\(\int \frac {1}{(a+b \cos ^2(x))^{5/2}} \, dx\) [72]

Optimal result

Integrand size = 12, antiderivative size = 177 \[ \int \frac {1}{\left (a+b \cos ^2(x)\right )^{5/2}} \, dx=\frac {2 (2 a+b) \sqrt {a+b \cos ^2(x)} E\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{3 a^2 (a+b)^2 \sqrt {\frac {a+b \cos ^2(x)}{a}}}-\frac {\sqrt {\frac {a+b \cos ^2(x)}{a}} \operatorname {EllipticF}\left (\frac {\pi }{2}+x,-\frac {b}{a}\right )}{3 a (a+b) \sqrt {a+b \cos ^2(x)}}-\frac {b \cos (x) \sin (x)}{3 a (a+b) \left (a+b \cos ^2(x)\right )^{3/2}}-\frac {2 b (2 a+b) \cos (x) \sin (x)}{3 a^2 (a+b)^2 \sqrt {a+b \cos ^2(x)}} \] Output:

2/3*(2*a+b)*(a+b*cos(x)^2)^(1/2)*EllipticE(cos(x),(-b/a)^(1/2))/a^2/(a+b)^ 
2/((a+b*cos(x)^2)/a)^(1/2)-1/3*((a+b*cos(x)^2)/a)^(1/2)*InverseJacobiAM(1/ 
2*Pi+x,(-b/a)^(1/2))/a/(a+b)/(a+b*cos(x)^2)^(1/2)-1/3*b*cos(x)*sin(x)/a/(a 
+b)/(a+b*cos(x)^2)^(3/2)-2/3*b*(2*a+b)*cos(x)*sin(x)/a^2/(a+b)^2/(a+b*cos( 
x)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (a+b \cos ^2(x)\right )^{5/2}} \, dx=\frac {2 (a+b)^2 (2 a+b) \left (\frac {2 a+b+b \cos (2 x)}{a+b}\right )^{3/2} E\left (x\left |\frac {b}{a+b}\right .\right )-a (a+b)^2 \left (\frac {2 a+b+b \cos (2 x)}{a+b}\right )^{3/2} \operatorname {EllipticF}\left (x,\frac {b}{a+b}\right )-\sqrt {2} b \left (5 a^2+5 a b+b^2+b (2 a+b) \cos (2 x)\right ) \sin (2 x)}{3 a^2 (a+b)^2 (2 a+b+b \cos (2 x))^{3/2}} \] Input:

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

Output:

(2*(a + b)^2*(2*a + b)*((2*a + b + b*Cos[2*x])/(a + b))^(3/2)*EllipticE[x, 
 b/(a + b)] - a*(a + b)^2*((2*a + b + b*Cos[2*x])/(a + b))^(3/2)*EllipticF 
[x, b/(a + b)] - Sqrt[2]*b*(5*a^2 + 5*a*b + b^2 + b*(2*a + b)*Cos[2*x])*Si 
n[2*x])/(3*a^2*(a + b)^2*(2*a + b + b*Cos[2*x])^(3/2))
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {3042, 3663, 25, 3042, 3652, 3042, 3651, 3042, 3657, 3042, 3656, 3662, 3042, 3661}

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.

\(\displaystyle \int \frac {1}{\left (a+b \cos ^2(x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (a+b \sin \left (x+\frac {\pi }{2}\right )^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 3663

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3652

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3651

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3657

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3656

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

\(\Big \downarrow \) 3662

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3661

\(\displaystyle \frac {\frac {\frac {2 (2 a+b) \sqrt {a+b \cos ^2(x)} E\left (x+\frac {\pi }{2}|-\frac {b}{a}\right )}{\sqrt {\frac {b \cos ^2(x)}{a}+1}}-\frac {a (a+b) \sqrt {\frac {b \cos ^2(x)}{a}+1} \operatorname {EllipticF}\left (x+\frac {\pi }{2},-\frac {b}{a}\right )}{\sqrt {a+b \cos ^2(x)}}}{a (a+b)}-\frac {2 b (2 a+b) \sin (x) \cos (x)}{a (a+b) \sqrt {a+b \cos ^2(x)}}}{3 a (a+b)}-\frac {b \sin (x) \cos (x)}{3 a (a+b) \left (a+b \cos ^2(x)\right )^{3/2}}\)

Input:

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

Output:

-1/3*(b*Cos[x]*Sin[x])/(a*(a + b)*(a + b*Cos[x]^2)^(3/2)) + (((2*(2*a + b) 
*Sqrt[a + b*Cos[x]^2]*EllipticE[Pi/2 + x, -(b/a)])/Sqrt[1 + (b*Cos[x]^2)/a 
] - (a*(a + b)*Sqrt[1 + (b*Cos[x]^2)/a]*EllipticF[Pi/2 + x, -(b/a)])/Sqrt[ 
a + b*Cos[x]^2])/(a*(a + b)) - (2*b*(2*a + b)*Cos[x]*Sin[x])/(a*(a + b)*Sq 
rt[a + b*Cos[x]^2]))/(3*a*(a + b))
 

Defintions of rubi rules used

rule 25

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

rule 3042

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

rule 3651

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + 
 (f_.)*(x_)]^2], x_Symbol] :> Simp[B/b   Int[Sqrt[a + b*Sin[e + f*x]^2], x] 
, x] + Simp[(A*b - a*B)/b   Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /; Fre 
eQ[{a, b, e, f, A, B}, x]
 

rule 3652

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b - a*B))*Cos[e + f*x]*Sin[e + f*x 
]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(a + b)*(p + 1))), x] - Simp[1/(2* 
a*(a + b)*(p + 1))   Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*( 
p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; 
 FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]
 

rule 3656

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a 
]/f)*EllipticE[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
 

rule 3657

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[Sqrt[a 
+ b*Sin[e + f*x]^2]/Sqrt[1 + b*(Sin[e + f*x]^2/a)]   Int[Sqrt[1 + (b*Sin[e 
+ f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] &&  !GtQ[a, 0]
 

rule 3661

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1/(S 
qrt[a]*f))*EllipticF[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 
 0]
 

rule 3662

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[Sqrt[ 
1 + b*(Sin[e + f*x]^2/a)]/Sqrt[a + b*Sin[e + f*x]^2]   Int[1/Sqrt[1 + (b*Si 
n[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] &&  !GtQ[a, 0]
 

rule 3663

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C 
os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a 
+ b))), x] + Simp[1/(2*a*(p + 1)*(a + b))   Int[(a + b*Sin[e + f*x]^2)^(p + 
 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] 
 /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(395\) vs. \(2(154)=308\).

Time = 0.86 (sec) , antiderivative size = 396, normalized size of antiderivative = 2.24

method	result	size
default	\(\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \cos \left (x \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b \cos \left (x \right )^{2}+\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \cos \left (x \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2} \cos \left (x \right )^{2}-4 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \cos \left (x \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b \cos \left (x \right )^{2}-2 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \cos \left (x \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2} \cos \left (x \right )^{2}+4 a \,b^{2} \cos \left (x \right )^{5}+2 b^{3} \cos \left (x \right )^{5}+\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \cos \left (x \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \cos \left (x \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b -4 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \cos \left (x \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}-2 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \cos \left (x \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +5 a^{2} b \cos \left (x \right )^{3}-a \,b^{2} \cos \left (x \right )^{3}-2 b^{3} \cos \left (x \right )^{3}-5 a^{2} b \cos \left (x \right )-3 a \,b^{2} \cos \left (x \right )}{3 \left (a +b \cos \left (x \right )^{2}\right )^{\frac {3}{2}} a^{2} \left (a +b \right )^{2} \sin \left (x \right )}\)	\(396\)

Input:

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

Output:

1/3*((sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^(1/2)*EllipticF(cos(x),(-b/a)^(1/ 
2))*a^2*b*cos(x)^2+(sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^(1/2)*EllipticF(cos 
(x),(-b/a)^(1/2))*a*b^2*cos(x)^2-4*(sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^(1/ 
2)*EllipticE(cos(x),(-b/a)^(1/2))*a^2*b*cos(x)^2-2*(sin(x)^2)^(1/2)*((a+b* 
cos(x)^2)/a)^(1/2)*EllipticE(cos(x),(-b/a)^(1/2))*a*b^2*cos(x)^2+4*a*b^2*c 
os(x)^5+2*b^3*cos(x)^5+(sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^(1/2)*EllipticF 
(cos(x),(-b/a)^(1/2))*a^3+(sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^(1/2)*Ellipt 
icF(cos(x),(-b/a)^(1/2))*a^2*b-4*(sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^(1/2) 
*EllipticE(cos(x),(-b/a)^(1/2))*a^3-2*(sin(x)^2)^(1/2)*((a+b*cos(x)^2)/a)^ 
(1/2)*EllipticE(cos(x),(-b/a)^(1/2))*a^2*b+5*a^2*b*cos(x)^3-a*b^2*cos(x)^3 
-2*b^3*cos(x)^3-5*a^2*b*cos(x)-3*a*b^2*cos(x))/(a+b*cos(x)^2)^(3/2)/a^2/(a 
+b)^2/sin(x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.22 (sec) , antiderivative size = 1213, normalized size of antiderivative = 6.85 \[ \int \frac {1}{\left (a+b \cos ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/3*((4*I*a^4*b + 4*I*a^3*b^2 + I*a^2*b^3 + (4*I*a^2*b^3 + 4*I*a*b^4 + I* 
b^5)*cos(x)^4 + 2*(4*I*a^3*b^2 + 4*I*a^2*b^3 + I*a*b^4)*cos(x)^2 - 2*(2*I* 
a^3*b^2 + I*a^2*b^3 + (2*I*a*b^4 + I*b^5)*cos(x)^4 - 2*(-2*I*a^2*b^3 - I*a 
*b^4)*cos(x)^2)*sqrt((a^2 + a*b)/b^2))*sqrt(b)*sqrt((2*b*sqrt((a^2 + a*b)/ 
b^2) - 2*a - b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) - 2*a 
 - b)/b)*(cos(x) + I*sin(x))), (8*a^2 + 8*a*b + b^2 + 4*(2*a*b + b^2)*sqrt 
((a^2 + a*b)/b^2))/b^2) + (-4*I*a^4*b - 4*I*a^3*b^2 - I*a^2*b^3 + (-4*I*a^ 
2*b^3 - 4*I*a*b^4 - I*b^5)*cos(x)^4 + 2*(-4*I*a^3*b^2 - 4*I*a^2*b^3 - I*a* 
b^4)*cos(x)^2 - 2*(-2*I*a^3*b^2 - I*a^2*b^3 + (-2*I*a*b^4 - I*b^5)*cos(x)^ 
4 - 2*(2*I*a^2*b^3 + I*a*b^4)*cos(x)^2)*sqrt((a^2 + a*b)/b^2))*sqrt(b)*sqr 
t((2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)*elliptic_e(arcsin(sqrt((2*b*sqr 
t((a^2 + a*b)/b^2) - 2*a - b)/b)*(cos(x) - I*sin(x))), (8*a^2 + 8*a*b + b^ 
2 + 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (-6*I*a^5 - 13*I*a^4*b - 
 9*I*a^3*b^2 - 2*I*a^2*b^3 + (-6*I*a^3*b^2 - 13*I*a^2*b^3 - 9*I*a*b^4 - 2* 
I*b^5)*cos(x)^4 + 2*(-6*I*a^4*b - 13*I*a^3*b^2 - 9*I*a^2*b^3 - 2*I*a*b^4)* 
cos(x)^2 - 2*(3*I*a^4*b + I*a^3*b^2 + (3*I*a^2*b^3 + I*a*b^4)*cos(x)^4 - 2 
*(-3*I*a^3*b^2 - I*a^2*b^3)*cos(x)^2)*sqrt((a^2 + a*b)/b^2))*sqrt(b)*sqrt( 
(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt( 
(a^2 + a*b)/b^2) - 2*a - b)/b)*(cos(x) + I*sin(x))), (8*a^2 + 8*a*b + b^2 
+ 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + (6*I*a^5 + 13*I*a^4*b +...
 

Sympy [F]

\[ \int \frac {1}{\left (a+b \cos ^2(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b \cos ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

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

Output:

Integral((a + b*cos(x)**2)**(-5/2), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {1}{\left (a+b \cos ^2(x)\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

integrate((b*cos(x)^2 + a)^(-5/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b \cos ^2(x)\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

integrate((b*cos(x)^2 + a)^(-5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \cos ^2(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,{\cos \left (x\right )}^2+a\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a+b \cos ^2(x)\right )^{5/2}} \, dx=\int \frac {\sqrt {\cos \left (x \right )^{2} b +a}}{\cos \left (x \right )^{6} b^{3}+3 \cos \left (x \right )^{4} a \,b^{2}+3 \cos \left (x \right )^{2} a^{2} b +a^{3}}d x \] Input:

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

Output:

int(sqrt(cos(x)**2*b + a)/(cos(x)**6*b**3 + 3*cos(x)**4*a*b**2 + 3*cos(x)* 
*2*a**2*b + a**3),x)