\(\int \frac {\sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx\) [16]

Optimal result

Integrand size = 27, antiderivative size = 213 \[ \int \frac {\sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\frac {2 \sqrt {c+d} \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d \sec (e+f x)}}{\sqrt {c+d}}\right ),\frac {c+d}{c-d}\right ) \sqrt {\frac {d (1-\sec (e+f x))}{c+d}} \sqrt {-\frac {d (1+\sec (e+f x))}{c-d}}}{a f}+\frac {2 (a c-b d) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\arcsin \left (\frac {\sqrt {1-\sec (e+f x)}}{\sqrt {2}}\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sec (e+f x)}{c+d}} \tan (e+f x)}{a (a+b) f \sqrt {c+d \sec (e+f x)} \sqrt {-\tan ^2(e+f x)}} \] Output:

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

Mathematica [A] (verified)

Time = 10.69 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx=\frac {4 \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {\cos (e+f x)}{1+\cos (e+f x)}} \sqrt {\frac {d+c \cos (e+f x)}{(c+d) (1+\cos (e+f x))}} \left (-\left ((a+b) (c-d) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {c-d}{c+d}\right )\right )+2 (a c-b d) \operatorname {EllipticPi}\left (\frac {-a+b}{a+b},\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {c-d}{c+d}\right )\right ) \sqrt {c+d \sec (e+f x)}}{(a-b) (a+b) f (d+c \cos (e+f x))} \] Input:

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

Output:

(4*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(d + c*Co 
s[e + f*x])/((c + d)*(1 + Cos[e + f*x]))]*(-((a + b)*(c - d)*EllipticF[Arc 
Sin[Tan[(e + f*x)/2]], (c - d)/(c + d)]) + 2*(a*c - b*d)*EllipticPi[(-a + 
b)/(a + b), ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)])*Sqrt[c + d*Sec[e + 
 f*x]])/((a - b)*(a + b)*f*(d + c*Cos[e + f*x]))
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3042, 3308, 3042, 4457, 3042, 4319, 4461}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3308

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4457

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4319

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

\(\Big \downarrow \) 4461

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

Input:

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

Output:

(2*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))])/(a*f) + (2*(a*c - b*d)*EllipticPi[(2*a)/(a 
+ b), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*d)/(c + d)]*Sqrt[(c + d*S 
ec[e + f*x])/(c + d)]*Tan[e + f*x])/(a*(a + b)*f*Sqrt[c + d*Sec[e + f*x]]* 
Sqrt[-Tan[e + f*x]^2])
 

Defintions of rubi rules used

rule 3042

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

rule 3308

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_)*((a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)])^(m_.), x_Symbol] :> Int[(b + a*Csc[e + f*x])^m*((c + d*Csc[e + 
f*x])^n/Csc[e + f*x]^m), x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !Integer 
Q[n] && IntegerQ[m]
 

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 4457

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

rule 4461

Int[csc[(e_.) + (f_.)*(x_)]/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(cs 
c[(e_.) + (f_.)*(x_)]*(d_.) + (c_))), x_Symbol] :> Simp[-2*(Cot[e + f*x]/(f 
*(c + d)*Sqrt[a + b*Csc[e + f*x]]*Sqrt[-Cot[e + f*x]^2]))*Sqrt[(a + b*Csc[e 
 + f*x])/(a + b)]*EllipticPi[2*(d/(c + d)), ArcSin[Sqrt[1 - Csc[e + f*x]]/S 
qrt[2]], 2*(b/(a + b))], 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]
 

Maple [A] (verified)

Time = 7.80 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.49

method	result	size
default	\(\frac {2 \left (\operatorname {EllipticF}\left (-\csc \left (f x +e \right )+\cot \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) a c -\operatorname {EllipticF}\left (-\csc \left (f x +e \right )+\cot \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) a d +\operatorname {EllipticF}\left (-\csc \left (f x +e \right )+\cot \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) b c -\operatorname {EllipticF}\left (-\csc \left (f x +e \right )+\cot \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) b d -2 \operatorname {EllipticPi}\left (-\csc \left (f x +e \right )+\cot \left (f x +e \right ), -\frac {a -b}{a +b}, \sqrt {\frac {c -d}{c +d}}\right ) a c +2 \operatorname {EllipticPi}\left (-\csc \left (f x +e \right )+\cot \left (f x +e \right ), -\frac {a -b}{a +b}, \sqrt {\frac {c -d}{c +d}}\right ) b d \right ) \left (1+\cos \left (f x +e \right )\right ) \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (c +d \right )}}\, \sqrt {c +d \sec \left (f x +e \right )}}{f \left (d +c \cos \left (f x +e \right )\right ) \left (a -b \right ) \left (a +b \right )}\)	\(318\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(sec(e + f*x)*d + c)/(cos(e + f*x)*b + a),x)