x) \left (-4 b \left (45 a^4-291 b^2 a^2+280 b^4\right ) \cos ^2(c+d x) \left (a^2-b^2\right )^2+5 b \left (24 a^4-169 b^2 a^2+168 b^4\right ) \left (a^2-b^2\right )^2+7 a b^2 \left (5 a^2-8 b^2\right ) \cos (c+d x) \left (a^2-b^2\right )^2\right )}{b+a \cos (c+d x)}dx}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (-4 b \left (45 a^4-291 b^2 a^2+280 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a^2-b^2\right )^2+5 b \left (24 a^4-169 b^2 a^2+168 b^4\right ) \left (a^2-b^2\right )^2+7 a b^2 \left (5 a^2-8 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a^2-b^2\right )^2\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \left (\frac {\int -\frac {\cos (c+d x) \left (a \left (207 a^2-280 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x) b^3-15 \left (a^2-b^2\right )^2 \left (43 a^4-244 b^2 a^2+224 b^4\right ) \cos ^2(c+d x) b^2+8 \left (a^2-b^2\right )^2 \left (45 a^4-291 b^2 a^2+280 b^4\right ) b^2\right )}{b+a \cos (c+d x)}dx}{3 a}-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \left (-\frac {\int \frac {\cos (c+d x) \left (a \left (207 a^2-280 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x) b^3-15 \left (a^2-b^2\right )^2 \left (43 a^4-244 b^2 a^2+224 b^4\right ) \cos ^2(c+d x) b^2+8 \left (a^2-b^2\right )^2 \left (45 a^4-291 b^2 a^2+280 b^4\right ) b^2\right )}{b+a \cos (c+d x)}dx}{3 a}-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \left (-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (a \left (207 a^2-280 b^2\right ) \left (a^2-b^2\right )^2 \sin \left (c+d x+\frac {\pi }{2}\right ) b^3-15 \left (a^2-b^2\right )^2 \left (43 a^4-244 b^2 a^2+224 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^2+8 \left (a^2-b^2\right )^2 \left (45 a^4-291 b^2 a^2+280 b^4\right ) b^2\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{3 a}-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \left (-\frac {\frac {\int -\frac {-8 \left (a^2-b^2\right )^2 \left (213 a^4-985 b^2 a^2+840 b^4\right ) \cos ^2(c+d x) b^3+15 \left (a^2-b^2\right )^2 \left (43 a^4-244 b^2 a^2+224 b^4\right ) b^3-a \left (a^2-b^2\right )^2 \left (75 a^4-996 b^2 a^2+1120 b^4\right ) \cos (c+d x) b^2}{b+a \cos (c+d x)}dx}{2 a}-\frac {15 b^2 \left (a^2-b^2\right )^2 \left (43 a^4-244 a^2 b^2+224 b^4\right ) \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \left (-\frac {-\frac {\int \frac {-8 \left (a^2-b^2\right )^2 \left (213 a^4-985 b^2 a^2+840 b^4\right ) \cos ^2(c+d x) b^3+15 \left (a^2-b^2\right )^2 \left (43 a^4-244 b^2 a^2+224 b^4\right ) b^3-a \left (a^2-b^2\right )^2 \left (75 a^4-996 b^2 a^2+1120 b^4\right ) \cos (c+d x) b^2}{b+a \cos (c+d x)}dx}{2 a}-\frac {15 b^2 \left (43 a^4-244 a^2 b^2+224 b^4\right ) \left (a^2-b^2\right )^2 \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \left (-\frac {-\frac {\int \frac {-8 \left (a^2-b^2\right )^2 \left (213 a^4-985 b^2 a^2+840 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^3+15 \left (a^2-b^2\right )^2 \left (43 a^4-244 b^2 a^2+224 b^4\right ) b^3-a \left (a^2-b^2\right )^2 \left (75 a^4-996 b^2 a^2+1120 b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^2}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a}-\frac {15 b^2 \left (43 a^4-244 a^2 b^2+224 b^4\right ) \left (a^2-b^2\right )^2 \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \left (-\frac {-\frac {\frac {\int \frac {15 \left (a b^3 \left (a^2-b^2\right )^2 \left (43 a^4-244 b^2 a^2+224 b^4\right )-b^2 \left (a^2-b^2\right )^2 \left (5 a^6-180 b^2 a^4+600 b^4 a^2-448 b^6\right ) \cos (c+d x)\right )}{b+a \cos (c+d x)}dx}{a}-\frac {8 b^3 \left (a^2-b^2\right )^2 \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (43 a^4-244 a^2 b^2+224 b^4\right ) \left (a^2-b^2\right )^2 \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \left (-\frac {-\frac {\frac {15 \int \frac {a b^3 \left (a^2-b^2\right )^2 \left (43 a^4-244 b^2 a^2+224 b^4\right )-b^2 \left (a^2-b^2\right )^2 \left (5 a^6-180 b^2 a^4+600 b^4 a^2-448 b^6\right ) \cos (c+d x)}{b+a \cos (c+d x)}dx}{a}-\frac {8 b^3 \left (a^2-b^2\right )^2 \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (43 a^4-244 a^2 b^2+224 b^4\right ) \left (a^2-b^2\right )^2 \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \left (-\frac {-\frac {\frac {15 \int \frac {a b^3 \left (a^2-b^2\right )^2 \left (43 a^4-244 b^2 a^2+224 b^4\right )-b^2 \left (a^2-b^2\right )^2 \left (5 a^6-180 b^2 a^4+600 b^4 a^2-448 b^6\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {8 b^3 \left (a^2-b^2\right )^2 \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (43 a^4-244 a^2 b^2+224 b^4\right ) \left (a^2-b^2\right )^2 \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {8 b^3 \left (a^2-b^2\right )^3 \left (6 a^4-47 a^2 b^2+56 b^4\right ) \int \frac {1}{b+a \cos (c+d x)}dx}{a}-\frac {b^2 x \left (a^2-b^2\right )^2 \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right )}{a}\right )}{a}-\frac {8 b^3 \left (a^2-b^2\right )^2 \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (43 a^4-244 a^2 b^2+224 b^4\right ) \left (a^2-b^2\right )^2 \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {8 b^3 \left (a^2-b^2\right )^3 \left (6 a^4-47 a^2 b^2+56 b^4\right ) \int \frac {1}{b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a}-\frac {b^2 x \left (a^2-b^2\right )^2 \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right )}{a}\right )}{a}-\frac {8 b^3 \left (a^2-b^2\right )^2 \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (43 a^4-244 a^2 b^2+224 b^4\right ) \left (a^2-b^2\right )^2 \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {\frac {2 \left (-\frac {3 \left (-\frac {-\frac {\frac {15 \left (\frac {16 b^3 \left (a^2-b^2\right )^3 \left (6 a^4-47 a^2 b^2+56 b^4\right ) \int \frac {1}{-\left ((a-b) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+a+b}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}-\frac {b^2 x \left (a^2-b^2\right )^2 \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right )}{a}\right )}{a}-\frac {8 b^3 \left (a^2-b^2\right )^2 \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{a d}}{2 a}-\frac {15 b^2 \left (43 a^4-244 a^2 b^2+224 b^4\right ) \left (a^2-b^2\right )^2 \sin (c+d x) \cos (c+d x)}{2 a d}}{3 a}-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}\right )}{4 a}-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}\right )}{a \left (a^2-b^2\right )}+\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}}{a \left (a^2-b^2\right )}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}}{60 a^2 b^2}+\frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{a d (a \cos (c+d x)+b)^2}+\frac {\frac {3 \left (a^2-b^2\right ) \left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{a d (a \cos (c+d x)+b)}+\frac {2 \left (-\frac {5 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac {3 \left (-\frac {4 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{3 a d}-\frac {-\frac {15 b^2 \left (43 a^4-244 a^2 b^2+224 b^4\right ) \left (a^2-b^2\right )^2 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {\frac {15 \left (\frac {16 b^3 \left (a^2-b^2\right )^3 \left (6 a^4-47 a^2 b^2+56 b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a d \sqrt {a-b} \sqrt {a+b}}-\frac {b^2 x \left (a^2-b^2\right )^2 \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right )}{a}\right )}{a}-\frac {8 b^3 \left (a^2-b^2\right )^2 \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{a d}}{2 a}}{3 a}\right )}{4 a}\right )}{a \left (a^2-b^2\right )}}{a \left (a^2-b^2\right )}}{60 a^2 b^2}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}\) |
-1/4*(Cos[c + d*x]^4*Sin[c + d*x])/(b*d*(b + a*Cos[c + d*x])^2) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(10*b^2*d*(b + a*Cos[c + d*x])^2) + (4*b*Cos[c + d *x]^6*Sin[c + d*x])/(15*a^2*d*(b + a*Cos[c + d*x])^2) - (Cos[c + d*x]^7*Si n[c + d*x])/(6*a*d*(b + a*Cos[c + d*x])^2) + (((9*a^4 - 60*a^2*b^2 + 56*b^ 4)*Cos[c + d*x]^5*Sin[c + d*x])/(a*d*(b + a*Cos[c + d*x])^2) + ((3*(a^2 - b^2)*(15*a^4 - 110*a^2*b^2 + 112*b^4)*Cos[c + d*x]^4*Sin[c + d*x])/(a*d*(b + a*Cos[c + d*x])) + (2*((-5*(a^2 - b^2)^2*(24*a^4 - 169*a^2*b^2 + 168*b^ 4)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - (3*((-4*b*(a^2 - b^2)^2*(45*a^4 - 291*a^2*b^2 + 280*b^4)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d) - ((-15*b^2* (a^2 - b^2)^2*(43*a^4 - 244*a^2*b^2 + 224*b^4)*Cos[c + d*x]*Sin[c + d*x])/ (2*a*d) - ((15*(-((b^2*(a^2 - b^2)^2*(5*a^6 - 180*a^4*b^2 + 600*a^2*b^4 - 448*b^6)*x)/a) + (16*b^3*(a^2 - b^2)^3*(6*a^4 - 47*a^2*b^2 + 56*b^4)*ArcTa nh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b] *d)))/a - (8*b^3*(a^2 - b^2)^2*(213*a^4 - 985*a^2*b^2 + 840*b^4)*Sin[c + d *x])/(a*d))/(2*a))/(3*a)))/(4*a)))/(a*(a^2 - b^2)))/(a*(a^2 - b^2)))/(60*a ^2*b^2)
3.3.28.3.1 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/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*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])))
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 4.42 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.08
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {2 \left (\left (\frac {5}{16} a^{6}+3 a^{5} b -\frac {21}{4} a^{4} b^{2}-20 a^{3} b^{3}+\frac {15}{2} a^{2} b^{4}+21 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (19 a^{5} b -\frac {87}{4} a^{4} b^{2}+\frac {45}{2} a^{2} b^{4}+105 a \,b^{5}+\frac {85}{48} a^{6}-\frac {340}{3} a^{3} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (\frac {258}{5} a^{5} b -\frac {33}{2} a^{4} b^{2}-240 a^{3} b^{3}+15 a^{2} b^{4}+210 a \,b^{5}+\frac {33}{8} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-\frac {33}{8} a^{6}+\frac {33}{2} a^{4} b^{2}-15 a^{2} b^{4}+\frac {258}{5} a^{5} b -240 a^{3} b^{3}+210 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (19 a^{5} b +\frac {87}{4} a^{4} b^{2}-\frac {340}{3} a^{3} b^{3}-\frac {45}{2} a^{2} b^{4}+105 a \,b^{5}-\frac {85}{48} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (3 a^{5} b -20 a^{3} b^{3}+21 a \,b^{5}-\frac {5}{16} a^{6}+\frac {21}{4} a^{4} b^{2}-\frac {15}{2} a^{2} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {\left (5 a^{6}-180 a^{4} b^{2}+600 a^{2} b^{4}-448 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{9}}+\frac {2 b \left (a +b \right ) \left (a -b \right ) \left (\frac {\left (\frac {5}{2} a^{3} b^{2}-7 a \,b^{4}-3 a^{4} b +\frac {15}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {5}{2} a^{3} b^{2}-7 a \,b^{4}+3 a^{4} b -\frac {15}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (6 a^{4}-47 a^{2} b^{2}+56 b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{9}}}{d}\) | \(582\) |
| default | \(\frac {\frac {\frac {2 \left (\left (\frac {5}{16} a^{6}+3 a^{5} b -\frac {21}{4} a^{4} b^{2}-20 a^{3} b^{3}+\frac {15}{2} a^{2} b^{4}+21 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (19 a^{5} b -\frac {87}{4} a^{4} b^{2}+\frac {45}{2} a^{2} b^{4}+105 a \,b^{5}+\frac {85}{48} a^{6}-\frac {340}{3} a^{3} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (\frac {258}{5} a^{5} b -\frac {33}{2} a^{4} b^{2}-240 a^{3} b^{3}+15 a^{2} b^{4}+210 a \,b^{5}+\frac {33}{8} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-\frac {33}{8} a^{6}+\frac {33}{2} a^{4} b^{2}-15 a^{2} b^{4}+\frac {258}{5} a^{5} b -240 a^{3} b^{3}+210 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (19 a^{5} b +\frac {87}{4} a^{4} b^{2}-\frac {340}{3} a^{3} b^{3}-\frac {45}{2} a^{2} b^{4}+105 a \,b^{5}-\frac {85}{48} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (3 a^{5} b -20 a^{3} b^{3}+21 a \,b^{5}-\frac {5}{16} a^{6}+\frac {21}{4} a^{4} b^{2}-\frac {15}{2} a^{2} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {\left (5 a^{6}-180 a^{4} b^{2}+600 a^{2} b^{4}-448 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{9}}+\frac {2 b \left (a +b \right ) \left (a -b \right ) \left (\frac {\left (\frac {5}{2} a^{3} b^{2}-7 a \,b^{4}-3 a^{4} b +\frac {15}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {5}{2} a^{3} b^{2}-7 a \,b^{4}+3 a^{4} b -\frac {15}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {\left (6 a^{4}-47 a^{2} b^{2}+56 b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{9}}}{d}\) | \(582\) |
| risch | \(\frac {5 x}{16 a^{3}}-\frac {45 x \,b^{2}}{4 a^{5}}+\frac {75 x \,b^{4}}{2 a^{7}}-\frac {28 x \,b^{6}}{a^{9}}-\frac {\sin \left (6 d x +6 c \right )}{192 a^{3} d}+\frac {3 \sin \left (4 d x +4 c \right )}{64 d \,a^{3}}+\frac {3 b \sin \left (5 d x +5 c \right )}{80 a^{4} d}-\frac {3 \sin \left (4 d x +4 c \right ) b^{2}}{16 d \,a^{5}}+\frac {15 i {\mathrm e}^{2 i \left (d x +c \right )}}{128 a^{3} d}-\frac {15 i {\mathrm e}^{-2 i \left (d x +c \right )}}{128 a^{3} d}+\frac {i b^{2} \left (7 a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}-23 a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+16 a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+6 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-27 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+30 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}+17 a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-61 a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}+44 a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}+6 a^{6}-21 a^{4} b^{2}+15 a^{2} b^{4}\right )}{a^{9} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2}}+\frac {5 i b^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{12 a^{6} d}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{2}}{2 a^{5} d}-\frac {15 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{4}}{8 a^{7} d}+\frac {33 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{16 a^{4} d}-\frac {45 i b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{4 a^{6} d}+\frac {21 i b^{5} {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{8} d}+\frac {7 i b \,{\mathrm e}^{3 i \left (d x +c \right )}}{32 a^{4} d}-\frac {5 i b^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{12 a^{6} d}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} b^{2}}{2 a^{5} d}+\frac {15 i {\mathrm e}^{2 i \left (d x +c \right )} b^{4}}{8 a^{7} d}-\frac {33 i b \,{\mathrm e}^{i \left (d x +c \right )}}{16 a^{4} d}+\frac {45 i b^{3} {\mathrm e}^{i \left (d x +c \right )}}{4 a^{6} d}-\frac {21 i b^{5} {\mathrm e}^{i \left (d x +c \right )}}{2 a^{8} d}-\frac {3 \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{d \,a^{5}}+\frac {47 \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{2 d \,a^{7}}-\frac {28 \sqrt {a^{2}-b^{2}}\, b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{d \,a^{9}}+\frac {3 \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{d \,a^{5}}-\frac {47 \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{2 d \,a^{7}}+\frac {28 \sqrt {a^{2}-b^{2}}\, b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{d \,a^{9}}-\frac {7 i b \,{\mathrm e}^{-3 i \left (d x +c \right )}}{32 a^{4} d}\) | \(969\) |
1/d*(2/a^9*(((5/16*a^6+3*a^5*b-21/4*a^4*b^2-20*a^3*b^3+15/2*a^2*b^4+21*a*b ^5)*tan(1/2*d*x+1/2*c)^11+(19*a^5*b-87/4*a^4*b^2+45/2*a^2*b^4+105*a*b^5+85 /48*a^6-340/3*a^3*b^3)*tan(1/2*d*x+1/2*c)^9+(258/5*a^5*b-33/2*a^4*b^2-240* a^3*b^3+15*a^2*b^4+210*a*b^5+33/8*a^6)*tan(1/2*d*x+1/2*c)^7+(-33/8*a^6+33/ 2*a^4*b^2-15*a^2*b^4+258/5*a^5*b-240*a^3*b^3+210*a*b^5)*tan(1/2*d*x+1/2*c) ^5+(19*a^5*b+87/4*a^4*b^2-340/3*a^3*b^3-45/2*a^2*b^4+105*a*b^5-85/48*a^6)* tan(1/2*d*x+1/2*c)^3+(3*a^5*b-20*a^3*b^3+21*a*b^5-5/16*a^6+21/4*a^4*b^2-15 /2*a^2*b^4)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^6+1/16*(5*a^6-180 *a^4*b^2+600*a^2*b^4-448*b^6)*arctan(tan(1/2*d*x+1/2*c)))+2*b*(a+b)*(a-b)/ a^9*(((5/2*a^3*b^2-7*a*b^4-3*a^4*b+15/2*a^2*b^3)*tan(1/2*d*x+1/2*c)^3+(5/2 *a^3*b^2-7*a*b^4+3*a^4*b-15/2*a^2*b^3)*tan(1/2*d*x+1/2*c))/(tan(1/2*d*x+1/ 2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2-1/2*(6*a^4-47*a^2*b^2+56*b^4)/((a-b )*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2))))
Time = 0.38 (sec) , antiderivative size = 1057, normalized size of antiderivative = 1.96 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
[1/240*(15*(5*a^8 - 180*a^6*b^2 + 600*a^4*b^4 - 448*a^2*b^6)*d*x*cos(d*x + c)^2 + 30*(5*a^7*b - 180*a^5*b^3 + 600*a^3*b^5 - 448*a*b^7)*d*x*cos(d*x + c) + 15*(5*a^6*b^2 - 180*a^4*b^4 + 600*a^2*b^6 - 448*b^8)*d*x + 60*(6*a^4 *b^3 - 47*a^2*b^5 + 56*b^7 + (6*a^6*b - 47*a^4*b^3 + 56*a^2*b^5)*cos(d*x + c)^2 + 2*(6*a^5*b^2 - 47*a^3*b^4 + 56*a*b^6)*cos(d*x + c))*sqrt(a^2 - b^2 )*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^ 2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - (40*a^8*cos(d*x + c)^7 - 64*a^7*b*cos(d*x + c )^6 - 1704*a^5*b^3 + 7880*a^3*b^5 - 6720*a*b^7 - 2*(65*a^8 - 56*a^6*b^2)*c os(d*x + c)^5 + 4*(67*a^7*b - 56*a^5*b^3)*cos(d*x + c)^4 + (165*a^8 - 694* a^6*b^2 + 560*a^4*b^4)*cos(d*x + c)^3 - 2*(387*a^7*b - 1444*a^5*b^3 + 1120 *a^3*b^5)*cos(d*x + c)^2 - (2763*a^6*b^2 - 12100*a^4*b^4 + 10080*a^2*b^6)* cos(d*x + c))*sin(d*x + c))/(a^11*d*cos(d*x + c)^2 + 2*a^10*b*d*cos(d*x + c) + a^9*b^2*d), 1/240*(15*(5*a^8 - 180*a^6*b^2 + 600*a^4*b^4 - 448*a^2*b^ 6)*d*x*cos(d*x + c)^2 + 30*(5*a^7*b - 180*a^5*b^3 + 600*a^3*b^5 - 448*a*b^ 7)*d*x*cos(d*x + c) + 15*(5*a^6*b^2 - 180*a^4*b^4 + 600*a^2*b^6 - 448*b^8) *d*x - 120*(6*a^4*b^3 - 47*a^2*b^5 + 56*b^7 + (6*a^6*b - 47*a^4*b^3 + 56*a ^2*b^5)*cos(d*x + c)^2 + 2*(6*a^5*b^2 - 47*a^3*b^4 + 56*a*b^6)*cos(d*x + c ))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - (40*a^8*cos(d*x + c)^7 - 64*a^7*b*cos(d*x + c)^6 ...
\[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\sin ^{6}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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*a^2-4*b^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 1030 vs. \(2 (508) = 1016\).
Time = 0.49 (sec) , antiderivative size = 1030, normalized size of antiderivative = 1.91 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
1/240*(15*(5*a^6 - 180*a^4*b^2 + 600*a^2*b^4 - 448*b^6)*(d*x + c)/a^9 - 24 0*(6*a^6*b - 53*a^4*b^3 + 103*a^2*b^5 - 56*b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/(sqrt(-a^2 + b^2)*a^9) - 240*(6*a^5*b^2*tan(1 /2*d*x + 1/2*c)^3 - 5*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 21*a^3*b^4*tan(1/2* d*x + 1/2*c)^3 + 19*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 15*a*b^6*tan(1/2*d*x + 1/2*c)^3 - 14*b^7*tan(1/2*d*x + 1/2*c)^3 - 6*a^5*b^2*tan(1/2*d*x + 1/2*c ) - 5*a^4*b^3*tan(1/2*d*x + 1/2*c) + 21*a^3*b^4*tan(1/2*d*x + 1/2*c) + 19* a^2*b^5*tan(1/2*d*x + 1/2*c) - 15*a*b^6*tan(1/2*d*x + 1/2*c) - 14*b^7*tan( 1/2*d*x + 1/2*c))/((a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^2*a^8) + 2*(75*a^5*tan(1/2*d*x + 1/2*c)^11 + 720*a^4*b*tan(1/2*d*x + 1/2*c)^11 - 1260*a^3*b^2*tan(1/2*d*x + 1/2*c)^11 - 4800*a^2*b^3*tan(1/2* d*x + 1/2*c)^11 + 1800*a*b^4*tan(1/2*d*x + 1/2*c)^11 + 5040*b^5*tan(1/2*d* x + 1/2*c)^11 + 425*a^5*tan(1/2*d*x + 1/2*c)^9 + 4560*a^4*b*tan(1/2*d*x + 1/2*c)^9 - 5220*a^3*b^2*tan(1/2*d*x + 1/2*c)^9 - 27200*a^2*b^3*tan(1/2*d*x + 1/2*c)^9 + 5400*a*b^4*tan(1/2*d*x + 1/2*c)^9 + 25200*b^5*tan(1/2*d*x + 1/2*c)^9 + 990*a^5*tan(1/2*d*x + 1/2*c)^7 + 12384*a^4*b*tan(1/2*d*x + 1/2* c)^7 - 3960*a^3*b^2*tan(1/2*d*x + 1/2*c)^7 - 57600*a^2*b^3*tan(1/2*d*x + 1 /2*c)^7 + 3600*a*b^4*tan(1/2*d*x + 1/2*c)^7 + 50400*b^5*tan(1/2*d*x + 1/2* c)^7 - 990*a^5*tan(1/2*d*x + 1/2*c)^5 + 12384*a^4*b*tan(1/2*d*x + 1/2*c...
Time = 18.19 (sec) , antiderivative size = 3975, normalized size of antiderivative = 7.37 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]
((tan(c/2 + (d*x)/2)^3*(10080*a*b^6 + 454*a^6*b - 55*a^7 + 9408*b^7 - 9688 *a^2*b^5 - 12212*a^3*b^4 + 608*a^4*b^3 + 2969*a^5*b^2))/(24*a^8) + (tan(c/ 2 + (d*x)/2)^13*(454*a^6*b - 10080*a*b^6 + 55*a^7 + 9408*b^7 - 9688*a^2*b^ 5 + 12212*a^3*b^4 + 608*a^4*b^3 - 2969*a^5*b^2))/(24*a^8) + (tan(c/2 + (d* x)/2)^5*(90720*a*b^6 + 2154*a^6*b - 215*a^7 + 141120*b^7 - 163240*a^2*b^5 - 107220*a^3*b^4 + 32224*a^4*b^3 + 22673*a^5*b^2))/(120*a^8) + (tan(c/2 + (d*x)/2)^11*(2154*a^6*b - 90720*a*b^6 + 215*a^7 + 141120*b^7 - 163240*a^2* b^5 + 107220*a^3*b^4 + 32224*a^4*b^3 - 22673*a^5*b^2))/(120*a^8) + (tan(c/ 2 + (d*x)/2)^7*(50400*a*b^6 - 4994*a^6*b + 2545*a^7 + 235200*b^7 - 287000* a^2*b^5 - 58820*a^3*b^4 + 74752*a^4*b^3 + 11173*a^5*b^2))/(120*a^8) - (tan (c/2 + (d*x)/2)^9*(50400*a*b^6 + 4994*a^6*b + 2545*a^7 - 235200*b^7 + 2870 00*a^2*b^5 - 58820*a^3*b^4 - 74752*a^4*b^3 + 11173*a^5*b^2))/(120*a^8) + ( tan(c/2 + (d*x)/2)^15*(a - b)*(224*a*b^5 + 43*a^5*b + 5*a^6 - 448*b^6 + 60 0*a^2*b^4 - 244*a^3*b^3 - 180*a^4*b^2))/(8*a^8) - (tan(c/2 + (d*x)/2)*(2*a *b + a^2 + b^2)*(224*a*b^4 - 48*a^4*b + 5*a^5 - 448*b^5 + 376*a^2*b^3 - 13 2*a^3*b^2))/(8*a^8))/(d*(2*a*b - tan(c/2 + (d*x)/2)^8*(10*a^2 - 70*b^2) + tan(c/2 + (d*x)/2)^2*(12*a*b + 4*a^2 + 8*b^2) + tan(c/2 + (d*x)/2)^14*(4*a ^2 - 12*a*b + 8*b^2) + tan(c/2 + (d*x)/2)^4*(28*a*b + 4*a^2 + 28*b^2) + ta n(c/2 + (d*x)/2)^12*(4*a^2 - 28*a*b + 28*b^2) + tan(c/2 + (d*x)/2)^6*(28*a *b - 4*a^2 + 56*b^2) - tan(c/2 + (d*x)/2)^10*(28*a*b + 4*a^2 - 56*b^2) ...