\(\int \sqrt {3+4 \cos (c+d x)} \sec (c+d x) \, dx\) [513]

Optimal result

Integrand size = 21, antiderivative size = 48 \[ \int \sqrt {3+4 \cos (c+d x)} \sec (c+d x) \, dx=\frac {8 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}+\frac {6 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d} \] Output:

8/7*InverseJacobiAM(1/2*d*x+1/2*c,2/7*14^(1/2))*7^(1/2)/d+6/7*EllipticPi(s 
in(1/2*d*x+1/2*c),2,2/7*14^(1/2))*7^(1/2)/d
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.85 \[ \int \sqrt {3+4 \cos (c+d x)} \sec (c+d x) \, dx=\frac {8 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )+6 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d} \] Input:

Integrate[Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x],x]
 

Output:

(8*EllipticF[(c + d*x)/2, 8/7] + 6*EllipticPi[2, (c + d*x)/2, 8/7])/(Sqrt[ 
7]*d)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3282, 3042, 3140, 3284}

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.

\(\displaystyle \int \sqrt {4 \cos (c+d x)+3} \sec (c+d x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 3282

\(\displaystyle 4 \int \frac {1}{\sqrt {4 \cos (c+d x)+3}}dx+3 \int \frac {\sec (c+d x)}{\sqrt {4 \cos (c+d x)+3}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 \int \frac {1}{\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx+3 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx\)

\(\Big \downarrow \) 3140

\(\displaystyle 3 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx+\frac {8 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {8 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}+\frac {6 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}\)

Input:

Int[Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x],x]
 

Output:

(8*EllipticF[(c + d*x)/2, 8/7])/(Sqrt[7]*d) + (6*EllipticPi[2, (c + d*x)/2 
, 8/7])/(Sqrt[7]*d)
 

Defintions of rubi rules used

rule 3042

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

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3282

Int[Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]/((a_.) + (b_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[d/b   Int[1/Sqrt[c + d*Sin[e + f*x]], x], x 
] + Simp[(b*c - a*d)/b   Int[1/((a + b*Sin[e + f*x])*Sqrt[c + d*Sin[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 3284

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 
2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(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] && GtQ[c + d, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(47)=94\).

Time = 2.40 (sec) , antiderivative size = 158, normalized size of antiderivative = 3.29

method	result	size
default	\(-\frac {2 \sqrt {\left (8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (4 \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )-3 \operatorname {EllipticPi}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, 2 \sqrt {2}\right )\right )}{\sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(158\)

Input:

int((3+4*cos(d*x+c))^(1/2)*sec(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

-2*((8*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/ 
2*c)^2)^(1/2)*(1-8*cos(1/2*d*x+1/2*c)^2)^(1/2)*(4*EllipticF(cos(1/2*d*x+1/ 
2*c),2*2^(1/2))-3*EllipticPi(cos(1/2*d*x+1/2*c),2,2*2^(1/2)))/(-8*sin(1/2* 
d*x+1/2*c)^4+7*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(8*cos(1/2*d 
*x+1/2*c)^2-1)^(1/2)/d
 

Fricas [F]

\[ \int \sqrt {3+4 \cos (c+d x)} \sec (c+d x) \, dx=\int { \sqrt {4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right ) \,d x } \] Input:

integrate((3+4*cos(d*x+c))^(1/2)*sec(d*x+c),x, algorithm="fricas")
 

Output:

integral(sqrt(4*cos(d*x + c) + 3)*sec(d*x + c), x)
 

Sympy [F]

\[ \int \sqrt {3+4 \cos (c+d x)} \sec (c+d x) \, dx=\int \sqrt {4 \cos {\left (c + d x \right )} + 3} \sec {\left (c + d x \right )}\, dx \] Input:

integrate((3+4*cos(d*x+c))**(1/2)*sec(d*x+c),x)
 

Output:

Integral(sqrt(4*cos(c + d*x) + 3)*sec(c + d*x), x)
 

Maxima [F]

\[ \int \sqrt {3+4 \cos (c+d x)} \sec (c+d x) \, dx=\int { \sqrt {4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right ) \,d x } \] Input:

integrate((3+4*cos(d*x+c))^(1/2)*sec(d*x+c),x, algorithm="maxima")
 

Output:

integrate(sqrt(4*cos(d*x + c) + 3)*sec(d*x + c), x)
 

Giac [F]

\[ \int \sqrt {3+4 \cos (c+d x)} \sec (c+d x) \, dx=\int { \sqrt {4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right ) \,d x } \] Input:

integrate((3+4*cos(d*x+c))^(1/2)*sec(d*x+c),x, algorithm="giac")
 

Output:

integrate(sqrt(4*cos(d*x + c) + 3)*sec(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {3+4 \cos (c+d x)} \sec (c+d x) \, dx=\int \frac {\sqrt {4\,\cos \left (c+d\,x\right )+3}}{\cos \left (c+d\,x\right )} \,d x \] Input:

int((4*cos(c + d*x) + 3)^(1/2)/cos(c + d*x),x)
 

Output:

int((4*cos(c + d*x) + 3)^(1/2)/cos(c + d*x), x)
 

Reduce [F]

\[ \int \sqrt {3+4 \cos (c+d x)} \sec (c+d x) \, dx=\int \sqrt {4 \cos \left (d x +c \right )+3}\, \sec \left (d x +c \right )d x \] Input:

int((3+4*cos(d*x+c))^(1/2)*sec(d*x+c),x)
 

Output:

int(sqrt(4*cos(c + d*x) + 3)*sec(c + d*x),x)