3.3.12 \(\int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, dx\) [212]

3.3.12.1 Optimal result

Integrand size = 29, antiderivative size = 919 \[ \int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (3 b c^3-7 a c^2 d+b c d^2+3 a d^3\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{3 c^2 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 \sqrt {a+b} \left (b^2 c^3 (3 c+d)-2 a b c^2 \left (3 c^2+2 c d-d^2\right )+a^2 d \left (9 c^3-2 c^2 d-6 c d^2+3 d^3\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{3 c^3 (c-d)^2 (c+d)^{3/2} (b c-a d) f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 a \sqrt {a+b} \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\cos (e+f x))}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x))^{3/2} \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt {a+b \sec (e+f x)}}{c^3 \sqrt {c+d} f \sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}-\frac {2 d (b c-a d) \sqrt {a+b \sec (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x)) \sqrt {c+d \sec (e+f x)}} \]

output

-2/3*d*(-a*d+b*c)*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)*(-7*a*c^2*d+3*a*d^3+3*b*c^3+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)/f/(b+a*cos(f*x+e))^(1/2)/(c+d*se 
c(f*x+e))^(1/2)-2/3*(b^2*c^3*(3*c+d)-2*a*b*c^2*(3*c^2+2*c*d-d^2)+a^2*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*co 
s(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^3/f/(c+d)^ 
(1/2)/(b+a*cos(f*x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2)
 

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

Leaf count is larger than twice the leaf count of optimal. \(1960\) vs. \(2(919)=1838\).

Time = 7.17 (sec) , antiderivative size = 1960, normalized size of antiderivative = 2.13 \[ \int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, dx =\text {Too large to display} \]

input

Integrate[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(5/2),x]
 

output

((d + c*Cos[e + f*x])^3*Sec[e + f*x]*(a + b*Sec[e + f*x])^(3/2)*((2*(-(b*c 
*d*Sin[e + f*x]) + a*d^2*Sin[e + f*x]))/(3*c*(c^2 - d^2)*(d + c*Cos[e + f* 
x])^2) + (2*(3*b*c^3*Sin[e + f*x] - 7*a*c^2*d*Sin[e + f*x] + b*c*d^2*Sin[e 
 + f*x] + 3*a*d^3*Sin[e + f*x]))/(3*c*(c^2 - d^2)^2*(d + c*Cos[e + f*x]))) 
)/(f*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^(5/2)) + ((d + c*Cos[e + f* 
x])^(5/2)*Sec[e + f*x]*(a + b*Sec[e + f*x])^(3/2)*((4*(b*c - a*d)*(3*a*b*c 
^3 + a^2*c^2*d - 4*b^2*c^2*d + a*b*c*d^2 - a^2*d^3)*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^2*c^3 - 3*b^2*c^3 + 4*a*b*c^2*d + 
 a^2*c*d^2 - b^2*c*d^2 - 4*a*b*d^3)*((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)*(b + ...
 

3.3.12.3 Rubi [A] (verified)

Time = 3.21 (sec) , antiderivative size = 850, 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, 3468, 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.

\(\displaystyle \int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx\)

\(\Big \downarrow \) 4430

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {\cos (e+f x) (b+a \cos (e+f x))^{3/2}}{(d+c \cos (e+f x))^{5/2}}dx}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right ) \left (b+a \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3468

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3532

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {\int \frac {3 b^2 c^4-4 a b d c^3-\left (\left (9 c^2 d-6 d^3\right ) a^2-b \left (6 c^3-2 c d^2\right ) a+b^2 c^2 d\right ) \cos (e+f x) c-a^2 d^2 \left (2 c^2-3 d^2\right )}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c^2}+\frac {3 a^2 \left (c^2-d^2\right ) \int \frac {\sqrt {d+c \cos (e+f x)}}{\sqrt {b+a \cos (e+f x)}}dx}{c^2}}{3 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {\int \frac {3 b^2 c^4-4 a b d c^3-\left (\left (9 c^2 d-6 d^3\right ) a^2-b \left (6 c^3-2 c d^2\right ) a+b^2 c^2 d\right ) \sin \left (e+f x+\frac {\pi }{2}\right ) c-a^2 d^2 \left (2 c^2-3 d^2\right )}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}+\frac {3 a^2 \left (c^2-d^2\right ) \int \frac {\sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c^2}}{3 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3290

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {\int \frac {3 b^2 c^4-4 a b d c^3-\left (\left (9 c^2 d-6 d^3\right ) a^2-b \left (6 c^3-2 c d^2\right ) a+b^2 c^2 d\right ) \sin \left (e+f x+\frac {\pi }{2}\right ) c-a^2 d^2 \left (2 c^2-3 d^2\right )}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c^2}-\frac {6 a \sqrt {a+b} \left (c^2-d^2\right ) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}}{3 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3477

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {\frac {c (b c-a d) \left (-7 a c^2 d+3 a d^3+3 b c^3+b c d^2\right ) \int \frac {\cos (e+f x)+1}{\sqrt {b+a \cos (e+f x)} (d+c \cos (e+f x))^{3/2}}dx}{c-d}-\frac {\left (a^2 d \left (9 c^3-2 c^2 d-6 c d^2+3 d^3\right )-2 a b c^2 \left (3 c^2+2 c d-d^2\right )+b^2 c^3 (3 c+d)\right ) \int \frac {1}{\sqrt {b+a \cos (e+f x)} \sqrt {d+c \cos (e+f x)}}dx}{c-d}}{c^2}-\frac {6 a \sqrt {a+b} \left (c^2-d^2\right ) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}}{3 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {\frac {c (b c-a d) \left (-7 a c^2 d+3 a d^3+3 b c^3+b c d^2\right ) \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )+1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c-d}-\frac {\left (a^2 d \left (9 c^3-2 c^2 d-6 c d^2+3 d^3\right )-2 a b c^2 \left (3 c^2+2 c d-d^2\right )+b^2 c^3 (3 c+d)\right ) \int \frac {1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \sqrt {d+c \sin \left (e+f x+\frac {\pi }{2}\right )}}dx}{c-d}}{c^2}-\frac {6 a \sqrt {a+b} \left (c^2-d^2\right ) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}}{3 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3297

\(\displaystyle \frac {\sqrt {a+b \sec (e+f x)} \sqrt {c \cos (e+f x)+d} \left (\frac {\frac {\frac {c (b c-a d) \left (-7 a c^2 d+3 a d^3+3 b c^3+b c d^2\right ) \int \frac {\sin \left (e+f x+\frac {\pi }{2}\right )+1}{\sqrt {b+a \sin \left (e+f x+\frac {\pi }{2}\right )} \left (d+c \sin \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{c-d}-\frac {2 \sqrt {a+b} \left (a^2 d \left (9 c^3-2 c^2 d-6 c d^2+3 d^3\right )-2 a b c^2 \left (3 c^2+2 c d-d^2\right )+b^2 c^3 (3 c+d)\right ) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}}{c^2}-\frac {6 a \sqrt {a+b} \left (c^2-d^2\right ) \csc (e+f x) (c \cos (e+f x)+d) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (c \cos (e+f x)+d)}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (c \cos (e+f x)+d)}} \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 f \sqrt {c+d}}}{3 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sin (e+f x) \sqrt {a \cos (e+f x)+b}}{3 c f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^{3/2}}\right )}{\sqrt {a \cos (e+f x)+b} \sqrt {c+d \sec (e+f x)}}\)

\(\Big \downarrow \) 3475

\(\displaystyle \frac {\sqrt {d+c \cos (e+f x)} \sqrt {a+b \sec (e+f x)} \left (\frac {\frac {-\frac {2 (a-b) \sqrt {a+b} c \left (3 b c^3-7 a d c^2+b d^2 c+3 a d^3\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{(c-d) \sqrt {c+d} (b c-a d) f}-\frac {2 \sqrt {a+b} \left (b^2 (3 c+d) c^3-2 a b \left (3 c^2+2 d c-d^2\right ) c^2+a^2 d \left (9 c^3-2 d c^2-6 d^2 c+3 d^3\right )\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{(c-d) \sqrt {c+d} (b c-a d) f}}{c^2}-\frac {6 a \sqrt {a+b} \left (c^2-d^2\right ) \sqrt {-\frac {(b c-a d) (1-\cos (e+f x))}{(a+b) (d+c \cos (e+f x))}} \sqrt {-\frac {(b c-a d) (\cos (e+f x)+1)}{(a-b) (d+c \cos (e+f x))}} (d+c \cos (e+f x)) \csc (e+f x) \operatorname {EllipticPi}\left (\frac {(a+b) c}{a (c+d)},\arcsin \left (\frac {\sqrt {c+d} \sqrt {b+a \cos (e+f x)}}{\sqrt {a+b} \sqrt {d+c \cos (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{c^2 \sqrt {c+d} f}}{3 c \left (c^2-d^2\right )}-\frac {2 d (b c-a d) \sqrt {b+a \cos (e+f x)} \sin (e+f x)}{3 c \left (c^2-d^2\right ) f (d+c \cos (e+f x))^{3/2}}\right )}{\sqrt {b+a \cos (e+f x)} \sqrt {c+d \sec (e+f x)}}\)

input

Int[(a + b*Sec[e + f*x])^(3/2)/(c + d*Sec[e + f*x])^(5/2),x]
 

output

(Sqrt[d + c*Cos[e + f*x]]*Sqrt[a + b*Sec[e + f*x]]*((((-2*(a - b)*Sqrt[a + 
 b]*c*(3*b*c^3 - 7*a*c^2*d + b*c*d^2 + 3*a*d^3)*Sqrt[-(((b*c - a*d)*(1 - C 
os[e + f*x]))/((a + b)*(d + c*Cos[e + f*x])))]*Sqrt[-(((b*c - a*d)*(1 + Co 
s[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)*f) - (2*Sqrt[a + b]*(b^2*c^3*(3*c + d) - 2*a*b*c 
^2*(3*c^2 + 2*c*d - d^2) + a^2*d*(9*c^3 - 2*c^2*d - 6*c*d^2 + 3*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]*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*a*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*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))/(3*c*(c^2 - d^2)) - (2*d*(b*c - a*d)*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...
 

3.3.12.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 3468

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]
 

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 3532

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]
 

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]
 

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

Leaf count of result is larger than twice the leaf count of optimal. \(15313\) vs. \(2(840)=1680\).

Time = 16.55 (sec) , antiderivative size = 15314, normalized size of antiderivative = 16.66

method	result	size
default	\(\text {Expression too large to display}\)	\(15314\)

input

int((a+b*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))^(5/2),x,method=_RETURNVERBOSE)
 

output

result too large to display
 

3.3.12.5 Fricas [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, dx=\text {Timed out} \]

input

integrate((a+b*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))^(5/2),x, algorithm="fric 
as")
 

output

Timed out
 

3.3.12.6 Sympy [F]

\[ \int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, dx=\int \frac {\left (a + b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\left (c + d \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]

input

integrate((a+b*sec(f*x+e))**(3/2)/(c+d*sec(f*x+e))**(5/2),x)
 

output

Integral((a + b*sec(e + f*x))**(3/2)/(c + d*sec(e + f*x))**(5/2), x)
 

3.3.12.7 Maxima [F]

\[ \int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate((a+b*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))^(5/2),x, algorithm="maxi 
ma")
 

output

integrate((b*sec(f*x + e) + a)^(3/2)/(d*sec(f*x + e) + c)^(5/2), x)
 

3.3.12.8 Giac [F]

\[ \int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, dx=\int { \frac {{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate((a+b*sec(f*x+e))^(3/2)/(c+d*sec(f*x+e))^(5/2),x, algorithm="giac 
")
 

output

integrate((b*sec(f*x + e) + a)^(3/2)/(d*sec(f*x + e) + c)^(5/2), x)
 

3.3.12.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (e+f x))^{3/2}}{(c+d \sec (e+f x))^{5/2}} \, dx=\text {Hanged} \]

input

int((a + b/cos(e + f*x))^(3/2)/(c + d/cos(e + f*x))^(5/2),x)
 

output

\text{Hanged}