\(\int \sqrt {3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx\) [514]

Optimal result

Integrand size = 23, antiderivative size = 95 \[ \int \sqrt {3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx=-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d}+\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}+\frac {4 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}+\frac {\sqrt {3+4 \cos (c+d x)} \tan (c+d x)}{d} \] Output:

-EllipticE(sin(1/2*d*x+1/2*c),2/7*14^(1/2))*7^(1/2)/d+3/7*InverseJacobiAM( 
1/2*d*x+1/2*c,2/7*14^(1/2))*7^(1/2)/d+4/7*EllipticPi(sin(1/2*d*x+1/2*c),2, 
2/7*14^(1/2))*7^(1/2)/d+(3+4*cos(d*x+c))^(1/2)*tan(d*x+c)/d
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.45 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.65 \[ \int \sqrt {3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx=\frac {6 \sqrt {7} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {8}{7}\right )+\frac {i \sqrt {7} \left (21 E\left (i \text {arcsinh}\left (\sqrt {3+4 \cos (c+d x)}\right )|-\frac {1}{7}\right )-12 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {3+4 \cos (c+d x)}\right ),-\frac {1}{7}\right )-8 \operatorname {EllipticPi}\left (-\frac {1}{3},i \text {arcsinh}\left (\sqrt {3+4 \cos (c+d x)}\right ),-\frac {1}{7}\right )\right ) \sin (c+d x)}{\sqrt {\sin ^2(c+d x)}}+21 \sqrt {3+4 \cos (c+d x)} \tan (c+d x)}{21 d} \] Input:

Integrate[Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]^2,x]
 

Output:

(6*Sqrt[7]*EllipticPi[2, (c + d*x)/2, 8/7] + (I*Sqrt[7]*(21*EllipticE[I*Ar 
cSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7] - 12*EllipticF[I*ArcSinh[Sqrt[3 + 4 
*Cos[c + d*x]]], -1/7] - 8*EllipticPi[-1/3, I*ArcSinh[Sqrt[3 + 4*Cos[c + d 
*x]]], -1/7])*Sin[c + d*x])/Sqrt[Sin[c + d*x]^2] + 21*Sqrt[3 + 4*Cos[c + d 
*x]]*Tan[c + d*x])/(21*d)
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 3275, 27, 3042, 3539, 25, 3042, 3132, 3481, 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.

\(\displaystyle \int \sqrt {4 \cos (c+d x)+3} \sec ^2(c+d x) \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3275

\(\displaystyle \int \frac {2 \left (1-\cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {4 \cos (c+d x)+3}}dx+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {\left (1-\cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {4 \cos (c+d x)+3}}dx+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{d}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3539

\(\displaystyle 2 \left (-\frac {1}{4} \int \sqrt {4 \cos (c+d x)+3}dx-\frac {1}{4} \int -\frac {(3 \cos (c+d x)+4) \sec (c+d x)}{\sqrt {4 \cos (c+d x)+3}}dx\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (\frac {1}{4} \int \frac {(3 \cos (c+d x)+4) \sec (c+d x)}{\sqrt {4 \cos (c+d x)+3}}dx-\frac {1}{4} \int \sqrt {4 \cos (c+d x)+3}dx\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{d}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

\(\displaystyle 2 \left (\frac {1}{4} \int \frac {3 \sin \left (c+d x+\frac {\pi }{2}\right )+4}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{d}\)

\(\Big \downarrow \) 3481

\(\displaystyle 2 \left (\frac {1}{4} \left (3 \int \frac {1}{\sqrt {4 \cos (c+d x)+3}}dx+4 \int \frac {\sec (c+d x)}{\sqrt {4 \cos (c+d x)+3}}dx\right )-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \left (\frac {1}{4} \left (3 \int \frac {1}{\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx+4 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx\right )-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{d}\)

\(\Big \downarrow \) 3140

\(\displaystyle 2 \left (\frac {1}{4} \left (4 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx+\frac {6 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}\right )-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{d}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{d}+2 \left (\frac {1}{4} \left (\frac {6 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}+\frac {8 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}\right )-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{2 d}\right )\)

Input:

Int[Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]^2,x]
 

Output:

2*(-1/2*(Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/d + ((6*EllipticF[(c + d*x)/ 
2, 8/7])/(Sqrt[7]*d) + (8*EllipticPi[2, (c + d*x)/2, 8/7])/(Sqrt[7]*d))/4) 
 + (Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(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 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 3275

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*(a + b*Sin[e + f*x] 
)^(m + 1)*((c + d*Sin[e + f*x])^n/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m 
 + 1)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^ 
(n - 1)*Simp[a*c*(m + 1) + b*d*n + (a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] 
 - b*d*(m + n + 2)*Sin[e + f*x]^2, 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] && LtQ[m, 
 -1] && LtQ[0, n, 1] && IntegersQ[2*m, 2*n]
 

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]
 

rule 3481

Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ 
B/d   Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d   Int[(a + b* 
Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, 
 B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3539

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(349\) vs. \(2(92)=184\).

Time = 3.15 (sec) , antiderivative size = 350, normalized size of antiderivative = 3.68

method	result	size
default	\(-\frac {\sqrt {-\left (1-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-\frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}+\frac {3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )}{\sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}+\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )}{\sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}-\frac {4 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \operatorname {EllipticPi}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, 2 \sqrt {2}\right )}{\sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(350\)

Input:

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

Output:

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

Fricas [F]

\[ \int \sqrt {3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx=\int { \sqrt {4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{2} \,d x } \] Input:

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

Output:

integral(sqrt(4*cos(d*x + c) + 3)*sec(d*x + c)^2, x)
 

Sympy [F]

\[ \int \sqrt {3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx=\int \sqrt {4 \cos {\left (c + d x \right )} + 3} \sec ^{2}{\left (c + d x \right )}\, dx \] Input:

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

Output:

Integral(sqrt(4*cos(c + d*x) + 3)*sec(c + d*x)**2, x)
 

Maxima [F]

\[ \int \sqrt {3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx=\int { \sqrt {4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{2} \,d x } \] Input:

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

Output:

integrate(sqrt(4*cos(d*x + c) + 3)*sec(d*x + c)^2, x)
 

Giac [F]

\[ \int \sqrt {3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx=\int { \sqrt {4 \, \cos \left (d x + c\right ) + 3} \sec \left (d x + c\right )^{2} \,d x } \] Input:

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

Output:

integrate(sqrt(4*cos(d*x + c) + 3)*sec(d*x + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx=\int \frac {\sqrt {4\,\cos \left (c+d\,x\right )+3}}{{\cos \left (c+d\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx=\int \sqrt {4 \cos \left (d x +c \right )+3}\, \sec \left (d x +c \right )^{2}d x \] Input:

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

Output:

int(sqrt(4*cos(c + d*x) + 3)*sec(c + d*x)**2,x)