\(\int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\) [607]

Optimal result

Integrand size = 25, antiderivative size = 271 \[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {2 (a-b) b \sqrt {a+b} \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3 a^2 d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{3 a d}+\frac {2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)} \] Output:

2/3*(a-b)*b*(a+b)^(1/2)*cot(d*x+c)*EllipticE((a+b*cos(d*x+c))^(1/2)/(a+b)^ 
(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2 
)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a^2/d+2/3*(a-b)*(a+b)^(1/2)*cot(d*x+c)*El 
lipticF((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b)) 
^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/a/d+ 
2/3*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/d/cos(d*x+c)^(3/2)
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 9.04 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\frac {-2 b (a+b) \cos ^{\frac {3}{2}}(c+d x) \sqrt {1+\cos (c+d x)} \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right )+2 a (a+b) \cos ^{\frac {3}{2}}(c+d x) \sqrt {1+\cos (c+d x)} \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right )+\left (2 a^2+a b+b^2+2 a (a+2 b) \cos (c+d x)+b (a+b) \cos (2 (c+d x))\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{3 a d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \] Input:

Integrate[Sqrt[a + b*Cos[c + d*x]]/Cos[c + d*x]^(5/2),x]
 

Output:

(-2*b*(a + b)*Cos[c + d*x]^(3/2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c 
+ d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], 
(-a + b)/(a + b)] + 2*a*(a + b)*Cos[c + d*x]^(3/2)*Sqrt[1 + Cos[c + d*x]]* 
Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[T 
an[(c + d*x)/2]], (-a + b)/(a + b)] + (2*a^2 + a*b + b^2 + 2*a*(a + 2*b)*C 
os[c + d*x] + b*(a + b)*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(3*a*d*Cos[c + 
 d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3275, 27, 3042, 3477, 3042, 3295, 3473}

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.

\(\displaystyle \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx\)

\(\Big \downarrow \) 3275

\(\displaystyle \frac {2}{3} \int \frac {b+a \cos (c+d x)}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {b+a \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \int \frac {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3477

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3295

\(\displaystyle \frac {1}{3} \left (b \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}\right )+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\)

\(\Big \downarrow \) 3473

\(\displaystyle \frac {1}{3} \left (\frac {2 b (a-b) \sqrt {a+b} \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}\right )+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d \cos ^{\frac {3}{2}}(c+d x)}\)

Input:

Int[Sqrt[a + b*Cos[c + d*x]]/Cos[c + d*x]^(5/2),x]
 

Output:

((2*(a - b)*b*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + 
 d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - 
 Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a^2*d) + ( 
2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x 
]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec 
[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d))/3 + (2*S 
qrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))
 

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 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 3295

Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f 
_.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr 
t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli 
pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] 
], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] 
&& PosQ[(a + b)/d]
 

rule 3473

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) 
^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* 
(c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] 
)/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + 
d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], 
x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && 
PosQ[(c + d)/b]
 

rule 3477

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S 
imp[(A - B)/(a - b)   Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* 
x]]), x], x] - Simp[(A*b - a*B)/(a - b)   Int[(1 + Sin[e + f*x])/((a + b*Si 
n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e 
, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 
0] && NeQ[A, B]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(548\) vs. \(2(239)=478\).

Time = 15.07 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.03

method	result	size
default	\(\frac {2 \left (\sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {a +\cos \left (d x +c \right ) b}{\left (\cos \left (d x +c \right )+1\right ) \left (a +b \right )}}\, a b \operatorname {EllipticE}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) \left (\cos \left (d x +c \right )^{3}+2 \cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )\right )+\sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {a +\cos \left (d x +c \right ) b}{\left (\cos \left (d x +c \right )+1\right ) \left (a +b \right )}}\, b^{2} \operatorname {EllipticE}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) \left (\cos \left (d x +c \right )^{3}+2 \cos \left (d x +c \right )^{2}+\cos \left (d x +c \right )\right )+\sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {a +\cos \left (d x +c \right ) b}{\left (\cos \left (d x +c \right )+1\right ) \left (a +b \right )}}\, a^{2} \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) \left (-\cos \left (d x +c \right )^{3}-2 \cos \left (d x +c \right )^{2}-\cos \left (d x +c \right )\right )+\sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {a +\cos \left (d x +c \right ) b}{\left (\cos \left (d x +c \right )+1\right ) \left (a +b \right )}}\, a b \operatorname {EllipticF}\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) \left (-\cos \left (d x +c \right )^{3}-2 \cos \left (d x +c \right )^{2}-\cos \left (d x +c \right )\right )+\left (\cos \left (d x +c \right )+1\right ) \sin \left (d x +c \right ) a^{2}+\sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (\cos \left (d x +c \right )+2\right ) a b +b^{2} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )\right ) \sqrt {a +\cos \left (d x +c \right ) b}}{3 d \cos \left (d x +c \right )^{\frac {3}{2}} \left (b \cos \left (d x +c \right )^{2}+a \cos \left (d x +c \right )+\cos \left (d x +c \right ) b +a \right ) a}\)	\(549\)

Input:

int((a+cos(d*x+c)*b)^(1/2)/cos(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

2/3/d*((cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/ 
(a+b))^(1/2)*a*b*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(co 
s(d*x+c)^3+2*cos(d*x+c)^2+cos(d*x+c))+(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(( 
a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*b^2*EllipticE(cot(d*x+c)-csc(d 
*x+c),(-(a-b)/(a+b))^(1/2))*(cos(d*x+c)^3+2*cos(d*x+c)^2+cos(d*x+c))+(cos( 
d*x+c)/(cos(d*x+c)+1))^(1/2)*((a+cos(d*x+c)*b)/(cos(d*x+c)+1)/(a+b))^(1/2) 
*a^2*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(-cos(d*x+c)^3- 
2*cos(d*x+c)^2-cos(d*x+c))+(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*((a+cos(d*x+c 
)*b)/(cos(d*x+c)+1)/(a+b))^(1/2)*a*b*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a- 
b)/(a+b))^(1/2))*(-cos(d*x+c)^3-2*cos(d*x+c)^2-cos(d*x+c))+(cos(d*x+c)+1)* 
sin(d*x+c)*a^2+sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)+2)*a*b+b^2*cos(d*x+c)^2*s 
in(d*x+c))*(a+cos(d*x+c)*b)^(1/2)/cos(d*x+c)^(3/2)/(b*cos(d*x+c)^2+a*cos(d 
*x+c)+cos(d*x+c)*b+a)/a
 

Fricas [F]

\[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {\sqrt {b \cos \left (d x + c\right ) + a}}{\cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^(1/2)/cos(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*cos(d*x + c) + a)/cos(d*x + c)^(5/2), x)
 

Sympy [F]

\[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {\sqrt {a + b \cos {\left (c + d x \right )}}}{\cos ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \] Input:

integrate((a+b*cos(d*x+c))**(1/2)/cos(d*x+c)**(5/2),x)
 

Output:

Integral(sqrt(a + b*cos(c + d*x))/cos(c + d*x)**(5/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {\sqrt {b \cos \left (d x + c\right ) + a}}{\cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^(1/2)/cos(d*x+c)^(5/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*cos(d*x + c) + a)/cos(d*x + c)^(5/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {\sqrt {b \cos \left (d x + c\right ) + a}}{\cos \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^(1/2)/cos(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*cos(d*x + c) + a)/cos(d*x + c)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {\sqrt {a+b\,\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int((a + b*cos(c + d*x))^(1/2)/cos(c + d*x)^(5/2),x)
 

Output:

int((a + b*cos(c + d*x))^(1/2)/cos(c + d*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \] Input:

int((a+b*cos(d*x+c))^(1/2)/cos(d*x+c)^(5/2),x)
 

Output:

int((sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)))/cos(c + d*x)**3,x)