3.2.46 \(\int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx\) [146]

3.2.46.1 Optimal result

Integrand size = 27, antiderivative size = 319 \[ \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\frac {2 \sqrt {c+d} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right ),\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a (c-d) f}-\frac {2 \sqrt {c+d} \cot (e+f x) \operatorname {EllipticPi}\left (\frac {c+d}{c},\arcsin \left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right ),\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a c f}-\frac {E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a (c-d) f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}} \]

output

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

3.2.46.2 Mathematica [A] (verified)

Time = 8.24 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.57 \[ \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=-\frac {4 \cos ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {d+c \cos (e+f x)}{(c+d) (1+\cos (e+f x))}} \left ((c+d) E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right )+2 (c-2 d) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {c-d}{c+d}\right )+4 (-c+d) \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {c-d}{c+d}\right )\right ) \sec ^2(e+f x) \left (\frac {1}{1+\sec (e+f x)}\right )^{3/2}}{a (c-d) f \sqrt {c+d \sec (e+f x)}} \]

input

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

output

(-4*Cos[(e + f*x)/2]^4*Sqrt[(d + c*Cos[e + f*x])/((c + d)*(1 + Cos[e + f*x 
]))]*((c + d)*EllipticE[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)] + 2*(c 
- 2*d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)] + 4*(-c + d)*E 
llipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)])*Sec[e + f*x]^2*( 
(1 + Sec[e + f*x])^(-1))^(3/2))/(a*(c - d)*f*Sqrt[c + d*Sec[e + f*x]])
 

3.2.46.3 Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4417, 25, 3042, 4409, 3042, 4271, 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.

\(\displaystyle \int \frac {1}{(a \sec (e+f x)+a) \sqrt {c+d \sec (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right ) \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 4417

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4409

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4271

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

\(\Big \downarrow \) 4319

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

\(\Big \downarrow \) 4456

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

input

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

output

-((EllipticE[ArcSin[Tan[e + f*x]/(1 + Sec[e + f*x])], (c - d)/(c + d)]*Sqr 
t[(1 + Sec[e + f*x])^(-1)]*Sqrt[c + d*Sec[e + f*x]])/(a*(c - d)*f*Sqrt[(c 
+ d*Sec[e + f*x])/((c + d)*(1 + Sec[e + f*x]))])) + ((2*a*Sqrt[c + d]*Cot[ 
e + f*x]*EllipticF[ArcSin[Sqrt[c + d*Sec[e + f*x]]/Sqrt[c + d]], (c + d)/( 
c - d)]*Sqrt[(d*(1 - Sec[e + f*x]))/(c + d)]*Sqrt[-((d*(1 + Sec[e + f*x])) 
/(c - d))])/f - (2*a*(c - d)*Sqrt[c + d]*Cot[e + f*x]*EllipticPi[(c + d)/c 
, ArcSin[Sqrt[c + d*Sec[e + f*x]]/Sqrt[c + d]], (c + d)/(c - d)]*Sqrt[(d*( 
1 - Sec[e + f*x]))/(c + d)]*Sqrt[-((d*(1 + Sec[e + f*x]))/(c - d))])/(c*f) 
)/(a^2*(c - d))
 

3.2.46.3.1 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 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 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 4417

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

3.2.46.4 Maple [A] (verified)

Time = 7.30 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.90

method	result	size
default	\(\frac {\left (\cos \left (f x +e \right )+1\right ) \left (2 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) c -4 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) d +c \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right )+d \operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right )-4 c \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {c -d}{c +d}}\right )+4 \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {c -d}{c +d}}\right ) d \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {c +d \sec \left (f x +e \right )}}{a f \left (c -d \right ) \left (d +c \cos \left (f x +e \right )\right )}\)	\(287\)

input

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

output

1/a/f/(c-d)*(cos(f*x+e)+1)*(2*EllipticF(cot(f*x+e)-csc(f*x+e),((c-d)/(c+d) 
)^(1/2))*c-4*EllipticF(cot(f*x+e)-csc(f*x+e),((c-d)/(c+d))^(1/2))*d+c*Elli 
pticE(cot(f*x+e)-csc(f*x+e),((c-d)/(c+d))^(1/2))+d*EllipticE(cot(f*x+e)-cs 
c(f*x+e),((c-d)/(c+d))^(1/2))-4*c*EllipticPi(cot(f*x+e)-csc(f*x+e),-1,((c- 
d)/(c+d))^(1/2))+4*EllipticPi(cot(f*x+e)-csc(f*x+e),-1,((c-d)/(c+d))^(1/2) 
)*d)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e))/(cos(f*x+ 
e)+1))^(1/2)*(c+d*sec(f*x+e))^(1/2)/(d+c*cos(f*x+e))
 

3.2.46.5 Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+a \sec (e+f x)) \sqrt {c+d \sec (e+f x)}} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.2.46.6 Sympy [F]

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

input

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

output

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

3.2.46.7 Maxima [F]

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

input

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

output

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

3.2.46.8 Giac [F]

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

input

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

output

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

3.2.46.9 Mupad [F(-1)]

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

input

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

output

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