Integrand size = 29, antiderivative size = 622 \[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx=-\frac {2 (a-b) \sqrt {a+b} d^2 \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{c (c-d) \sqrt {c+d} (b c-a d)^2 f}-\frac {2 \sqrt {a+b} (2 c-d) d \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{c^2 (c-d) \sqrt {c+d} (b c-a d) f}-\frac {2 \sqrt {a+b} \cot (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{a c^2 \sqrt {c+d} f} \]
-2*(a-b)*d^2*cot(f*x+e)*EllipticE((c+d)^(1/2)*(a+b*sec(f*x+e))^(1/2)/(a+b) ^(1/2)/(c+d*sec(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(c+d*sec(f* x+e))*(a+b)^(1/2)*((-a*d+b*c)*(1-sec(f*x+e))/(a+b)/(c+d*sec(f*x+e)))^(1/2) *(-(-a*d+b*c)*(1+sec(f*x+e))/(a-b)/(c+d*sec(f*x+e)))^(1/2)/c/(c-d)/(-a*d+b *c)^2/f/(c+d)^(1/2)-2*(2*c-d)*d*cot(f*x+e)*EllipticF((c+d)^(1/2)*(a+b*sec( f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sec(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d)) ^(1/2))*(c+d*sec(f*x+e))*(a+b)^(1/2)*((-a*d+b*c)*(1-sec(f*x+e))/(a+b)/(c+d *sec(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sec(f*x+e))/(a-b)/(c+d*sec(f*x+e)))^(1 /2)/c^2/(c-d)/(-a*d+b*c)/f/(c+d)^(1/2)-2*cot(f*x+e)*EllipticPi((c+d)^(1/2) *(a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2)/(c+d*sec(f*x+e))^(1/2),(a+b)*c/a/(c+d) ,((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*(c+d*sec(f*x+e))*(a+b)^(1/2)*((-a*d+b*c) *(1-sec(f*x+e))/(a+b)/(c+d*sec(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sec(f*x+e))/ (a-b)/(c+d*sec(f*x+e)))^(1/2)/a/c^2/f/(c+d)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(1761\) vs. \(2(622)=1244\).
Time = 19.32 (sec) , antiderivative size = 1761, normalized size of antiderivative = 2.83 \[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx =\text {Too large to display} \]
(Sqrt[b + a*Cos[e + f*x]]*(d + c*Cos[e + f*x])^(3/2)*Sec[e + f*x]^2*((-4*b *c*d*(b*c - a*d)*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) *(b*c^2 - a*c*d - 2*b*d^2)*((Sqrt[((c + d)*Cot[(e + f*x)/2]^2)/(c - d)]*Sq rt[((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)*(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])*C sc[(e + f*x)/2]^2)/(b*c - a*d)]*Csc[e + f*x]*EllipticPi[(b*c - a*d)/((a + b)*c), 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*Sqrt[b + a*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])) - 2*a*d^2*( (Sqrt[(-a + b)/(a + b)]*(a + b)*Cos[(e + f*x)/2]*Sqrt[d + c*Cos[e + f*x...
Time = 2.13 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {3042, 4430, 3042, 3533, 25, 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.
\(\displaystyle \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )} \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4430 |
\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {\cos ^2(e+f x)}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )^2}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 3533 |
\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\int -\frac {d^2+2 c \cos (e+f x) d}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c^2}+\frac {\int \frac {\sqrt {d+c \cos (e+f x)}}{\sqrt {b+a \cos (e+f x)}}dx}{c^2}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\int \frac {\sqrt {d+c \cos (e+f x)}}{\sqrt {b+a \cos (e+f x)}}dx}{c^2}-\frac {\int \frac {d^2+2 c \cos (e+f x) d}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c^2}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\int \frac {\sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c^2}-\frac {\int \frac {d^2+2 c \sin \left (e+f x+\frac {\pi }{2}\right ) d}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 3290 |
\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (-\frac {\int \frac {d^2+2 c \sin \left (e+f x+\frac {\pi }{2}\right ) d}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}-\frac {2 \sqrt {a+b} \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{a c^2 f \sqrt {c+d}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 3477 |
\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (-\frac {\frac {d (2 c-d) \int \frac {1}{\sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}dx}{c-d}-\frac {c d^2 \int \frac {\cos (e+f x)+1}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c-d}}{c^2}-\frac {2 \sqrt {a+b} \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{a c^2 f \sqrt {c+d}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (-\frac {\frac {d (2 c-d) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c-d}-\frac {c d^2 \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )+1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c-d}}{c^2}-\frac {2 \sqrt {a+b} \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{a c^2 f \sqrt {c+d}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 3297 |
\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (-\frac {\frac {2 d \sqrt {a+b} (2 c-d) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}-\frac {c d^2 \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )+1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c-d}}{c^2}-\frac {2 \sqrt {a+b} \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{a c^2 f \sqrt {c+d}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
\(\Big \downarrow \) 3475 |
\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (-\frac {\frac {2 c d^2 (a-b) \sqrt {a+b} \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)^2}+\frac {2 d \sqrt {a+b} (2 c-d) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}}{c^2}-\frac {2 \sqrt {a+b} \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{a c^2 f \sqrt {c+d}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\) |
(Sqrt[d + c*Cos[e + f*x]]*(-(((2*(a - b)*Sqrt[a + b]*c*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*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)^2*f) + (2*Sqrt[a + b]*(2*c - d)*d*Sqr t[-(((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) - (2*Sqrt[a + b] *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]*EllipticPi[((a + b)*c)/(a*(c + d)), ArcSi n[(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))])/(a*c^2*Sqrt[c + d]*f))*Sqrt[ a + b*Sec[e + f*x]])/(Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])
3.3.20.3.1 Defintions of rubi rules used
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_.) + (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 - 2*a*b*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, 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. \(2866\) vs. \(2(577)=1154\).
Time = 17.18 (sec) , antiderivative size = 2867, normalized size of antiderivative = 4.61
-2/f/((a-b)/(a+b))^(1/2)/(a*d-b*c)/(c-d)/(c+d)/c*((a*(1-cos(f*x+e))^2*csc( f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/((1-cos(f*x+e))^2*csc(f*x+e) ^2-1))^(1/2)*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)*((c*(1-cos(f*x+e))^2*csc(f* x+e)^2-d*(1-cos(f*x+e))^2*csc(f*x+e)^2-c-d)/((1-cos(f*x+e))^2*csc(f*x+e)^2 -1))^(1/2)*(((a-b)/(a+b))^(1/2)*a*c*d^2*(1-cos(f*x+e))^3*csc(f*x+e)^3-((a- b)/(a+b))^(1/2)*a*d^3*(1-cos(f*x+e))^3*csc(f*x+e)^3-((a-b)/(a+b))^(1/2)*b* c*d^2*(1-cos(f*x+e))^3*csc(f*x+e)^3+((a-b)/(a+b))^(1/2)*b*d^3*(1-cos(f*x+e ))^3*csc(f*x+e)^3+2*(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2* csc(f*x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos (f*x+e))^2*csc(f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticPi(((a-b)/(a+b))^(1/2)*( -cot(f*x+e)+csc(f*x+e)),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/ 2))*a*c^2*d-2*(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f* x+e)^2-a-b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e ))^2*csc(f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(-cot(f *x+e)+csc(f*x+e)),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/2))*a* d^3-2*(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a -b)/(a+b))^(1/2)*(-(c*(1-cos(f*x+e))^2*csc(f*x+e)^2-d*(1-cos(f*x+e))^2*csc (f*x+e)^2-c-d)/(c+d))^(1/2)*EllipticPi(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+cs c(f*x+e)),-(a+b)/(a-b),((c-d)/(c+d))^(1/2)/((a-b)/(a+b))^(1/2))*b*c^3+2*(- (a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(...
\[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(b*sec(f*x + e) + a)*sqrt(d*sec(f*x + e) + c)/(b*d^2*sec(f*x + e)^3 + a*c^2 + (2*b*c*d + a*d^2)*sec(f*x + e)^2 + (b*c^2 + 2*a*c*d)*sec( f*x + e)), x)
\[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx=\int \frac {1}{\sqrt {a + b \sec {\left (e + f x \right )}} \left (c + d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx=\int \frac {1}{\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]