Integrand size = 29, antiderivative size = 1428 \[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{9/2}} \, dx=\frac {2 (a-b) \sqrt {a+b} \left (2 b^3 c^3 d \left (133 c^4+62 c^2 d^2-3 d^4\right )+2 a^2 b c d \left (406 c^6+73 c^4 d^2+132 c^2 d^4-35 d^6\right )-a b^2 c^2 \left (245 c^6+852 c^4 d^2+41 c^2 d^4+14 d^6\right )-a^3 \left (582 c^6 d^2-485 c^4 d^4+392 c^2 d^6-105 d^8\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{105 c^4 (c-d)^4 (c+d)^{7/2} (b c-a d)^2 f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b^3 c^4 \left (35 c^4+231 c^3 d+67 c^2 d^2+57 c d^3-6 d^4\right )-a b^2 c^3 \left (245 c^5+413 c^4 d+439 c^3 d^2+53 c^2 d^3-12 c d^4+14 d^5\right )+a^2 b c^2 \left (315 c^6+497 c^5 d+219 c^4 d^2-73 c^3 d^3+208 c^2 d^4+56 c d^5-70 d^6\right )-a^3 d \left (525 c^7+57 c^6 d-699 c^5 d^2+214 c^4 d^3+672 c^3 d^4-280 c^2 d^5-210 c d^6+105 d^7\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{105 c^5 (c-d)^4 (c+d)^{7/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 a^2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^5 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 d^2 (b+a \cos (e+f x))^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^3 \sqrt {c+d \sec (e+f x)}}-\frac {2 d \left (14 b c^3-19 a c^2 d-2 b c d^2+7 a d^3\right ) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{35 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}-\frac {2 \left (2 a b c d \left (91 c^4-2 c^2 d^2+7 d^4\right )-a^2 d^2 \left (162 c^4-101 c^2 d^2+35 d^4\right )-b^2 \left (35 c^6+67 c^4 d^2-6 c^2 d^4\right )\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{105 c^3 \left (c^2-d^2\right )^3 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}} \]
2/7*d^2*(b+a*cos(f*x+e))^2*sin(f*x+e)*(a+b*sec(f*x+e))^(1/2)/c/(c^2-d^2)/f /(d+c*cos(f*x+e))^3/(c+d*sec(f*x+e))^(1/2)-2/35*d*(-19*a*c^2*d+7*a*d^3+14* b*c^3-2*b*c*d^2)*(b+a*cos(f*x+e))*sin(f*x+e)*(a+b*sec(f*x+e))^(1/2)/c^2/(c ^2-d^2)^2/f/(d+c*cos(f*x+e))^2/(c+d*sec(f*x+e))^(1/2)-2/105*(2*a*b*c*d*(91 *c^4-2*c^2*d^2+7*d^4)-a^2*d^2*(162*c^4-101*c^2*d^2+35*d^4)-b^2*(35*c^6+67* c^4*d^2-6*c^2*d^4))*sin(f*x+e)*(a+b*sec(f*x+e))^(1/2)/c^3/(c^2-d^2)^3/f/(d +c*cos(f*x+e))/(c+d*sec(f*x+e))^(1/2)+2/105*(a-b)*(2*b^3*c^3*d*(133*c^4+62 *c^2*d^2-3*d^4)+2*a^2*b*c*d*(406*c^6+73*c^4*d^2+132*c^2*d^4-35*d^6)-a*b^2* c^2*(245*c^6+852*c^4*d^2+41*c^2*d^4+14*d^6)-a^3*(582*c^6*d^2-485*c^4*d^4+3 92*c^2*d^6-105*d^8))*(d+c*cos(f*x+e))^(3/2)*csc(f*x+e)*EllipticE((c+d)^(1/ 2)*(b+a*cos(f*x+e))^(1/2)/(a+b)^(1/2)/(d+c*cos(f*x+e))^(1/2),((a+b)*(c-d)/ (a-b)/(c+d))^(1/2))*(a+b)^(1/2)*(-(-a*d+b*c)*(1-cos(f*x+e))/(a+b)/(d+c*cos (f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+cos(f*x+e))/(a-b)/(d+c*cos(f*x+e)))^(1/2)* (a+b*sec(f*x+e))^(1/2)/c^4/(c-d)^4/(c+d)^(7/2)/(-a*d+b*c)^2/f/(b+a*cos(f*x +e))^(1/2)/(c+d*sec(f*x+e))^(1/2)+2/105*(b^3*c^4*(35*c^4+231*c^3*d+67*c^2* d^2+57*c*d^3-6*d^4)-a*b^2*c^3*(245*c^5+413*c^4*d+439*c^3*d^2+53*c^2*d^3-12 *c*d^4+14*d^5)+a^2*b*c^2*(315*c^6+497*c^5*d+219*c^4*d^2-73*c^3*d^3+208*c^2 *d^4+56*c*d^5-70*d^6)-a^3*d*(525*c^7+57*c^6*d-699*c^5*d^2+214*c^4*d^3+672* c^3*d^4-280*c^2*d^5-210*c*d^6+105*d^7))*(d+c*cos(f*x+e))^(3/2)*csc(f*x+e)* EllipticF((c+d)^(1/2)*(b+a*cos(f*x+e))^(1/2)/(a+b)^(1/2)/(d+c*cos(f*x+e...
Leaf count is larger than twice the leaf count of optimal. \(2979\) vs. \(2(1428)=2856\).
Time = 9.06 (sec) , antiderivative size = 2979, normalized size of antiderivative = 2.09 \[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{9/2}} \, dx=\text {Result too large to show} \]
((d + c*Cos[e + f*x])^5*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^(5/2)*((2*(b^2 *c^2*d^2*Sin[e + f*x] - 2*a*b*c*d^3*Sin[e + f*x] + a^2*d^4*Sin[e + f*x]))/ (7*c^3*(c^2 - d^2)*(d + c*Cos[e + f*x])^4) + (2*(-14*b^2*c^4*d*Sin[e + f*x ] + 43*a*b*c^3*d^2*Sin[e + f*x] - 29*a^2*c^2*d^3*Sin[e + f*x] + 2*b^2*c^2* d^3*Sin[e + f*x] - 19*a*b*c*d^4*Sin[e + f*x] + 17*a^2*d^5*Sin[e + f*x]))/( 35*c^3*(c^2 - d^2)^2*(d + c*Cos[e + f*x])^3) + (2*(35*b^2*c^6*Sin[e + f*x] - 224*a*b*c^5*d*Sin[e + f*x] + 234*a^2*c^4*d^2*Sin[e + f*x] + 67*b^2*c^4* d^2*Sin[e + f*x] + 52*a*b*c^3*d^3*Sin[e + f*x] - 209*a^2*c^2*d^4*Sin[e + f *x] - 6*b^2*c^2*d^4*Sin[e + f*x] - 20*a*b*c*d^5*Sin[e + f*x] + 71*a^2*d^6* Sin[e + f*x]))/(105*c^3*(c^2 - d^2)^3*(d + c*Cos[e + f*x])^2) + (2*(245*a* b^2*c^8*Sin[e + f*x] - 812*a^2*b*c^7*d*Sin[e + f*x] - 266*b^3*c^7*d*Sin[e + f*x] + 582*a^3*c^6*d^2*Sin[e + f*x] + 852*a*b^2*c^6*d^2*Sin[e + f*x] - 1 46*a^2*b*c^5*d^3*Sin[e + f*x] - 124*b^3*c^5*d^3*Sin[e + f*x] - 485*a^3*c^4 *d^4*Sin[e + f*x] + 41*a*b^2*c^4*d^4*Sin[e + f*x] - 264*a^2*b*c^3*d^5*Sin[ e + f*x] + 6*b^3*c^3*d^5*Sin[e + f*x] + 392*a^3*c^2*d^6*Sin[e + f*x] + 14* a*b^2*c^2*d^6*Sin[e + f*x] + 70*a^2*b*c*d^7*Sin[e + f*x] - 105*a^3*d^8*Sin [e + f*x]))/(105*c^3*(b*c - a*d)*(c^2 - d^2)^4*(d + c*Cos[e + f*x]))))/(f* (b + a*Cos[e + f*x])^2*(c + d*Sec[e + f*x])^(9/2)) + ((d + c*Cos[e + f*x]) ^(9/2)*Sec[e + f*x]^2*(a + b*Sec[e + f*x])^(5/2)*((4*(b*c - a*d)*(-70*a^2* b^2*c^8 - 35*b^4*c^8 - 77*a^3*b*c^7*d + 427*a*b^3*c^7*d + 162*a^4*c^6*d...
Time = 6.43 (sec) , antiderivative size = 1353, normalized size of antiderivative = 0.95, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.655, Rules used = {3042, 4430, 3042, 3527, 27, 3042, 3526, 27, 3042, 3526, 27, 3042, 3532, 3042, 3290, 3477, 3042, 3297, 3475}
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.
| \(\displaystyle \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}{\left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4430 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {\cos ^2(e+f x) (b+a \cos (e+f x))^{5/2}}{(d+c \cos (e+f x))^{9/2}}dx}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )^2 \left (b+a \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}{\left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{9/2}}dx}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 \int -\frac {(b+a \cos (e+f x))^{3/2} \left (-7 a \left (c^2-d^2\right ) \cos ^2(e+f x)-\left (7 b c^2-7 a d c-2 b d^2\right ) \cos (e+f x)+d (7 b c-5 a d)\right )}{2 (d+c \cos (e+f x))^{7/2}}dx}{7 c \left (c^2-d^2\right )}+\frac {2 d^2 \sin (e+f x) (a \cos (e+f x)+b)^{5/2}}{7 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{7/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) (a \cos (e+f x)+b)^{5/2}}{7 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{7/2}}-\frac {\int \frac {(b+a \cos (e+f x))^{3/2} \left (-7 a \left (c^2-d^2\right ) \cos ^2(e+f x)-\left (7 b c^2-7 a d c-2 b d^2\right ) \cos (e+f x)+d (7 b c-5 a d)\right )}{(d+c \cos (e+f x))^{7/2}}dx}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) (a \cos (e+f x)+b)^{5/2}}{7 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{7/2}}-\frac {\int \frac {\left (b+a \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2} \left (-7 a \left (c^2-d^2\right ) \sin \left (e+f x+\frac {\pi }{2}\right )^2+\left (-7 b c^2+7 a d c+2 b d^2\right ) \sin \left (e+f x+\frac {\pi }{2}\right )+d (7 b c-5 a d)\right )}{\left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) (a \cos (e+f x)+b)^{5/2}}{7 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{7/2}}-\frac {\frac {2 \int -\frac {\sqrt {b+a \cos (e+f x)} \left (35 b^2 c^4-112 a b d c^3+57 a^2 d^2 c^2+25 b^2 d^2 c^2+16 a b d^3 c-21 a^2 d^4+35 a^2 \left (c^2-d^2\right )^2 \cos ^2(e+f x)-2 \left (5 \left (7 c^3 d-c d^3\right ) a^2-b \left (35 c^4+6 d^2 c^2+7 d^4\right ) a+3 b^2 c d \left (7 c^2-d^2\right )\right ) \cos (e+f x)\right )}{2 (d+c \cos (e+f x))^{5/2}}dx}{5 c \left (c^2-d^2\right )}+\frac {2 d \left (-19 a c^2 d+7 a d^3+14 b c^3-2 b c d^2\right ) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) (a \cos (e+f x)+b)^{5/2}}{7 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{7/2}}-\frac {\frac {2 d \left (-19 a c^2 d+7 a d^3+14 b c^3-2 b c d^2\right ) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}-\frac {\int \frac {\sqrt {b+a \cos (e+f x)} \left (35 b^2 c^4-112 a b d c^3+57 a^2 d^2 c^2+25 b^2 d^2 c^2+16 a b d^3 c-21 a^2 d^4+35 a^2 \left (c^2-d^2\right )^2 \cos ^2(e+f x)-2 \left (5 \left (7 c^3 d-c d^3\right ) a^2-b \left (35 c^4+6 d^2 c^2+7 d^4\right ) a+3 b^2 c d \left (7 c^2-d^2\right )\right ) \cos (e+f x)\right )}{(d+c \cos (e+f x))^{5/2}}dx}{5 c \left (c^2-d^2\right )}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) (a \cos (e+f x)+b)^{5/2}}{7 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{7/2}}-\frac {\frac {2 d \left (-19 a c^2 d+7 a d^3+14 b c^3-2 b c d^2\right ) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}-\frac {\int \frac {\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (35 b^2 c^4-112 a b d c^3+57 a^2 d^2 c^2+25 b^2 d^2 c^2+16 a b d^3 c-21 a^2 d^4+35 a^2 \left (c^2-d^2\right )^2 \sin \left (e+f x+\frac {\pi }{2}\right )^2-2 \left (5 \left (7 c^3 d-c d^3\right ) a^2-b \left (35 c^4+6 d^2 c^2+7 d^4\right ) a+3 b^2 c d \left (7 c^2-d^2\right )\right ) \sin \left (e+f x+\frac {\pi }{2}\right )\right )}{\left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{5 c \left (c^2-d^2\right )}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) (a \cos (e+f x)+b)^{5/2}}{7 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{7/2}}-\frac {\frac {2 d \left (-19 a c^2 d+7 a d^3+14 b c^3-2 b c d^2\right ) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}-\frac {\frac {2 \int -\frac {-105 \left (c^2-d^2\right )^3 \cos ^2(e+f x) a^3-\left (35 d^6-101 c^2 d^4+162 c^4 d^2\right ) a^3+b c d \left (497 c^4-73 d^2 c^2+56 d^4\right ) a^2-b^2 c^2 \left (245 c^4+439 d^2 c^2-12 d^4\right ) a+3 b^3 c^3 d \left (77 c^2+19 d^2\right )-\left (-\left (\left (315 d c^5-69 d^3 c^3+42 d^5 c\right ) a^3\right )+b \left (315 c^6+219 d^2 c^4+208 d^4 c^2-70 d^6\right ) a^2-b^2 c d \left (413 c^4+53 d^2 c^2+14 d^4\right ) a+b^3 c^2 \left (35 c^4+67 d^2 c^2-6 d^4\right )\right ) \cos (e+f x)}{2 \sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{3 c \left (c^2-d^2\right )}-\frac {2 \left (-a^2 d^2 \left (162 c^4-101 c^2 d^2+35 d^4\right )+2 a b c d \left (91 c^4-2 c^2 d^2+7 d^4\right )-\left (b^2 \left (35 c^6+67 c^4 d^2-6 c^2 d^4\right )\right )\right ) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}}{5 c \left (c^2-d^2\right )}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) (a \cos (e+f x)+b)^{5/2}}{7 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{7/2}}-\frac {\frac {2 d \left (-19 a c^2 d+7 a d^3+14 b c^3-2 b c d^2\right ) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}-\frac {-\frac {\int \frac {-105 \left (c^2-d^2\right )^3 \cos ^2(e+f x) a^3-\left (35 d^6-101 c^2 d^4+162 c^4 d^2\right ) a^3+b c d \left (497 c^4-73 d^2 c^2+56 d^4\right ) a^2-b^2 c^2 \left (245 c^4+439 d^2 c^2-12 d^4\right ) a+3 b^3 c^3 d \left (77 c^2+19 d^2\right )-\left (-\left (\left (315 d c^5-69 d^3 c^3+42 d^5 c\right ) a^3\right )+b \left (315 c^6+219 d^2 c^4+208 d^4 c^2-70 d^6\right ) a^2-b^2 c d \left (413 c^4+53 d^2 c^2+14 d^4\right ) a+b^3 c^2 \left (35 c^4+67 d^2 c^2-6 d^4\right )\right ) \cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{3 c \left (c^2-d^2\right )}-\frac {2 \left (-a^2 d^2 \left (162 c^4-101 c^2 d^2+35 d^4\right )+2 a b c d \left (91 c^4-2 c^2 d^2+7 d^4\right )-\left (b^2 \left (35 c^6+67 c^4 d^2-6 c^2 d^4\right )\right )\right ) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}}{5 c \left (c^2-d^2\right )}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) (a \cos (e+f x)+b)^{5/2}}{7 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{7/2}}-\frac {\frac {2 d \left (-19 a c^2 d+7 a d^3+14 b c^3-2 b c d^2\right ) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}-\frac {-\frac {\int \frac {-105 \left (c^2-d^2\right )^3 \sin \left (e+f x+\frac {\pi }{2}\right )^2 a^3-\left (35 d^6-101 c^2 d^4+162 c^4 d^2\right ) a^3+b c d \left (497 c^4-73 d^2 c^2+56 d^4\right ) a^2-b^2 c^2 \left (245 c^4+439 d^2 c^2-12 d^4\right ) a+3 b^3 c^3 d \left (77 c^2+19 d^2\right )+\left (\left (315 d c^5-69 d^3 c^3+42 d^5 c\right ) a^3-b \left (315 c^6+219 d^2 c^4+208 d^4 c^2-70 d^6\right ) a^2+b^2 c d \left (413 c^4+53 d^2 c^2+14 d^4\right ) a-b^3 c^2 \left (35 c^4+67 d^2 c^2-6 d^4\right )\right ) \sin \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 c \left (c^2-d^2\right )}-\frac {2 \left (-a^2 d^2 \left (162 c^4-101 c^2 d^2+35 d^4\right )+2 a b c d \left (91 c^4-2 c^2 d^2+7 d^4\right )-\left (b^2 \left (35 c^6+67 c^4 d^2-6 c^2 d^4\right )\right )\right ) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}}{5 c \left (c^2-d^2\right )}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) (a \cos (e+f x)+b)^{5/2}}{7 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{7/2}}-\frac {\frac {2 d \left (-19 a c^2 d+7 a d^3+14 b c^3-2 b c d^2\right ) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}-\frac {-\frac {\frac {\int \frac {3 b^3 d \left (77 c^2+19 d^2\right ) c^5-a b^2 \left (245 c^4+439 d^2 c^2-12 d^4\right ) c^4+a^2 b d \left (497 c^4-73 d^2 c^2+56 d^4\right ) c^3-\left (-3 \left (-70 d^7+224 c^2 d^5-233 c^4 d^3+175 c^6 d\right ) a^3+b c \left (315 c^6+219 d^2 c^4+208 d^4 c^2-70 d^6\right ) a^2-b^2 c^2 d \left (413 c^4+53 d^2 c^2+14 d^4\right ) a+b^3 c^3 \left (35 c^4+67 d^2 c^2-6 d^4\right )\right ) \cos (e+f x) c-a^3 \left (105 d^8-280 c^2 d^6+214 c^4 d^4+57 c^6 d^2\right )}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c^2}-\frac {105 a^3 \left (c^2-d^2\right )^3 \int \frac {\sqrt {d+c \cos (e+f x)}}{\sqrt {b+a \cos (e+f x)}}dx}{c^2}}{3 c \left (c^2-d^2\right )}-\frac {2 \left (-a^2 d^2 \left (162 c^4-101 c^2 d^2+35 d^4\right )+2 a b c d \left (91 c^4-2 c^2 d^2+7 d^4\right )-\left (b^2 \left (35 c^6+67 c^4 d^2-6 c^2 d^4\right )\right )\right ) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}}{5 c \left (c^2-d^2\right )}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) (a \cos (e+f x)+b)^{5/2}}{7 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{7/2}}-\frac {\frac {2 d \left (-19 a c^2 d+7 a d^3+14 b c^3-2 b c d^2\right ) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}-\frac {-\frac {\frac {\int \frac {3 b^3 d \left (77 c^2+19 d^2\right ) c^5-a b^2 \left (245 c^4+439 d^2 c^2-12 d^4\right ) c^4+a^2 b d \left (497 c^4-73 d^2 c^2+56 d^4\right ) c^3-\left (-3 \left (-70 d^7+224 c^2 d^5-233 c^4 d^3+175 c^6 d\right ) a^3+b c \left (315 c^6+219 d^2 c^4+208 d^4 c^2-70 d^6\right ) a^2-b^2 c^2 d \left (413 c^4+53 d^2 c^2+14 d^4\right ) a+b^3 c^3 \left (35 c^4+67 d^2 c^2-6 d^4\right )\right ) \sin \left (e+f x+\frac {\pi }{2}\right ) c-a^3 \left (105 d^8-280 c^2 d^6+214 c^4 d^4+57 c^6 d^2\right )}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}-\frac {105 a^3 \left (c^2-d^2\right )^3 \int \frac {\sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c^2}}{3 c \left (c^2-d^2\right )}-\frac {2 \left (-a^2 d^2 \left (162 c^4-101 c^2 d^2+35 d^4\right )+2 a b c d \left (91 c^4-2 c^2 d^2+7 d^4\right )-\left (b^2 \left (35 c^6+67 c^4 d^2-6 c^2 d^4\right )\right )\right ) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}}{5 c \left (c^2-d^2\right )}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {2 d^2 (b+a \cos (e+f x))^{5/2} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{7/2}}-\frac {\frac {2 d \left (14 b c^3-19 a d c^2-2 b d^2 c+7 a d^3\right ) (b+a \cos (e+f x))^{3/2} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{5/2}}-\frac {-\frac {2 \left (-\left (\left (35 c^6+67 d^2 c^4-6 d^4 c^2\right ) b^2\right )+2 a c d \left (91 c^4-2 d^2 c^2+7 d^4\right ) b-a^2 d^2 \left (162 c^4-101 d^2 c^2+35 d^4\right )\right ) \sqrt {b+a \cos (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{3/2}}-\frac {\frac {210 a^2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \left (c^2-d^2\right )^3}{c^2 \sqrt {c+d} f}+\frac {\int \frac {3 b^3 d \left (77 c^2+19 d^2\right ) c^5-a b^2 \left (245 c^4+439 d^2 c^2-12 d^4\right ) c^4+a^2 b d \left (497 c^4-73 d^2 c^2+56 d^4\right ) c^3-\left (-3 \left (-70 d^7+224 c^2 d^5-233 c^4 d^3+175 c^6 d\right ) a^3+b c \left (315 c^6+219 d^2 c^4+208 d^4 c^2-70 d^6\right ) a^2-b^2 c^2 d \left (413 c^4+53 d^2 c^2+14 d^4\right ) a+b^3 c^3 \left (35 c^4+67 d^2 c^2-6 d^4\right )\right ) \sin \left (e+f x+\frac {\pi }{2}\right ) c-a^3 \left (105 d^8-280 c^2 d^6+214 c^4 d^4+57 c^6 d^2\right )}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}}{3 c \left (c^2-d^2\right )}}{5 c \left (c^2-d^2\right )}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {2 d^2 (b+a \cos (e+f x))^{5/2} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{7/2}}-\frac {\frac {2 d \left (14 b c^3-19 a d c^2-2 b d^2 c+7 a d^3\right ) (b+a \cos (e+f x))^{3/2} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{5/2}}-\frac {-\frac {2 \left (-\left (\left (35 c^6+67 d^2 c^4-6 d^4 c^2\right ) b^2\right )+2 a c d \left (91 c^4-2 d^2 c^2+7 d^4\right ) b-a^2 d^2 \left (162 c^4-101 d^2 c^2+35 d^4\right )\right ) \sqrt {b+a \cos (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{3/2}}-\frac {\frac {210 a^2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \left (c^2-d^2\right )^3}{c^2 \sqrt {c+d} f}+\frac {\frac {c \left (-\left (\left (-105 d^8+392 c^2 d^6-485 c^4 d^4+582 c^6 d^2\right ) a^3\right )+2 b c d \left (406 c^6+73 d^2 c^4+132 d^4 c^2-35 d^6\right ) a^2-b^2 c^2 \left (245 c^6+852 d^2 c^4+41 d^4 c^2+14 d^6\right ) a+2 b^3 c^3 d \left (133 c^4+62 d^2 c^2-3 d^4\right )\right ) \int \frac {\cos (e+f x)+1}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c-d}-\frac {\left (b^3 \left (35 c^4+231 d c^3+67 d^2 c^2+57 d^3 c-6 d^4\right ) c^4-a b^2 \left (245 c^5+413 d c^4+439 d^2 c^3+53 d^3 c^2-12 d^4 c+14 d^5\right ) c^3+a^2 b \left (315 c^6+497 d c^5+219 d^2 c^4-73 d^3 c^3+208 d^4 c^2+56 d^5 c-70 d^6\right ) c^2-a^3 d \left (525 c^7+57 d c^6-699 d^2 c^5+214 d^3 c^4+672 d^4 c^3-280 d^5 c^2-210 d^6 c+105 d^7\right )\right ) \int \frac {1}{\sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}dx}{c-d}}{c^2}}{3 c \left (c^2-d^2\right )}}{5 c \left (c^2-d^2\right )}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {2 d^2 (b+a \cos (e+f x))^{5/2} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{7/2}}-\frac {\frac {2 d \left (14 b c^3-19 a d c^2-2 b d^2 c+7 a d^3\right ) (b+a \cos (e+f x))^{3/2} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{5/2}}-\frac {-\frac {2 \left (-\left (\left (35 c^6+67 d^2 c^4-6 d^4 c^2\right ) b^2\right )+2 a c d \left (91 c^4-2 d^2 c^2+7 d^4\right ) b-a^2 d^2 \left (162 c^4-101 d^2 c^2+35 d^4\right )\right ) \sqrt {b+a \cos (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{3/2}}-\frac {\frac {210 a^2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \left (c^2-d^2\right )^3}{c^2 \sqrt {c+d} f}+\frac {\frac {c \left (-\left (\left (-105 d^8+392 c^2 d^6-485 c^4 d^4+582 c^6 d^2\right ) a^3\right )+2 b c d \left (406 c^6+73 d^2 c^4+132 d^4 c^2-35 d^6\right ) a^2-b^2 c^2 \left (245 c^6+852 d^2 c^4+41 d^4 c^2+14 d^6\right ) a+2 b^3 c^3 d \left (133 c^4+62 d^2 c^2-3 d^4\right )\right ) \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )+1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c-d}-\frac {\left (b^3 \left (35 c^4+231 d c^3+67 d^2 c^2+57 d^3 c-6 d^4\right ) c^4-a b^2 \left (245 c^5+413 d c^4+439 d^2 c^3+53 d^3 c^2-12 d^4 c+14 d^5\right ) c^3+a^2 b \left (315 c^6+497 d c^5+219 d^2 c^4-73 d^3 c^3+208 d^4 c^2+56 d^5 c-70 d^6\right ) c^2-a^3 d \left (525 c^7+57 d c^6-699 d^2 c^5+214 d^3 c^4+672 d^4 c^3-280 d^5 c^2-210 d^6 c+105 d^7\right )\right ) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c-d}}{c^2}}{3 c \left (c^2-d^2\right )}}{5 c \left (c^2-d^2\right )}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {2 d^2 (b+a \cos (e+f x))^{5/2} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{7/2}}-\frac {\frac {2 d \left (14 b c^3-19 a d c^2-2 b d^2 c+7 a d^3\right ) (b+a \cos (e+f x))^{3/2} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{5/2}}-\frac {-\frac {2 \left (-\left (\left (35 c^6+67 d^2 c^4-6 d^4 c^2\right ) b^2\right )+2 a c d \left (91 c^4-2 d^2 c^2+7 d^4\right ) b-a^2 d^2 \left (162 c^4-101 d^2 c^2+35 d^4\right )\right ) \sqrt {b+a \cos (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{3/2}}-\frac {\frac {210 a^2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \left (c^2-d^2\right )^3}{c^2 \sqrt {c+d} f}+\frac {\frac {c \left (-\left (\left (-105 d^8+392 c^2 d^6-485 c^4 d^4+582 c^6 d^2\right ) a^3\right )+2 b c d \left (406 c^6+73 d^2 c^4+132 d^4 c^2-35 d^6\right ) a^2-b^2 c^2 \left (245 c^6+852 d^2 c^4+41 d^4 c^2+14 d^6\right ) a+2 b^3 c^3 d \left (133 c^4+62 d^2 c^2-3 d^4\right )\right ) \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )+1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c-d}-\frac {2 \sqrt {a+b} \left (b^3 \left (35 c^4+231 d c^3+67 d^2 c^2+57 d^3 c-6 d^4\right ) c^4-a b^2 \left (245 c^5+413 d c^4+439 d^2 c^3+53 d^3 c^2-12 d^4 c+14 d^5\right ) c^3+a^2 b \left (315 c^6+497 d c^5+219 d^2 c^4-73 d^3 c^3+208 d^4 c^2+56 d^5 c-70 d^6\right ) c^2-a^3 d \left (525 c^7+57 d c^6-699 d^2 c^5+214 d^3 c^4+672 d^4 c^3-280 d^5 c^2-210 d^6 c+105 d^7\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{(c-d) \sqrt {c+d} (b c-a d) f}}{c^2}}{3 c \left (c^2-d^2\right )}}{5 c \left (c^2-d^2\right )}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {2 d^2 (b+a \cos (e+f x))^{5/2} \sin (e+f x)}{7 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{7/2}}-\frac {\frac {2 d \left (14 b c^3-19 a d c^2-2 b d^2 c+7 a d^3\right ) (b+a \cos (e+f x))^{3/2} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{5/2}}-\frac {-\frac {\frac {210 a^2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \left (c^2-d^2\right )^3}{c^2 \sqrt {c+d} f}+\frac {-\frac {2 (a-b) \sqrt {a+b} c \left (-\left (\left (-105 d^8+392 c^2 d^6-485 c^4 d^4+582 c^6 d^2\right ) a^3\right )+2 b c d \left (406 c^6+73 d^2 c^4+132 d^4 c^2-35 d^6\right ) a^2-b^2 c^2 \left (245 c^6+852 d^2 c^4+41 d^4 c^2+14 d^6\right ) a+2 b^3 c^3 d \left (133 c^4+62 d^2 c^2-3 d^4\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{(c-d) \sqrt {c+d} (b c-a d)^2 f}-\frac {2 \sqrt {a+b} \left (b^3 \left (35 c^4+231 d c^3+67 d^2 c^2+57 d^3 c-6 d^4\right ) c^4-a b^2 \left (245 c^5+413 d c^4+439 d^2 c^3+53 d^3 c^2-12 d^4 c+14 d^5\right ) c^3+a^2 b \left (315 c^6+497 d c^5+219 d^2 c^4-73 d^3 c^3+208 d^4 c^2+56 d^5 c-70 d^6\right ) c^2-a^3 d \left (525 c^7+57 d c^6-699 d^2 c^5+214 d^3 c^4+672 d^4 c^3-280 d^5 c^2-210 d^6 c+105 d^7\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{(c-d) \sqrt {c+d} (b c-a d) f}}{c^2}}{3 c \left (c^2-d^2\right )}-\frac {2 \left (-\left (\left (35 c^6+67 d^2 c^4-6 d^4 c^2\right ) b^2\right )+2 a c d \left (91 c^4-2 d^2 c^2+7 d^4\right ) b-a^2 d^2 \left (162 c^4-101 d^2 c^2+35 d^4\right )\right ) \sqrt {b+a \cos (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{3/2}}}{5 c \left (c^2-d^2\right )}}{7 c \left (c^2-d^2\right )}\right )}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\) |
(Sqrt[d + c*Cos[e + f*x]]*Sqrt[a + b*Sec[e + f*x]]*((2*d^2*(b + a*Cos[e + f*x])^(5/2)*Sin[e + f*x])/(7*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^(7/2)) - ((2*d*(14*b*c^3 - 19*a*c^2*d - 2*b*c*d^2 + 7*a*d^3)*(b + a*Cos[e + f*x])^ (3/2)*Sin[e + f*x])/(5*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^(5/2)) - (-1/3 *(((-2*(a - b)*Sqrt[a + b]*c*(2*b^3*c^3*d*(133*c^4 + 62*c^2*d^2 - 3*d^4) + 2*a^2*b*c*d*(406*c^6 + 73*c^4*d^2 + 132*c^2*d^4 - 35*d^6) - a*b^2*c^2*(24 5*c^6 + 852*c^4*d^2 + 41*c^2*d^4 + 14*d^6) - a^3*(582*c^6*d^2 - 485*c^4*d^ 4 + 392*c^2*d^6 - 105*d^8))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b )*(d + c*Cos[e + f*x])))]*(d + c*Cos[e + f*x])*Csc[e + f*x]*EllipticE[ArcS in[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sqrt[a + b]*Sqrt[d + c*Cos[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))])/((c - d)*Sqrt[c + d]*(b*c - a*d)^2*f) - (2*Sqrt[a + b]*(b^3*c^4*(35*c^4 + 231*c^3*d + 67*c^2*d^2 + 57 *c*d^3 - 6*d^4) - a*b^2*c^3*(245*c^5 + 413*c^4*d + 439*c^3*d^2 + 53*c^2*d^ 3 - 12*c*d^4 + 14*d^5) + a^2*b*c^2*(315*c^6 + 497*c^5*d + 219*c^4*d^2 - 73 *c^3*d^3 + 208*c^2*d^4 + 56*c*d^5 - 70*d^6) - a^3*d*(525*c^7 + 57*c^6*d - 699*c^5*d^2 + 214*c^4*d^3 + 672*c^3*d^4 - 280*c^2*d^5 - 210*c*d^6 + 105*d^ 7))*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])) )]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Cos[e + f*x]))) ]*(d + c*Cos[e + f*x])*Csc[e + f*x]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[...
3.3.16.3.1 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[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[2*((a + b*Sin[e + f*x])/(d*f*Rt[(a + b)/ (c + d), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticPi[b*((c + d)/(d*(a + b))), ArcSin[Rt[(a + b)/( c + d), 2]*(Sqrt[c + d*Sin[e + f*x]]/Sqrt[a + b*Sin[e + f*x]])], (a - b)*(( c + d)/((a + b)*(c - d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(a + b)/(c + d)]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_ .) + (f_.)*(x_)]]), x_Symbol] :> Simp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d )*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e + f*x] )/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/ ((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(S qrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c - d)/((a - b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && N eQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.) *(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim p[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2 ]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]] /Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; Fr eeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e , f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && NeQ[A, B]
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_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(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, 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_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2 Int[Sqrt[a + b*Sin[e + f*x]] /Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2 Int[(A*b^2 - a^2*C + b*(b* B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*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]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d _.) + (c_))^(n_), x_Symbol] :> Simp[Sqrt[d + c*Sin[e + f*x]]*(Sqrt[a + b*Cs c[e + f*x]]/(Sqrt[b + a*Sin[e + f*x]]*Sqrt[c + d*Csc[e + f*x]])) Int[(b + a*Sin[e + f*x])^m*((d + c*Sin[e + f*x])^n/Sin[e + f*x]^(m + n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m + 1/ 2] && IntegerQ[n + 1/2] && LeQ[-2, m + n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(88655\) vs. \(2(1337)=2674\).
Time = 23.70 (sec) , antiderivative size = 88656, normalized size of antiderivative = 62.08
Timed out. \[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{9/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{9/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{9/2}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]
\[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{9/2}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{9/2}} \, dx=\text {Hanged} \]