\(\int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\) [609]

Optimal result

Integrand size = 25, antiderivative size = 389 \[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 (a-b) b \sqrt {a+b} \left (19 a^2+8 b^2\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (25 a^2+6 a b+8 b^2\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{105 a^3 d}+\frac {2 \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 b \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (25 a^2-4 b^2\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 a^2 d \cos ^{\frac {3}{2}}(c+d x)} \] Output:

2/105*(a-b)*b*(a+b)^(1/2)*(19*a^2+8*b^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^4/d+2/105*(a-b)*(a+b)^ 
(1/2)*(25*a^2+6*a*b+8*b^2)*cot(d*x+c)*EllipticF((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^3/d+2/7*(a+b*cos(d*x+c))^(1/2)*sin(d 
*x+c)/d/cos(d*x+c)^(7/2)+2/35*b*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/a/d/cos( 
d*x+c)^(5/2)+2/105*(25*a^2-4*b^2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/a^2/d/ 
cos(d*x+c)^(3/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.55 (sec) , antiderivative size = 1304, normalized size of antiderivative = 3.35 \[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:

Integrate[Sqrt[a + b*Cos[c + d*x]]/Cos[c + d*x]^(9/2),x]
 

Output:

((-4*a*(25*a^4 - 17*a^2*b^2 - 8*b^4)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a 
 + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*C 
os[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a 
 + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[( 
c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a 
*(-19*a^3*b - 8*a*b^3)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[ 
-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x]) 
*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + 
 d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^ 
4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)* 
Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2 
]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*El 
lipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/ 
Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[ 
a + b*Cos[c + d*x]])) + 2*(-19*a^2*b^2 - 8*b^4)*((I*Cos[(c + d*x)/2]*Sqrt[ 
a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x] 
]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x] 
]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + 
 b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d* 
x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + ...
 

Rubi [A] (verified)

Time = 1.67 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 3275, 27, 3042, 3534, 27, 3042, 3534, 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.

\(\displaystyle \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\)

\(\Big \downarrow \) 3275

\(\displaystyle \frac {2}{7} \int \frac {4 b \cos ^2(c+d x)+5 a \cos (c+d x)+b}{2 \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \int \frac {4 b \cos ^2(c+d x)+5 a \cos (c+d x)+b}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \int \frac {4 b \sin \left (c+d x+\frac {\pi }{2}\right )^2+5 a \sin \left (c+d x+\frac {\pi }{2}\right )+b}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {1}{7} \left (\frac {2 \int \frac {25 a^2+23 b \cos (c+d x) a-4 b^2+2 b^2 \cos ^2(c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {\int \frac {25 a^2+23 b \cos (c+d x) a-4 b^2+2 b^2 \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{5 a}+\frac {2 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {\int \frac {25 a^2+23 b \sin \left (c+d x+\frac {\pi }{2}\right ) a-4 b^2+2 b^2 \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}+\frac {2 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {1}{7} \left (\frac {\frac {2 \int \frac {b \left (19 a^2+8 b^2\right )+a \left (25 a^2+2 b^2\right ) \cos (c+d x)}{2 \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \left (25 a^2-4 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {\frac {\int \frac {b \left (19 a^2+8 b^2\right )+a \left (25 a^2+2 b^2\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \left (25 a^2-4 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {\frac {\int \frac {b \left (19 a^2+8 b^2\right )+a \left (25 a^2+2 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}+\frac {2 \left (25 a^2-4 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3477

\(\displaystyle \frac {1}{7} \left (\frac {\frac {b \left (19 a^2+8 b^2\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+(a-b) \left (25 a^2+6 a b+8 b^2\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx}{3 a}+\frac {2 \left (25 a^2-4 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {\frac {(a-b) \left (25 a^2+6 a b+8 b^2\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (19 a^2+8 b^2\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 a}+\frac {2 \left (25 a^2-4 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3295

\(\displaystyle \frac {1}{7} \left (\frac {\frac {b \left (19 a^2+8 b^2\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (a-b) \sqrt {a+b} \left (25 a^2+6 a b+8 b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}}{3 a}+\frac {2 \left (25 a^2-4 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3473

\(\displaystyle \frac {1}{7} \left (\frac {\frac {\frac {2 (a-b) \sqrt {a+b} \left (25 a^2+6 a b+8 b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d}+\frac {2 b (a-b) \sqrt {a+b} \left (19 a^2+8 b^2\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d}}{3 a}+\frac {2 \left (25 a^2-4 b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 a d \cos ^{\frac {3}{2}}(c+d x)}}{5 a}+\frac {2 b \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{5 a d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{7 d \cos ^{\frac {7}{2}}(c+d x)}\)

Input:

Int[Sqrt[a + b*Cos[c + d*x]]/Cos[c + d*x]^(9/2),x]
 

Output:

(2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*b 
*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) + (((2* 
(a - b)*b*Sqrt[a + b]*(19*a^2 + 8*b^2)*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]*(25*a^2 + 6*a*b + 8*b^2)*Cot[c + d*x]*E 
llipticF[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 + Se 
c[c + d*x]))/(a - b)])/(a*d))/(3*a) + (2*(25*a^2 - 4*b^2)*Sqrt[a + b*Cos[c 
 + d*x]]*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)))/(5*a))/7
 

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]
 

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])))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1139\) vs. \(2(345)=690\).

Time = 24.25 (sec) , antiderivative size = 1140, normalized size of antiderivative = 2.93

method	result	size
default	\(\text {Expression too large to display}\)	\(1140\)

Input:

int((a+cos(d*x+c)*b)^(1/2)/cos(d*x+c)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

-2/105/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^3*b*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2) 
)*(-19*cos(d*x+c)^5-38*cos(d*x+c)^4-19*cos(d*x+c)^3)+(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*b^2*Ellipti 
cE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(-19*cos(d*x+c)^5-38*cos(d* 
x+c)^4-19*cos(d*x+c)^3)+(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^3*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b 
)/(a+b))^(1/2))*(-8*cos(d*x+c)^5-16*cos(d*x+c)^4-8*cos(d*x+c)^3)+(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^4 
*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(-8*cos(d*x+c)^5-16 
*cos(d*x+c)^4-8*cos(d*x+c)^3)+(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^4*EllipticF(cot(d*x+c)-csc(d*x+c),(- 
(a-b)/(a+b))^(1/2))*(25*cos(d*x+c)^5+50*cos(d*x+c)^4+25*cos(d*x+c)^3)+(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^3*b*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*(19*cos(d*x+ 
c)^5+38*cos(d*x+c)^4+19*cos(d*x+c)^3)+(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*b^2*EllipticF(cot(d*x+c)-c 
sc(d*x+c),(-(a-b)/(a+b))^(1/2))*(2*cos(d*x+c)^5+4*cos(d*x+c)^4+2*cos(d*x+c 
)^3)+(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^3*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))*...
 

Fricas [F]

\[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {\sqrt {b \cos \left (d x + c\right ) + a}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^(1/2)/cos(d*x+c)^(9/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*cos(d*x + c) + a)/cos(d*x + c)^(9/2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:

integrate((a+b*cos(d*x+c))**(1/2)/cos(d*x+c)**(9/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {\sqrt {b \cos \left (d x + c\right ) + a}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^(1/2)/cos(d*x+c)^(9/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*cos(d*x + c) + a)/cos(d*x + c)^(9/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {\sqrt {b \cos \left (d x + c\right ) + a}}{\cos \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^(1/2)/cos(d*x+c)^(9/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*cos(d*x + c) + a)/cos(d*x + c)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {\sqrt {a+b\,\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^{9/2}} \,d x \] Input:

int((a + b*cos(c + d*x))^(1/2)/cos(c + d*x)^(9/2),x)
 

Output:

int((a + b*cos(c + d*x))^(1/2)/cos(c + d*x)^(9/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b \cos (c+d x)}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \] Input:

int((a+b*cos(d*x+c))^(1/2)/cos(d*x+c)^(9/2),x)
 

Output:

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