Integrand size = 29, antiderivative size = 1150 \[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{7/2}} \, dx=\frac {2 (a-b) \sqrt {a+b} \left (b^2 c^2 d \left (29 c^2+3 d^2\right )-a b c \left (35 c^4+34 c^2 d^2-5 d^4\right )+a^2 \left (58 c^4 d-41 c^2 d^3+15 d^5\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)}}{15 c^3 (c-d)^3 (c+d)^{5/2} (b c-a d) 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 (5 c^2+24 c d+3 d^2\right )-a b^2 c^3 \left (35 c^3+42 c^2 d+21 c d^2-2 d^3\right )+a^2 b c^2 \left (45 c^4+48 c^3 d+c^2 d^2-8 c d^3+10 d^4\right )-a^3 d \left (60 c^5-2 c^4 d-66 c^3 d^2+25 c^2 d^3+30 c d^4-15 d^5\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)}}{15 c^4 (c-d)^3 (c+d)^{5/2} (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^4 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 d (b c-a d) (b+a \cos (e+f x)) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2 \sqrt {c+d \sec (e+f x)}}+\frac {2 (b c-a d) \left (5 b c^3-13 a c^2 d+3 b c d^2+5 a d^3\right ) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{15 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}} \]
-2/5*d*(-a*d+b*c)*(b+a*cos(f*x+e))*sin(f*x+e)*(a+b*sec(f*x+e))^(1/2)/c/(c^ 2-d^2)/f/(d+c*cos(f*x+e))^2/(c+d*sec(f*x+e))^(1/2)+2/15*(-a*d+b*c)*(-13*a* c^2*d+5*a*d^3+5*b*c^3+3*b*c*d^2)*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))/(c+d*sec(f*x+e))^(1/2)+2/15*(a-b)*(b^2*c^2*d*( 29*c^2+3*d^2)-a*b*c*(35*c^4+34*c^2*d^2-5*d^4)+a^2*(58*c^4*d-41*c^2*d^3+15* d^5))*(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^3/(c-d)^3/(c+d)^(5/2)/(-a*d+b*c)/f/(b+a*cos(f*x+e))^(1/2)/(c+d*s ec(f*x+e))^(1/2)+2/15*(b^3*c^4*(5*c^2+24*c*d+3*d^2)-a*b^2*c^3*(35*c^3+42*c ^2*d+21*c*d^2-2*d^3)+a^2*b*c^2*(45*c^4+48*c^3*d+c^2*d^2-8*c*d^3+10*d^4)-a^ 3*d*(60*c^5-2*c^4*d-66*c^3*d^2+25*c^2*d^3+30*c*d^4-15*d^5))*(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))^(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)^3/ (c+d)^(5/2)/(-a*d+b*c)/f/(b+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2)-2*a ^2*(d+c*cos(f*x+e))^(3/2)*csc(f*x+e)*EllipticPi((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/a/(c+d),((a+b)*(c-d...
Leaf count is larger than twice the leaf count of optimal. \(2344\) vs. \(2(1150)=2300\).
Time = 8.15 (sec) , antiderivative size = 2344, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{7/2}} \, dx=\text {Result too large to show} \]
((d + c*Cos[e + f*x])^4*Sec[e + f*x]*(a + b*Sec[e + f*x])^(5/2)*((-2*(b^2* c^2*d*Sin[e + f*x] - 2*a*b*c*d^2*Sin[e + f*x] + a^2*d^3*Sin[e + f*x]))/(5* c^2*(c^2 - d^2)*(d + c*Cos[e + f*x])^3) + (2*(5*b^2*c^4*Sin[e + f*x] - 21* a*b*c^3*d*Sin[e + f*x] + 16*a^2*c^2*d^2*Sin[e + f*x] + 3*b^2*c^2*d^2*Sin[e + f*x] + 5*a*b*c*d^3*Sin[e + f*x] - 8*a^2*d^4*Sin[e + f*x]))/(15*c^2*(c^2 - d^2)^2*(d + c*Cos[e + f*x])^2) + (2*(35*a*b*c^5*Sin[e + f*x] - 58*a^2*c ^4*d*Sin[e + f*x] - 29*b^2*c^4*d*Sin[e + f*x] + 34*a*b*c^3*d^2*Sin[e + f*x ] + 41*a^2*c^2*d^3*Sin[e + f*x] - 3*b^2*c^2*d^3*Sin[e + f*x] - 5*a*b*c*d^4 *Sin[e + f*x] - 15*a^2*d^5*Sin[e + f*x]))/(15*c^2*(c^2 - d^2)^3*(d + c*Cos [e + f*x]))))/(f*(b + a*Cos[e + f*x])^2*(c + d*Sec[e + f*x])^(7/2)) + ((d + c*Cos[e + f*x])^(7/2)*Sec[e + f*x]*(a + b*Sec[e + f*x])^(5/2)*((4*(b*c - a*d)*(10*a^2*b*c^5 + 5*b^3*c^5 + 13*a^3*c^4*d - 48*a*b^2*c^4*d + 15*a^2*b *c^3*d^2 + 27*b^3*c^3*d^2 - 18*a^3*c^2*d^3 - 16*a*b^2*c^2*d^3 + 7*a^2*b*c* d^4 + 5*a^3*d^5)*Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)* (b + a*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticF[Ar cSin[Sqrt[((-a - b)*(d + c*Cos[e + f*x])*Csc[(e + f*x)/2]^2)/(b*c - a*d)]/ Sqrt[2]], (2*(b*c - a*d))/((a + b)*(c - d))]*Sin[(e + f*x)/2]^4)/((a + b)* (c + d)*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]]) + 4*(b*c - a*d) *(15*a^3*c^5 - 35*a*b^2*c^5 + 23*a^2*b*c^4*d + 29*b^3*c^4*d + 13*a^3*c^...
Time = 4.33 (sec) , antiderivative size = 1086, normalized size of antiderivative = 0.94, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.552, Rules used = {3042, 4430, 3042, 3468, 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.
| \(\displaystyle \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}{\left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4430 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {\cos (e+f x) (b+a \cos (e+f x))^{5/2}}{(d+c \cos (e+f x))^{7/2}}dx}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right ) \left (b+a \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}{\left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{7/2}}dx}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 \int \frac {\sqrt {b+a \cos (e+f x)} \left (5 a^2 \left (c^2-d^2\right ) \cos ^2(e+f x)-\left (5 c d a^2-2 b \left (5 c^2-d^2\right ) a+3 b^2 c d\right ) \cos (e+f x)+(5 b c-3 a d) (b c-a d)\right )}{2 (d+c \cos (e+f x))^{5/2}}dx}{5 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\int \frac {\sqrt {b+a \cos (e+f x)} \left (5 a^2 \left (c^2-d^2\right ) \cos ^2(e+f x)-\left (5 c d a^2-2 b \left (5 c^2-d^2\right ) a+3 b^2 c d\right ) \cos (e+f x)+(5 b c-3 a d) (b c-a d)\right )}{(d+c \cos (e+f x))^{5/2}}dx}{5 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\int \frac {\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (5 a^2 \left (c^2-d^2\right ) \sin \left (e+f x+\frac {\pi }{2}\right )^2+\left (-5 c d a^2+2 b \left (5 c^2-d^2\right ) a-3 b^2 c d\right ) \sin \left (e+f x+\frac {\pi }{2}\right )+(5 b c-3 a d) (b c-a d)\right )}{\left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{5 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {2 \int -\frac {-15 \left (c^2-d^2\right )^2 \cos ^2(e+f x) a^3+(b c-a d) \left (\left (13 c^2 d-5 d^3\right ) a^2-b \left (35 c^3-3 c d^2\right ) a+24 b^2 c^2 d\right )+\left (6 c d \left (5 c^2-d^2\right ) a^3-b \left (45 c^4+d^2 c^2+10 d^4\right ) a^2+2 b^2 c d \left (21 c^2-d^2\right ) a-b^3 c^2 \left (5 c^2+3 d^2\right )\right ) \cos (e+f x)}{2 \sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{3 c \left (c^2-d^2\right )}+\frac {2 (b c-a d) \left (-13 a c^2 d+5 a d^3+5 b c^3+3 b c d^2\right ) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}}{5 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {2 (b c-a d) \left (-13 a c^2 d+5 a d^3+5 b c^3+3 b c d^2\right ) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\int \frac {-15 \left (c^2-d^2\right )^2 \cos ^2(e+f x) a^3+(b c-a d) \left (\left (13 c^2 d-5 d^3\right ) a^2-b \left (35 c^3-3 c d^2\right ) a+24 b^2 c^2 d\right )+\left (6 \left (5 c^3 d-c d^3\right ) a^3-b \left (45 c^4+d^2 c^2+10 d^4\right ) a^2+2 b^2 c d \left (21 c^2-d^2\right ) a-b^3 c^2 \left (5 c^2+3 d^2\right )\right ) \cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{3 c \left (c^2-d^2\right )}}{5 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {2 (b c-a d) \left (-13 a c^2 d+5 a d^3+5 b c^3+3 b c d^2\right ) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\int \frac {-15 \left (c^2-d^2\right )^2 \sin \left (e+f x+\frac {\pi }{2}\right )^2 a^3+(b c-a d) \left (\left (13 c^2 d-5 d^3\right ) a^2-b \left (35 c^3-3 c d^2\right ) a+24 b^2 c^2 d\right )+\left (6 \left (5 c^3 d-c d^3\right ) a^3-b \left (45 c^4+d^2 c^2+10 d^4\right ) a^2+2 b^2 c d \left (21 c^2-d^2\right ) a-b^3 c^2 \left (5 c^2+3 d^2\right )\right ) \sin \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 c \left (c^2-d^2\right )}}{5 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {2 (b c-a d) \left (-13 a c^2 d+5 a d^3+5 b c^3+3 b c d^2\right ) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\frac {\int \frac {15 \left (c^2 d-d^3\right )^2 a^3+c^2 (b c-a d) \left (\left (13 c^2 d-5 d^3\right ) a^2-b \left (35 c^3-3 c d^2\right ) a+24 b^2 c^2 d\right )+c \left (6 \left (5 d^5-11 c^2 d^3+10 c^4 d\right ) a^3-b c \left (45 c^4+d^2 c^2+10 d^4\right ) a^2+2 b^2 c^2 d \left (21 c^2-d^2\right ) a-b^3 c^3 \left (5 c^2+3 d^2\right )\right ) \cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c^2}-\frac {15 a^3 \left (c^2-d^2\right )^2 \int \frac {\sqrt {d+c \cos (e+f x)}}{\sqrt {b+a \cos (e+f x)}}dx}{c^2}}{3 c \left (c^2-d^2\right )}}{5 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {2 (b c-a d) \left (-13 a c^2 d+5 a d^3+5 b c^3+3 b c d^2\right ) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\frac {\int \frac {15 \left (c^2 d-d^3\right )^2 a^3+c^2 (b c-a d) \left (\left (13 c^2 d-5 d^3\right ) a^2-b \left (35 c^3-3 c d^2\right ) a+24 b^2 c^2 d\right )+c \left (6 \left (5 d^5-11 c^2 d^3+10 c^4 d\right ) a^3-b c \left (45 c^4+d^2 c^2+10 d^4\right ) a^2+2 b^2 c^2 d \left (21 c^2-d^2\right ) a-b^3 c^3 \left (5 c^2+3 d^2\right )\right ) \sin \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}-\frac {15 a^3 \left (c^2-d^2\right )^2 \int \frac {\sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c^2}}{3 c \left (c^2-d^2\right )}}{5 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {2 (b c-a d) \left (-13 a c^2 d+5 a d^3+5 b c^3+3 b c d^2\right ) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\frac {\int \frac {15 \left (c^2 d-d^3\right )^2 a^3+c^2 (b c-a d) \left (\left (13 c^2 d-5 d^3\right ) a^2-b \left (35 c^3-3 c d^2\right ) a+24 b^2 c^2 d\right )+c \left (6 \left (5 d^5-11 c^2 d^3+10 c^4 d\right ) a^3-b c \left (45 c^4+d^2 c^2+10 d^4\right ) a^2+2 b^2 c^2 d \left (21 c^2-d^2\right ) a-b^3 c^3 \left (5 c^2+3 d^2\right )\right ) \sin \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}+\frac {30 a^2 \sqrt {a+b} \left (c^2-d^2\right )^2 \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}}{3 c \left (c^2-d^2\right )}}{5 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {2 (b c-a d) \left (-13 a c^2 d+5 a d^3+5 b c^3+3 b c d^2\right ) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\frac {-\frac {c (b c-a d) \left (-58 a^2 c^4 d+41 a^2 c^2 d^3-15 a^2 d^5+35 a b c^5+34 a b c^3 d^2-5 a b c d^4-29 b^2 c^4 d-3 b^2 c^2 d^3\right ) \int \frac {\cos (e+f x)+1}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c-d}-\frac {\left (-a^3 d \left (60 c^5-2 c^4 d-66 c^3 d^2+25 c^2 d^3+30 c d^4-15 d^5\right )+a^2 b c^2 \left (45 c^4+48 c^3 d+c^2 d^2-8 c d^3+10 d^4\right )-a b^2 c^3 \left (35 c^3+42 c^2 d+21 c d^2-2 d^3\right )+b^3 c^4 \left (5 c^2+24 c d+3 d^2\right )\right ) \int \frac {1}{\sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}dx}{c-d}}{c^2}+\frac {30 a^2 \sqrt {a+b} \left (c^2-d^2\right )^2 \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}}{3 c \left (c^2-d^2\right )}}{5 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) (a \cos (e+f x)+b)^{3/2}}{5 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{5/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {\frac {2 (b c-a d) \left (5 b c^3-13 a d c^2+3 b d^2 c+5 a d^3\right ) \sqrt {b+a \cos (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{3/2}}-\frac {\frac {30 a^2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \left (c^2-d^2\right )^2}{c^2 \sqrt {c+d} f}+\frac {-\frac {c (b c-a d) \left (35 a b c^5-58 a^2 d c^4-29 b^2 d c^4+34 a b d^2 c^3+41 a^2 d^3 c^2-3 b^2 d^3 c^2-5 a b d^4 c-15 a^2 d^5\right ) \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )+1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c-d}-\frac {\left (b^3 \left (5 c^2+24 d c+3 d^2\right ) c^4-a b^2 \left (35 c^3+42 d c^2+21 d^2 c-2 d^3\right ) c^3+a^2 b \left (45 c^4+48 d c^3+d^2 c^2-8 d^3 c+10 d^4\right ) c^2-a^3 d \left (60 c^5-2 d c^4-66 d^2 c^3+25 d^3 c^2+30 d^4 c-15 d^5\right )\right ) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c-d}}{c^2}}{3 c \left (c^2-d^2\right )}}{5 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) (b+a \cos (e+f x))^{3/2} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{5/2}}\right )}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {\frac {2 (b c-a d) \left (5 b c^3-13 a d c^2+3 b d^2 c+5 a d^3\right ) \sqrt {b+a \cos (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{3/2}}-\frac {\frac {30 a^2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \left (c^2-d^2\right )^2}{c^2 \sqrt {c+d} f}+\frac {-\frac {2 \sqrt {a+b} \left (b^3 \left (5 c^2+24 d c+3 d^2\right ) c^4-a b^2 \left (35 c^3+42 d c^2+21 d^2 c-2 d^3\right ) c^3+a^2 b \left (45 c^4+48 d c^3+d^2 c^2-8 d^3 c+10 d^4\right ) c^2-a^3 d \left (60 c^5-2 d c^4-66 d^2 c^3+25 d^3 c^2+30 d^4 c-15 d^5\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{(c-d) \sqrt {c+d} (b c-a d) f}-\frac {c (b c-a d) \left (35 a b c^5-58 a^2 d c^4-29 b^2 d c^4+34 a b d^2 c^3+41 a^2 d^3 c^2-3 b^2 d^3 c^2-5 a b d^4 c-15 a^2 d^5\right ) \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )+1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c-d}}{c^2}}{3 c \left (c^2-d^2\right )}}{5 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) (b+a \cos (e+f x))^{3/2} \sin (e+f x)}{5 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{5/2}}\right )}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {\frac {2 (b c-a d) \left (5 b c^3-13 a d c^2+3 b d^2 c+5 a d^3\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 {30 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 )^2}{c^2 \sqrt {c+d} f}+\frac {\frac {2 (a-b) \sqrt {a+b} c \left (35 a b c^5-58 a^2 d c^4-29 b^2 d c^4+34 a b d^2 c^3+41 a^2 d^3 c^2-3 b^2 d^3 c^2-5 a b d^4 c-15 a^2 d^5\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) f}-\frac {2 \sqrt {a+b} \left (b^3 \left (5 c^2+24 d c+3 d^2\right ) c^4-a b^2 \left (35 c^3+42 d c^2+21 d^2 c-2 d^3\right ) c^3+a^2 b \left (45 c^4+48 d c^3+d^2 c^2-8 d^3 c+10 d^4\right ) c^2-a^3 d \left (60 c^5-2 d c^4-66 d^2 c^3+25 d^3 c^2+30 d^4 c-15 d^5\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 )}-\frac {2 d (b c-a d) (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}}\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*(b*c - a*d)*(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*(35*a*b*c^5 - 58*a^2*c^4*d - 29*b^2*c^4*d + 34*a*b*c^3*d^2 + 41*a^2*c^2*d^3 - 3*b^2*c^2*d^3 - 5*a*b*c*d ^4 - 15*a^2*d^5)*Sqrt[-(((b*c - a*d)*(1 - Cos[e + f*x]))/((a + b)*(d + c*C os[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Cos[e + f*x]))/((a - b)*(d + c*Co s[e + f*x])))]*(d + c*Cos[e + f*x])*Csc[e + f*x]*EllipticE[ArcSin[(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)*f) - (2*Sqrt[a + b]*(b^3*c^4*(5*c^2 + 24*c*d + 3*d^2) - a*b^2*c^3*(35*c^3 + 42* c^2*d + 21*c*d^2 - 2*d^3) + a^2*b*c^2*(45*c^4 + 48*c^3*d + c^2*d^2 - 8*c*d ^3 + 10*d^4) - a^3*d*(60*c^5 - 2*c^4*d - 66*c^3*d^2 + 25*c^2*d^3 + 30*c*d^ 4 - 15*d^5))*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[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)*f))/c^2 + (30*a^2*Sqrt[a + b]*(c^2 - d^2)^2*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]*Elli...
3.3.15.3.1 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_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-(b*c - a*d))*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((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 - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a *B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, 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] && GtQ[m, 1] && LtQ[n, -1]
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_)] + (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. \(39099\) vs. \(2(1065)=2130\).
Time = 20.62 (sec) , antiderivative size = 39100, normalized size of antiderivative = 34.00
Timed out. \[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{7/2}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{7/2}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{7/2}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \sec (e+f x))^{5/2}}{(c+d \sec (e+f x))^{7/2}} \, dx=\text {Hanged} \]