3.3.20 \(\int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx\) [220]

3.3.20.1 Optimal result

Integrand size = 29, antiderivative size = 622 \[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx=-\frac {2 (a-b) \sqrt {a+b} d^2 \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{c (c-d) \sqrt {c+d} (b c-a d)^2 f}-\frac {2 \sqrt {a+b} (2 c-d) d \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{c^2 (c-d) \sqrt {c+d} (b c-a d) f}-\frac {2 \sqrt {a+b} \cot (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sec (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sec (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {\frac {(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{a c^2 \sqrt {c+d} f} \]

output

-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)
 

3.3.20.2 Mathematica [B] (warning: unable to verify)

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 \[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx =\text {Too large to display} \]

input

Integrate[1/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^(3/2)),x]
 

output

(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...
 

3.3.20.3 Rubi [A] (verified)

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

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (-\frac {\frac {2 c d^2 (a-b) \sqrt {a+b} \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)^2}+\frac {2 d \sqrt {a+b} (2 c-d) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}}{c^2}-\frac {2 \sqrt {a+b} \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{a c^2 f \sqrt {c+d}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

input

Int[1/(Sqrt[a + b*Sec[e + f*x]]*(c + d*Sec[e + f*x])^(3/2)),x]
 

output

(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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3290

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)]
 

rule 3297

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)]
 

rule 3475

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)]
 

rule 3477

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]
 

rule 3533

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]
 

rule 4430

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]
 

3.3.20.4 Maple [B] (warning: unable to verify)

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

method	result	size
default	\(\text {Expression too large to display}\)	\(2867\)

input

int(1/(c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOS 
E)
 

output

-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)/(...
 

3.3.20.5 Fricas [F]

\[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate(1/(c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="fr 
icas")
 

output

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)
 

3.3.20.6 Sympy [F]

\[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx=\int \frac {1}{\sqrt {a + b \sec {\left (e + f x \right )}} \left (c + d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

input

integrate(1/(c+d*sec(f*x+e))**(3/2)/(a+b*sec(f*x+e))**(1/2),x)
 

output

Integral(1/(sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))**(3/2)), x)
 

3.3.20.7 Maxima [F]

\[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate(1/(c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="ma 
xima")
 

output

integrate(1/(sqrt(b*sec(f*x + e) + a)*(d*sec(f*x + e) + c)^(3/2)), x)
 

3.3.20.8 Giac [F]

\[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {b \sec \left (f x + e\right ) + a} {\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate(1/(c+d*sec(f*x+e))^(3/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="gi 
ac")
 

output

integrate(1/(sqrt(b*sec(f*x + e) + a)*(d*sec(f*x + e) + c)^(3/2)), x)
 

3.3.20.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b \sec (e+f x)} (c+d \sec (e+f x))^{3/2}} \, dx=\int \frac {1}{\sqrt {a+\frac {b}{\cos \left (e+f\,x\right )}}\,{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]

input

int(1/((a + b/cos(e + f*x))^(1/2)*(c + d/cos(e + f*x))^(3/2)),x)
 

output

int(1/((a + b/cos(e + f*x))^(1/2)*(c + d/cos(e + f*x))^(3/2)), x)