3.3.76 \(\int \frac {\sec ^2(e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx\) [276]

3.3.76.1 Optimal result

Integrand size = 35, antiderivative size = 214 \[ \int \frac {\sec ^2(e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {2 a \sqrt {a+b} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{(a-b) b c f}-\frac {E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {a-b}{a+b}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {a+b \sec (e+f x)}}{(a-b) c f \sqrt {\frac {a+b \sec (e+f x)}{(a+b) (1+\sec (e+f x))}}} \]

output

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

3.3.76.2 Mathematica [A] (verified)

Time = 5.48 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.73 \[ \int \frac {\sec ^2(e+f x)}{\sqrt {a+b \sec (e+f x)} (c+c \sec (e+f x))} \, dx=\frac {4 \cos ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} \left ((a+b) E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right )-2 a \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right )\right )}{(-a+b) c f \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} (1+\cos (e+f x))^2 \sqrt {a+b \sec (e+f x)}} \]

input

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

output

(4*Cos[(e + f*x)/2]^4*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x] 
))]*((a + b)*EllipticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] - 2*a*El 
lipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]))/((-a + b)*c*f*Sqrt[Co 
s[e + f*x]/(1 + Cos[e + f*x])]*(1 + Cos[e + f*x])^2*Sqrt[a + b*Sec[e + f*x 
]])
 

3.3.76.3 Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 4464, 3042, 4319, 4456}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4464

\(\displaystyle \frac {a \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}}dx}{c (a-b)}-\frac {\int \frac {\sec (e+f x) \sqrt {a+b \sec (e+f x)}}{\sec (e+f x) c+c}dx}{a-b}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4319

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

\(\Big \downarrow \) 4456

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

input

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

output

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

3.3.76.3.1 Defintions of rubi rules used

rule 3042

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

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 4456

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

rule 4464

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

3.3.76.4 Maple [A] (verified)

Time = 8.77 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.90

method	result	size
default	\(-\frac {\left (\cos \left (f x +e \right )+1\right ) \left (-a \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )-b \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right )+2 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {a -b}{a +b}}\right ) a \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {b +a \cos \left (f x +e \right )}{\left (a +b \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {a +b \sec \left (f x +e \right )}}{c f \left (a -b \right ) \left (b +a \cos \left (f x +e \right )\right )}\)	\(192\)

input

int(sec(f*x+e)^2/(c+c*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x,method=_RETURNV 
ERBOSE)
 

output

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

3.3.76.5 Fricas [F]

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

input

integrate(sec(f*x+e)^2/(c+c*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x, algorith 
m="fricas")
 

output

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

3.3.76.6 Sympy [F]

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

input

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

output

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

3.3.76.7 Maxima [F]

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

input

integrate(sec(f*x+e)^2/(c+c*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x, algorith 
m="maxima")
 

output

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

3.3.76.8 Giac [F]

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

input

integrate(sec(f*x+e)^2/(c+c*sec(f*x+e))/(a+b*sec(f*x+e))^(1/2),x, algorith 
m="giac")
 

output

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

3.3.76.9 Mupad [F(-1)]

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

input

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

output

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