3.3.11 \(\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx\) [211]

3.3.11.1 Optimal result

Integrand size = 29, antiderivative size = 744 \[ \int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx=-\frac {2 (a-b) \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c (c-d) \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \sqrt {a+b} (b c-a (2 c-d)) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^2 (c-d) \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 a \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^2 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}} \]

output

-2*(a-b)*(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/(c-d)/f/(c+d)^(1/2)/(b+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x+e))^( 
1/2)-2*(b*c-a*(2*c-d))*(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*c 
os(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)/f/(c+d)^(1/2)/(b+a*cos(f*x+e))^(1/2)/(c 
+d*sec(f*x+e))^(1/2)-2*a*(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^2/f/(c+d)^(1/2)/(b+a*cos(f*x 
+e))^(1/2)/(c+d*sec(f*x+e))^(1/2)
 

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

Leaf count is larger than twice the leaf count of optimal. \(1750\) vs. \(2(744)=1488\).

Time = 19.22 (sec) , antiderivative size = 1750, normalized size of antiderivative = 2.35 \[ \int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx =\text {Too large to display} \]

input

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

output

(2*(d + c*Cos[e + f*x])*(a + b*Sec[e + f*x])^(3/2)*(-(b*c*Sin[e + f*x]) + 
a*d*Sin[e + f*x]))/((-c^2 + d^2)*f*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x 
])^(3/2)) + ((d + c*Cos[e + f*x])^(3/2)*(a + b*Sec[e + f*x])^(3/2)*((4*(b* 
c - a*d)*(a*b*c - b^2*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]*Elli 
pticF[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*(a^ 
2*c - b^2*c)*(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]*El 
lipticF[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]]) - (Sq 
rt[((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]*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)...
 

3.3.11.3 Rubi [A] (verified)

Time = 2.04 (sec) , antiderivative size = 691, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 4430, 3042, 3277, 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.

\(\displaystyle \int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 4430

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {(b+a \cos (e+f x))^{3/2}}{(d+c \cos (e+f x))^{3/2}}dx}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {\left (b+a \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3277

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {a^2 \int \frac {\sqrt {d+c \cos (e+f x)}}{\sqrt {b+a \cos (e+f x)}}dx}{c^2}+\frac {(b c-a d) \int \frac {b c+2 a \cos (e+f x) c+a d}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c^2}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {a^2 \int \frac {\sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c^2}+\frac {(b c-a d) \int \frac {b c+2 a \sin \left (e+f x+\frac {\pi }{2}\right ) c+a d}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3290

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {(b c-a d) \int \frac {b c+2 a \sin \left (e+f x+\frac {\pi }{2}\right ) c+a d}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}-\frac {2 a \sqrt {a+b} \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3477

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {(b c-a d) \left (\frac {c (b c-a d) \int \frac {\cos (e+f x)+1}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c-d}-\frac {(b c-a (2 c-d)) \int \frac {1}{\sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}dx}{c-d}\right )}{c^2}-\frac {2 a \sqrt {a+b} \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {(b c-a d) \left (\frac {c (b c-a d) \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )+1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c-d}-\frac {(b c-a (2 c-d)) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c-d}\right )}{c^2}-\frac {2 a \sqrt {a+b} \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3297

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {(b c-a d) \left (\frac {c (b c-a d) \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )+1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c-d}-\frac {2 \sqrt {a+b} (b c-a (2 c-d)) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}\right )}{c^2}-\frac {2 a \sqrt {a+b} \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3475

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

input

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

output

(Sqrt[d + c*Cos[e + f*x]]*(((b*c - a*d)*((-2*(a - b)*Sqrt[a + b]*c*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*Co 
s[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*c - a* 
(2*c - d))*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]*S 
qrt[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*a*Sqrt[a + b]*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]*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))*Sqrt[a + b*Sec[e + f*x]])/(Sqrt[b + a*Cos[e + f*x]]*Sqrt[c + d*Se 
c[e + f*x]])
 

3.3.11.3.1 Defintions of rubi rules used

rule 3042

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

rule 3277

Int[((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)/((a_.) + (b_.)*sin[(e_.) + 
 (f_.)*(x_)])^(3/2), x_Symbol] :> Simp[d^2/b^2   Int[Sqrt[a + b*Sin[e + f*x 
]]/Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[(b*c - a*d)/b^2   Int[Simp[b*c + 
 a*d + 2*b*d*Sin[e + f*x], 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}, x] && NeQ[b*c - a*d, 0] && Ne 
Q[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

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 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]
 

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

Leaf count of result is larger than twice the leaf count of optimal. \(3568\) vs. \(2(675)=1350\).

Time = 13.97 (sec) , antiderivative size = 3569, normalized size of antiderivative = 4.80

method	result	size
default	\(\text {Expression too large to display}\)	\(3569\)

input

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

output

-2/f/((a-b)/(a+b))^(1/2)/(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) 
*(-2*((a-b)/(a+b))^(1/2)*a*b*c*d*(-cot(f*x+e)+csc(f*x+e))+(-(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)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/ 
(a-b)/(c+d))^(1/2))*a^2*c*d-(-(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)*EllipticE(((a-b)/(a+b))^ 
(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*c^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)*EllipticE(((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)) 
,((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*a*b*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)*EllipticE( 
((a-b)/(a+b))^(1/2)*(-cot(f*x+e)+csc(f*x+e)),((a+b)*(c-d)/(a-b)/(c+d))^(1/ 
2))*b^2*c*d-(-(a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f...
 

3.3.11.5 Fricas [F]

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

input

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

output

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

3.3.11.6 Sympy [F]

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

input

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

output

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

3.3.11.7 Maxima [F]

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

input

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

output

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

3.3.11.8 Giac [F]

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

input

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

output

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

3.3.11.9 Mupad [F(-1)]

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

input

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

output

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