Integrand size = 29, antiderivative size = 899 \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\frac {2 (a-b) \sqrt {a+b} d \left (6 b c^3-7 a c^2 d-2 b c d^2+3 a d^3\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{3 c^2 (c-d)^2 (c+d)^{3/2} (b c-a d)^2 f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 \sqrt {a+b} \left (b c^2 \left (3 c^2+3 c d-2 d^2\right )-a d \left (9 c^3-2 c^2 d-6 c d^2+3 d^3\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^3 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}+\frac {2 d^2 \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}} \]
2/3*d^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))/( c+d*sec(f*x+e))^(1/2)+2/3*(a-b)*d*(-7*a*c^2*d+3*a*d^3+6*b*c^3-2*b*c*d^2)*( 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^2/(c-d)^2/(c+d)^(3/2)/(-a*d+b*c)^2/f/(b+a*cos(f*x+e))^(1/2)/(c+d*sec(f* x+e))^(1/2)+2/3*(b*c^2*(3*c^2+3*c*d-2*d^2)-a*d*(9*c^3-2*c^2*d-6*c*d^2+3*d^ 3))*(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^3/(c-d)^2/(c+d)^(3/2)/(-a*d+b*c)/f/(b+a*cos(f*x+e))^(1/2)/(c+d*sec (f*x+e))^(1/2)-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)/(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/f/(c+d)^(1/2)/(b+a*cos(f*x+e))^(1/ 2)/(c+d*sec(f*x+e))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(1990\) vs. \(2(899)=1798\).
Time = 7.33 (sec) , antiderivative size = 1990, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx =\text {Too large to display} \]
((d + c*Cos[e + f*x])^3*Sec[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]]*((2*d^2*Si n[e + f*x])/(3*c*(c^2 - d^2)*(d + c*Cos[e + f*x])^2) - (2*(6*b*c^3*d*Sin[e + f*x] - 7*a*c^2*d^2*Sin[e + f*x] - 2*b*c*d^3*Sin[e + f*x] + 3*a*d^4*Sin[ e + f*x]))/(3*c*(b*c - a*d)*(c^2 - d^2)^2*(d + c*Cos[e + f*x]))))/(f*(c + d*Sec[e + f*x])^(5/2)) + ((d + c*Cos[e + f*x])^(5/2)*Sec[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]]*((4*(b*c - a*d)*(3*b^2*c^4 - 3*a*b*c^3*d - a^2*c^2*d^2 + b^2*c^2*d^2 - a*b*c*d^3 + a^2*d^4)*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[ArcSin[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)*(3*a*b*c^4 - 3*a^2*c^3*d + 6*b^2*c^3*d - 7*a*b*c^2*d^ 2 - a^2*c*d^3 - 2*b^2*c*d^3 + 4*a*b*d^4)*((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[ArcSin[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]]) - (Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sqrt[((c + d)*...
Time = 3.03 (sec) , antiderivative size = 826, normalized size of antiderivative = 0.92, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {3042, 4430, 3042, 3527, 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.
| \(\displaystyle \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}{\left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4430 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {\cos ^2(e+f x) \sqrt {b+a \cos (e+f x)}}{(d+c \cos (e+f x))^{5/2}}dx}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )^2 \sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}{\left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 \int -\frac {-3 a \left (c^2-d^2\right ) \cos ^2(e+f x)-\left (3 b c^2-3 a d c-2 b d^2\right ) \cos (e+f x)+d (3 b c-a d)}{2 \sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{3 c \left (c^2-d^2\right )}+\frac {2 d^2 \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\int \frac {-3 a \left (c^2-d^2\right ) \cos ^2(e+f x)-\left (3 b c^2-3 a d c-2 b d^2\right ) \cos (e+f x)+d (3 b c-a d)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{3 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\int \frac {-3 a \left (c^2-d^2\right ) \sin \left (e+f x+\frac {\pi }{2}\right )^2+\left (-3 b c^2+3 a d c+2 b d^2\right ) \sin \left (e+f x+\frac {\pi }{2}\right )+d (3 b c-a d)}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\frac {\int \frac {d \left (3 b c^3+2 a d c^2-3 a d^3\right )-c (b c-3 a d) \left (3 c^2-2 d^2\right ) \cos (e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c^2}-\frac {3 a \left (c^2-d^2\right ) \int \frac {\sqrt {d+c \cos (e+f x)}}{\sqrt {b+a \cos (e+f x)}}dx}{c^2}}{3 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\frac {\int \frac {d \left (3 b c^3+2 a d c^2-3 a d^3\right )-c (b c-3 a d) \left (3 c^2-2 d^2\right ) \sin \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}-\frac {3 a \left (c^2-d^2\right ) \int \frac {\sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c^2}}{3 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3290 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\frac {\int \frac {d \left (3 b c^3+2 a d c^2-3 a d^3\right )-c (b c-3 a d) \left (3 c^2-2 d^2\right ) \sin \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}+\frac {6 \sqrt {a+b} \left (c^2-d^2\right ) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}}{3 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\frac {\frac {c d \left (-7 a c^2 d+3 a d^3+6 b c^3-2 b c d^2\right ) \int \frac {\cos (e+f x)+1}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c-d}-\frac {\left (c \left (3 c^2-2 d^2\right ) (b c-3 a d)+d \left (2 a c^2 d-3 a d^3+3 b c^3\right )\right ) \int \frac {1}{\sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}dx}{c-d}}{c^2}+\frac {6 \sqrt {a+b} \left (c^2-d^2\right ) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}}{3 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\frac {\frac {c d \left (-7 a c^2 d+3 a d^3+6 b c^3-2 b c d^2\right ) \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )+1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c-d}-\frac {\left (c \left (3 c^2-2 d^2\right ) (b c-3 a d)+d \left (2 a c^2 d-3 a d^3+3 b c^3\right )\right ) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c-d}}{c^2}+\frac {6 \sqrt {a+b} \left (c^2-d^2\right ) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}}{3 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3297 |
| \(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {2 d^2 \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}-\frac {\frac {\frac {c d \left (-7 a c^2 d+3 a d^3+6 b c^3-2 b c d^2\right ) \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )+1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c-d}-\frac {2 \sqrt {a+b} \left (c \left (3 c^2-2 d^2\right ) (b c-3 a d)+d \left (2 a c^2 d-3 a d^3+3 b c^3\right )\right ) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}}{c^2}+\frac {6 \sqrt {a+b} \left (c^2-d^2\right ) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}}{3 c \left (c^2-d^2\right )}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
| \(\Big \downarrow \) 3475 |
| \(\displaystyle \frac {\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {2 d^2 \sqrt {b+a \cos (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{3/2}}-\frac {\frac {-\frac {2 (a-b) \sqrt {a+b} c d \left (6 b c^3-7 a d c^2-2 b d^2 c+3 a d^3\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{(c-d) \sqrt {c+d} (b c-a d)^2 f}-\frac {2 \sqrt {a+b} \left (c (b c-3 a d) \left (3 c^2-2 d^2\right )+d \left (3 b c^3+2 a d c^2-3 a d^3\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{(c-d) \sqrt {c+d} (b c-a d) f}}{c^2}+\frac {6 \sqrt {a+b} \left (c^2-d^2\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 \sqrt {c+d} f}}{3 c \left (c^2-d^2\right )}\right )}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\) |
(Sqrt[d + c*Cos[e + f*x]]*Sqrt[a + b*Sec[e + f*x]]*(-1/3*(((-2*(a - b)*Sqr t[a + b]*c*d*(6*b*c^3 - 7*a*c^2*d - 2*b*c*d^2 + 3*a*d^3)*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[ArcSin[(Sqrt[c + d]*Sqrt[b + a*Cos[e + f*x]])/(Sq rt[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]*(c*(b*c - 3*a*d)* (3*c^2 - 2*d^2) + d*(3*b*c^3 + 2*a*c^2*d - 3*a*d^3))*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])*Cs c[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 + (6*Sqrt[a + b]*(c^2 - d^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*C os[e + f*x])*Csc[e + f*x]*EllipticPi[((a + b)*c)/(a*(c + d)), 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^2*Sqrt[c + d]*f))/(c*(c^2 - d^2) ) + (2*d^2*Sqrt[b + a*Cos[e + f*x]]*Sin[e + f*x])/(3*c*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^(3/2))))/(Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f...
3.3.10.3.1 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_.) + (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. \(18679\) vs. \(2(820)=1640\).
Time = 17.71 (sec) , antiderivative size = 18680, normalized size of antiderivative = 20.78
\[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
integral(sqrt(b*sec(f*x + e) + a)*sqrt(d*sec(f*x + e) + c)/(d^3*sec(f*x + e)^3 + 3*c*d^2*sec(f*x + e)^2 + 3*c^2*d*sec(f*x + e) + c^3), x)
\[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {a + b \sec {\left (e + f x \right )}}}{\left (c + d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {b \sec \left (f x + e\right ) + a}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+b \sec (e+f x)}}{(c+d \sec (e+f x))^{5/2}} \, dx=\text {Hanged} \]