3.3.55 \(\int \frac {1}{(a+b \sec (e+f x))^{3/2}} \, dx\) [255]

3.3.55.1 Optimal result

Integrand size = 14, antiderivative size = 347 \[ \int \frac {1}{(a+b \sec (e+f x))^{3/2}} \, dx=\frac {2 \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{a \sqrt {a+b} f}-\frac {2 \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{a \sqrt {a+b} f}-\frac {2 \sqrt {a+b} \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{a^2 f}+\frac {2 b^2 \tan (e+f x)}{a \left (a^2-b^2\right ) f \sqrt {a+b \sec (e+f x)}} \]

output

2*cot(f*x+e)*EllipticE((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1 
/2))*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b*(1+sec(f*x+e))/(a-b))^(1/2)/a/f/(a 
+b)^(1/2)-2*cot(f*x+e)*EllipticF((a+b*sec(f*x+e))^(1/2)/(a+b)^(1/2),((a+b) 
/(a-b))^(1/2))*(b*(1-sec(f*x+e))/(a+b))^(1/2)*(-b*(1+sec(f*x+e))/(a-b))^(1 
/2)/a/f/(a+b)^(1/2)-2*cot(f*x+e)*EllipticPi((a+b*sec(f*x+e))^(1/2)/(a+b)^( 
1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sec(f*x+e))/(a+b))^(1/ 
2)*(-b*(1+sec(f*x+e))/(a-b))^(1/2)/a^2/f+2*b^2*tan(f*x+e)/a/(a^2-b^2)/f/(a 
+b*sec(f*x+e))^(1/2)
 

3.3.55.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(972\) vs. \(2(347)=694\).

Time = 6.60 (sec) , antiderivative size = 972, normalized size of antiderivative = 2.80 \[ \int \frac {1}{(a+b \sec (e+f x))^{3/2}} \, dx=\frac {(b+a \cos (e+f x))^2 \sec ^2(e+f x) \left (\frac {2 b \sin (e+f x)}{a \left (-a^2+b^2\right )}+\frac {2 b^2 \sin (e+f x)}{a \left (a^2-b^2\right ) (b+a \cos (e+f x))}\right )}{f (a+b \sec (e+f x))^{3/2}}+\frac {2 (b+a \cos (e+f x))^{3/2} \sec ^{\frac {3}{2}}(e+f x) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}} \left (a b \tan \left (\frac {1}{2} (e+f x)\right )+b^2 \tan \left (\frac {1}{2} (e+f x)\right )-2 a b \tan ^3\left (\frac {1}{2} (e+f x)\right )+a b \tan ^5\left (\frac {1}{2} (e+f x)\right )-b^2 \tan ^5\left (\frac {1}{2} (e+f x)\right )+2 a^2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}-2 b^2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+2 a^2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}-2 b^2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+b (a+b) E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}-a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}\right )}{a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2} \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}{1-\tan ^2\left (\frac {1}{2} (e+f x)\right )}} \left (a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-b \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )} \]

input

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

output

((b + a*Cos[e + f*x])^2*Sec[e + f*x]^2*((2*b*Sin[e + f*x])/(a*(-a^2 + b^2) 
) + (2*b^2*Sin[e + f*x])/(a*(a^2 - b^2)*(b + a*Cos[e + f*x]))))/(f*(a + b* 
Sec[e + f*x])^(3/2)) + (2*(b + a*Cos[e + f*x])^(3/2)*Sec[e + f*x]^(3/2)*Sq 
rt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x 
)/2]^2)]*(a*b*Tan[(e + f*x)/2] + b^2*Tan[(e + f*x)/2] - 2*a*b*Tan[(e + f*x 
)/2]^3 + a*b*Tan[(e + f*x)/2]^5 - b^2*Tan[(e + f*x)/2]^5 + 2*a^2*EllipticP 
i[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(e + f*x)/2] 
^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - 
2*b^2*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sqrt[1 - T 
an[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2] 
^2)/(a + b)] + 2*a^2*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + 
 b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[( 
e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - 2*b^2*EllipticPi[-1, ArcS 
in[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e 
+ f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a 
 + b)] + b*(a + b)*EllipticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sq 
rt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e 
 + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - a*(a + b)*EllipticF[ArcSin 
[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan 
[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2...
 

3.3.55.3 Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {3042, 4272, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}

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.

\(\displaystyle \int \frac {1}{(a+b \sec (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 4272

\(\displaystyle \frac {2 b^2 \tan (e+f x)}{a f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}-\frac {2 \int -\frac {a^2-b \sec (e+f x) a-b^2-b^2 \sec ^2(e+f x)}{2 \sqrt {a+b \sec (e+f x)}}dx}{a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a^2-b \sec (e+f x) a-b^2-b^2 \sec ^2(e+f x)}{\sqrt {a+b \sec (e+f x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \tan (e+f x)}{a f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a^2-b \csc \left (e+f x+\frac {\pi }{2}\right ) a-b^2-b^2 \csc \left (e+f x+\frac {\pi }{2}\right )^2}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \tan (e+f x)}{a f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {\int \frac {a^2-b^2+\left (b^2-a b\right ) \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}}dx-b^2 \int \frac {\sec (e+f x) (\sec (e+f x)+1)}{\sqrt {a+b \sec (e+f x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \tan (e+f x)}{a f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a^2-b^2+\left (b^2-a b\right ) \csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx-b^2 \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \tan (e+f x)}{a f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}\)

\(\Big \downarrow \) 4409

\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sec (e+f x)}}dx+b^2 \left (-\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\right )-b (a-b) \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \tan (e+f x)}{a f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+b^2 \left (-\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\right )-b (a-b) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}+\frac {2 b^2 \tan (e+f x)}{a f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {b^2 \left (-\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\right )-b (a-b) \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a f}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \tan (e+f x)}{a f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {b^2 \left (-\int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \left (\csc \left (e+f x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a f}-\frac {2 (a-b) \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{f}}{a \left (a^2-b^2\right )}+\frac {2 b^2 \tan (e+f x)}{a f \left (a^2-b^2\right ) \sqrt {a+b \sec (e+f x)}}\)

\(\Big \downarrow \) 4492

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

input

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

output

((2*(a - b)*Sqrt[a + b]*Cot[e + f*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[e + f 
*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*S 
qrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/f - (2*(a - b)*Sqrt[a + b]*Cot[e + 
 f*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - 
 b)]*Sqrt[(b*(1 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a 
 - b))])/f - (2*Sqrt[a + b]*(a^2 - b^2)*Cot[e + f*x]*EllipticPi[(a + b)/a, 
 ArcSin[Sqrt[a + b*Sec[e + f*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 
 - Sec[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[e + f*x]))/(a - b))])/(a*f)) 
/(a*(a^2 - b^2)) + (2*b^2*Tan[e + f*x])/(a*(a^2 - b^2)*f*Sqrt[a + b*Sec[e 
+ f*x]])
 

3.3.55.3.1 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 4271

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a 
 + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) 
*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ 
c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[a^2 - b^2, 0]
 

rule 4272

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[ 
c + d*x]*((a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Sim 
p[1/(a*(n + 1)*(a^2 - b^2))   Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - 
b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x 
], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ 
erQ[2*n]
 

rule 4319

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* 
x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt 
[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4409

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ 
.) + (a_)], x_Symbol] :> Simp[c   Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + 
Simp[d   Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, 
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
 

rule 4492

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c 
sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a 
 + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e 
+ f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + 
 f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, 
 f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
 

rule 4546

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Int[(A + (B - C 
)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Simp[C   Int[Csc[e + f*x]*(( 
1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A 
, B, C}, x] && NeQ[a^2 - b^2, 0]
 

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

Leaf count of result is larger than twice the leaf count of optimal. \(1798\) vs. \(2(318)=636\).

Time = 6.48 (sec) , antiderivative size = 1799, normalized size of antiderivative = 5.18

method	result	size
default	\(\text {Expression too large to display}\)	\(1799\)

input

int(1/(a+b*sec(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
 

output

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

3.3.55.5 Fricas [F]

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

input

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

output

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

3.3.55.6 Sympy [F]

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

input

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

output

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

3.3.55.7 Maxima [F]

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

input

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

output

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

3.3.55.8 Giac [F]

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

input

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

output

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

3.3.55.9 Mupad [F(-1)]

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

input

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

output

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