\(\int \sqrt {3+4 \cos (c+d x)} \sec ^3(c+d x) \, dx\) [515]

Optimal result

Integrand size = 23, antiderivative size = 135 \[ \int \sqrt {3+4 \cos (c+d x)} \sec ^3(c+d x) \, dx=-\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{3 d}+\frac {3 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}+\frac {5 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {8}{7}\right )}{3 \sqrt {7} d}+\frac {\sqrt {3+4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3+4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d} \] Output:

-1/3*EllipticE(sin(1/2*d*x+1/2*c),2/7*14^(1/2))*7^(1/2)/d+3/7*InverseJacob 
iAM(1/2*d*x+1/2*c,2/7*14^(1/2))*7^(1/2)/d+5/21*EllipticPi(sin(1/2*d*x+1/2* 
c),2,2/7*14^(1/2))*7^(1/2)/d+1/3*(3+4*cos(d*x+c))^(1/2)*tan(d*x+c)/d+1/2*( 
3+4*cos(d*x+c))^(1/2)*sec(d*x+c)*tan(d*x+c)/d
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.52 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.44 \[ \int \sqrt {3+4 \cos (c+d x)} \sec ^3(c+d x) \, dx=\frac {\frac {12 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7}}+\frac {6 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7}}+\frac {2 i \left (21 E\left (i \text {arcsinh}\left (\sqrt {3+4 \cos (c+d x)}\right )|-\frac {1}{7}\right )-12 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {3+4 \cos (c+d x)}\right ),-\frac {1}{7}\right )-8 \operatorname {EllipticPi}\left (-\frac {1}{3},i \text {arcsinh}\left (\sqrt {3+4 \cos (c+d x)}\right ),-\frac {1}{7}\right )\right ) \sin (c+d x)}{3 \sqrt {7} \sqrt {\sin ^2(c+d x)}}+(3+2 \cos (c+d x)) \sqrt {3+4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{6 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3042, 3275, 3042, 3534, 3042, 3538, 27, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3275

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {\left (-4 \cos ^2(c+d x)+6 \cos (c+d x)+5\right ) \sec (c+d x)}{\sqrt {4 \cos (c+d x)+3}}dx+\frac {2 \sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {-4 \sin \left (c+d x+\frac {\pi }{2}\right )^2+6 \sin \left (c+d x+\frac {\pi }{2}\right )+5}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx+\frac {2 \sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}\)

\(\Big \downarrow \) 3538

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-\int \sqrt {4 \cos (c+d x)+3}dx-\frac {1}{4} \int -\frac {4 (9 \cos (c+d x)+5) \sec (c+d x)}{\sqrt {4 \cos (c+d x)+3}}dx\right )+\frac {2 \sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\int \frac {(9 \cos (c+d x)+5) \sec (c+d x)}{\sqrt {4 \cos (c+d x)+3}}dx-\int \sqrt {4 \cos (c+d x)+3}dx\right )+\frac {2 \sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\int \frac {9 \sin \left (c+d x+\frac {\pi }{2}\right )+5}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx-\int \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}dx\right )+\frac {2 \sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\int \frac {9 \sin \left (c+d x+\frac {\pi }{2}\right )+5}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx-\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d}\right )+\frac {2 \sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (9 \int \frac {1}{\sqrt {4 \cos (c+d x)+3}}dx+5 \int \frac {\sec (c+d x)}{\sqrt {4 \cos (c+d x)+3}}dx-\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d}\right )+\frac {2 \sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (9 \int \frac {1}{\sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx+5 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx-\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d}\right )+\frac {2 \sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (5 \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {4 \sin \left (c+d x+\frac {\pi }{2}\right )+3}}dx+\frac {18 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}-\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d}\right )+\frac {2 \sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}\right )+\frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {\sqrt {4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}+\frac {1}{2} \left (\frac {2 \sqrt {4 \cos (c+d x)+3} \tan (c+d x)}{3 d}+\frac {1}{3} \left (\frac {18 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}-\frac {2 \sqrt {7} E\left (\frac {1}{2} (c+d x)|\frac {8}{7}\right )}{d}+\frac {10 \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {8}{7}\right )}{\sqrt {7} d}\right )\right )\)

Input:

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

Output:

(Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (((-2*Sqrt[7] 
*EllipticE[(c + d*x)/2, 8/7])/d + (18*EllipticF[(c + d*x)/2, 8/7])/(Sqrt[7 
]*d) + (10*EllipticPi[2, (c + d*x)/2, 8/7])/(Sqrt[7]*d))/3 + (2*Sqrt[3 + 4 
*Cos[c + d*x]]*Tan[c + d*x])/(3*d))/2
 

Defintions of rubi rules used

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 3534

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 
NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

rule 3538

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (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 - b*B*d + a*C*d)*Sin[ 
e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre 
eQ[{a, b, c, d, e, f, A, B, 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. \(407\) vs. \(2(122)=244\).

Time = 4.30 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.02

method	result	size
default	\(-\frac {\sqrt {-\left (1-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}-\frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{3 \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}+\frac {3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )}{\sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}+\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2 \sqrt {2}\right )}{3 \sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}-\frac {5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {1-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \operatorname {EllipticPi}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, 2 \sqrt {2}\right )}{3 \sqrt {-8 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(408\)

Input:

int((3+4*cos(d*x+c))^(1/2)*sec(d*x+c)^3,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)*(-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)^2-2/3*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))+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)*EllipticE(cos(1/2*d*x+1/2*c),2*2^(1/2))-5/3* 
(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)*EllipticPi(cos(1/2*d*x+1/2*c),2 
,2*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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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