Integrand size = 12, antiderivative size = 123 \[ \int \frac {1}{\left (a+a \cos ^2(x)\right )^{5/2}} \, dx=\frac {\sqrt {a+a \cos ^2(x)} E\left (\left .\frac {\pi }{2}+x\right |-1\right )}{2 a^3 \sqrt {1+\cos ^2(x)}}-\frac {\sqrt {1+\cos ^2(x)} \operatorname {EllipticF}\left (\frac {\pi }{2}+x,-1\right )}{6 a^2 \sqrt {a+a \cos ^2(x)}}-\frac {\cos (x) \sin (x)}{6 a \left (a+a \cos ^2(x)\right )^{3/2}}-\frac {\cos (x) \sin (x)}{2 a^2 \sqrt {a+a \cos ^2(x)}} \] Output:
1/2*(a+a*cos(x)^2)^(1/2)*EllipticE(cos(x),I)/a^3/(1+cos(x)^2)^(1/2)-1/6*(1 +cos(x)^2)^(1/2)*InverseJacobiAM(1/2*Pi+x,I)/a^2/(a+a*cos(x)^2)^(1/2)-1/6* cos(x)*sin(x)/a/(a+a*cos(x)^2)^(3/2)-1/2*cos(x)*sin(x)/a^2/(a+a*cos(x)^2)^ (1/2)
Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (a+a \cos ^2(x)\right )^{5/2}} \, dx=\frac {12 (3+\cos (2 x))^{3/2} E\left (x\left |\frac {1}{2}\right .\right )-2 (3+\cos (2 x))^{3/2} \operatorname {EllipticF}\left (x,\frac {1}{2}\right )-22 \sin (2 x)-3 \sin (4 x)}{12 \sqrt {2} a (a (3+\cos (2 x)))^{3/2}} \] Input:
Integrate[(a + a*Cos[x]^2)^(-5/2),x]
Output:
(12*(3 + Cos[2*x])^(3/2)*EllipticE[x, 1/2] - 2*(3 + Cos[2*x])^(3/2)*Ellipt icF[x, 1/2] - 22*Sin[2*x] - 3*Sin[4*x])/(12*Sqrt[2]*a*(a*(3 + Cos[2*x]))^( 3/2))
Time = 0.80 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.02, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {3042, 3663, 25, 3042, 3652, 27, 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.
| \(\displaystyle \int \frac {1}{\left (a \cos ^2(x)+a\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \sin \left (x+\frac {\pi }{2}\right )^2+a\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3663 |
| \(\displaystyle -\frac {\int -\frac {5 a-a \cos ^2(x)}{\left (a \cos ^2(x)+a\right )^{3/2}}dx}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {5 a-a \cos ^2(x)}{\left (a \cos ^2(x)+a\right )^{3/2}}dx}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 a-a \sin \left (x+\frac {\pi }{2}\right )^2}{\left (a \sin \left (x+\frac {\pi }{2}\right )^2+a\right )^{3/2}}dx}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (3 \cos ^2(x) a^2+2 a^2\right )}{\sqrt {a \cos ^2(x)+a}}dx}{2 a^2}-\frac {3 \sin (x) \cos (x)}{\sqrt {a \cos ^2(x)+a}}}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {3 \cos ^2(x) a^2+2 a^2}{\sqrt {a \cos ^2(x)+a}}dx}{a^2}-\frac {3 \sin (x) \cos (x)}{\sqrt {a \cos ^2(x)+a}}}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {3 \sin \left (x+\frac {\pi }{2}\right )^2 a^2+2 a^2}{\sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx}{a^2}-\frac {3 \sin (x) \cos (x)}{\sqrt {a \cos ^2(x)+a}}}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 3651 |
| \(\displaystyle \frac {\frac {3 a \int \sqrt {a \cos ^2(x)+a}dx-a^2 \int \frac {1}{\sqrt {a \cos ^2(x)+a}}dx}{a^2}-\frac {3 \sin (x) \cos (x)}{\sqrt {a \cos ^2(x)+a}}}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 a \int \sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a}dx-a^2 \int \frac {1}{\sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx}{a^2}-\frac {3 \sin (x) \cos (x)}{\sqrt {a \cos ^2(x)+a}}}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 3657 |
| \(\displaystyle \frac {\frac {\frac {3 a \sqrt {a \cos ^2(x)+a} \int \sqrt {\cos ^2(x)+1}dx}{\sqrt {\cos ^2(x)+1}}-a^2 \int \frac {1}{\sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx}{a^2}-\frac {3 \sin (x) \cos (x)}{\sqrt {a \cos ^2(x)+a}}}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 a \sqrt {a \cos ^2(x)+a} \int \sqrt {\sin \left (x+\frac {\pi }{2}\right )^2+1}dx}{\sqrt {\cos ^2(x)+1}}-a^2 \int \frac {1}{\sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx}{a^2}-\frac {3 \sin (x) \cos (x)}{\sqrt {a \cos ^2(x)+a}}}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 3656 |
| \(\displaystyle \frac {\frac {\frac {3 a E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \sqrt {a \cos ^2(x)+a}}{\sqrt {\cos ^2(x)+1}}-a^2 \int \frac {1}{\sqrt {a \sin \left (x+\frac {\pi }{2}\right )^2+a}}dx}{a^2}-\frac {3 \sin (x) \cos (x)}{\sqrt {a \cos ^2(x)+a}}}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 3662 |
| \(\displaystyle \frac {\frac {\frac {3 a E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \sqrt {a \cos ^2(x)+a}}{\sqrt {\cos ^2(x)+1}}-\frac {a^2 \sqrt {\cos ^2(x)+1} \int \frac {1}{\sqrt {\cos ^2(x)+1}}dx}{\sqrt {a \cos ^2(x)+a}}}{a^2}-\frac {3 \sin (x) \cos (x)}{\sqrt {a \cos ^2(x)+a}}}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 a E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \sqrt {a \cos ^2(x)+a}}{\sqrt {\cos ^2(x)+1}}-\frac {a^2 \sqrt {\cos ^2(x)+1} \int \frac {1}{\sqrt {\sin \left (x+\frac {\pi }{2}\right )^2+1}}dx}{\sqrt {a \cos ^2(x)+a}}}{a^2}-\frac {3 \sin (x) \cos (x)}{\sqrt {a \cos ^2(x)+a}}}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
| \(\Big \downarrow \) 3661 |
| \(\displaystyle \frac {\frac {\frac {3 a E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \sqrt {a \cos ^2(x)+a}}{\sqrt {\cos ^2(x)+1}}-\frac {a^2 \sqrt {\cos ^2(x)+1} \operatorname {EllipticF}\left (x+\frac {\pi }{2},-1\right )}{\sqrt {a \cos ^2(x)+a}}}{a^2}-\frac {3 \sin (x) \cos (x)}{\sqrt {a \cos ^2(x)+a}}}{6 a^2}-\frac {\sin (x) \cos (x)}{6 a \left (a \cos ^2(x)+a\right )^{3/2}}\) |
Input:
Int[(a + a*Cos[x]^2)^(-5/2),x]
Output:
-1/6*(Cos[x]*Sin[x])/(a*(a + a*Cos[x]^2)^(3/2)) + (((3*a*Sqrt[a + a*Cos[x] ^2]*EllipticE[Pi/2 + x, -1])/Sqrt[1 + Cos[x]^2] - (a^2*Sqrt[1 + Cos[x]^2]* EllipticF[Pi/2 + x, -1])/Sqrt[a + a*Cos[x]^2])/a^2 - (3*Cos[x]*Sin[x])/Sqr t[a + a*Cos[x]^2])/(6*a^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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]
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]
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]
Time = 0.24 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.47
| method | result | size |
| default | \(-\frac {\sqrt {a \left (1+\cos \left (x \right )^{2}\right ) \sin \left (x \right )^{2}}\, \left (\sqrt {-\sin \left (x \right )^{2}+2}\, \operatorname {EllipticF}\left (\cos \left (x \right ), i\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sin \left (x \right )^{2}-3 \sqrt {-\sin \left (x \right )^{2}+2}\, \operatorname {EllipticE}\left (\cos \left (x \right ), i\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sin \left (x \right )^{2}-3 \sin \left (x \right )^{4} \cos \left (x \right )-2 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\sin \left (x \right )^{2}+2}\, \operatorname {EllipticF}\left (\cos \left (x \right ), i\right )+6 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\sin \left (x \right )^{2}+2}\, \operatorname {EllipticE}\left (\cos \left (x \right ), i\right )+7 \sin \left (x \right )^{2} \cos \left (x \right )\right ) \sqrt {-a \sin \left (x \right )^{4}+2 a \sin \left (x \right )^{2}}}{6 a^{3} \sin \left (x \right )^{3} \left (\sin \left (x \right )^{4}-4 \sin \left (x \right )^{2}+4\right ) \sqrt {a \left (1+\cos \left (x \right )^{2}\right )}}\) | \(181\) |
Input:
int(1/(a+a*cos(x)^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/6*(a*(1+cos(x)^2)*sin(x)^2)^(1/2)/a^3/sin(x)^3/(sin(x)^4-4*sin(x)^2+4)* ((-sin(x)^2+2)^(1/2)*EllipticF(cos(x),I)*(sin(x)^2)^(1/2)*sin(x)^2-3*(-sin (x)^2+2)^(1/2)*EllipticE(cos(x),I)*(sin(x)^2)^(1/2)*sin(x)^2-3*sin(x)^4*co s(x)-2*(sin(x)^2)^(1/2)*(-sin(x)^2+2)^(1/2)*EllipticF(cos(x),I)+6*(sin(x)^ 2)^(1/2)*(-sin(x)^2+2)^(1/2)*EllipticE(cos(x),I)+7*sin(x)^2*cos(x))*(-a*si n(x)^4+2*a*sin(x)^2)^(1/2)/(a*(1+cos(x)^2))^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (96) = 192\).
Time = 0.12 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.76 \[ \int \frac {1}{\left (a+a \cos ^2(x)\right )^{5/2}} \, dx=-\frac {3 \, {\left ({\left (-2 i \, \sqrt {2} + 3 i\right )} \cos \left (x\right )^{4} + 2 \, {\left (-2 i \, \sqrt {2} + 3 i\right )} \cos \left (x\right )^{2} - 2 i \, \sqrt {2} + 3 i\right )} \sqrt {a} \sqrt {2 \, \sqrt {2} - 3} E(\arcsin \left (\sqrt {2 \, \sqrt {2} - 3} {\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right )}\right )\,|\,12 \, \sqrt {2} + 17) + 3 \, {\left ({\left (2 i \, \sqrt {2} - 3 i\right )} \cos \left (x\right )^{4} + 2 \, {\left (2 i \, \sqrt {2} - 3 i\right )} \cos \left (x\right )^{2} + 2 i \, \sqrt {2} - 3 i\right )} \sqrt {a} \sqrt {2 \, \sqrt {2} - 3} E(\arcsin \left (\sqrt {2 \, \sqrt {2} - 3} {\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right )}\right )\,|\,12 \, \sqrt {2} + 17) + 2 \, {\left ({\left (-4 i \, \sqrt {2} - 15 i\right )} \cos \left (x\right )^{4} + 2 \, {\left (-4 i \, \sqrt {2} - 15 i\right )} \cos \left (x\right )^{2} - 4 i \, \sqrt {2} - 15 i\right )} \sqrt {a} \sqrt {2 \, \sqrt {2} - 3} F(\arcsin \left (\sqrt {2 \, \sqrt {2} - 3} {\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right )}\right )\,|\,12 \, \sqrt {2} + 17) + 2 \, {\left ({\left (4 i \, \sqrt {2} + 15 i\right )} \cos \left (x\right )^{4} + 2 \, {\left (4 i \, \sqrt {2} + 15 i\right )} \cos \left (x\right )^{2} + 4 i \, \sqrt {2} + 15 i\right )} \sqrt {a} \sqrt {2 \, \sqrt {2} - 3} F(\arcsin \left (\sqrt {2 \, \sqrt {2} - 3} {\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right )}\right )\,|\,12 \, \sqrt {2} + 17) + 2 \, \sqrt {a \cos \left (x\right )^{2} + a} {\left (3 \, \cos \left (x\right )^{3} + 4 \, \cos \left (x\right )\right )} \sin \left (x\right )}{12 \, {\left (a^{3} \cos \left (x\right )^{4} + 2 \, a^{3} \cos \left (x\right )^{2} + a^{3}\right )}} \] Input:
integrate(1/(a+a*cos(x)^2)^(5/2),x, algorithm="fricas")
Output:
-1/12*(3*((-2*I*sqrt(2) + 3*I)*cos(x)^4 + 2*(-2*I*sqrt(2) + 3*I)*cos(x)^2 - 2*I*sqrt(2) + 3*I)*sqrt(a)*sqrt(2*sqrt(2) - 3)*elliptic_e(arcsin(sqrt(2* sqrt(2) - 3)*(cos(x) + I*sin(x))), 12*sqrt(2) + 17) + 3*((2*I*sqrt(2) - 3* I)*cos(x)^4 + 2*(2*I*sqrt(2) - 3*I)*cos(x)^2 + 2*I*sqrt(2) - 3*I)*sqrt(a)* sqrt(2*sqrt(2) - 3)*elliptic_e(arcsin(sqrt(2*sqrt(2) - 3)*(cos(x) - I*sin( x))), 12*sqrt(2) + 17) + 2*((-4*I*sqrt(2) - 15*I)*cos(x)^4 + 2*(-4*I*sqrt( 2) - 15*I)*cos(x)^2 - 4*I*sqrt(2) - 15*I)*sqrt(a)*sqrt(2*sqrt(2) - 3)*elli ptic_f(arcsin(sqrt(2*sqrt(2) - 3)*(cos(x) + I*sin(x))), 12*sqrt(2) + 17) + 2*((4*I*sqrt(2) + 15*I)*cos(x)^4 + 2*(4*I*sqrt(2) + 15*I)*cos(x)^2 + 4*I* sqrt(2) + 15*I)*sqrt(a)*sqrt(2*sqrt(2) - 3)*elliptic_f(arcsin(sqrt(2*sqrt( 2) - 3)*(cos(x) - I*sin(x))), 12*sqrt(2) + 17) + 2*sqrt(a*cos(x)^2 + a)*(3 *cos(x)^3 + 4*cos(x))*sin(x))/(a^3*cos(x)^4 + 2*a^3*cos(x)^2 + a^3)
\[ \int \frac {1}{\left (a+a \cos ^2(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \cos ^{2}{\left (x \right )} + a\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(a+a*cos(x)**2)**(5/2),x)
Output:
Integral((a*cos(x)**2 + a)**(-5/2), x)
\[ \int \frac {1}{\left (a+a \cos ^2(x)\right )^{5/2}} \, dx=\int { \frac {1}{{\left (a \cos \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a+a*cos(x)^2)^(5/2),x, algorithm="maxima")
Output:
integrate((a*cos(x)^2 + a)^(-5/2), x)
\[ \int \frac {1}{\left (a+a \cos ^2(x)\right )^{5/2}} \, dx=\int { \frac {1}{{\left (a \cos \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a+a*cos(x)^2)^(5/2),x, algorithm="giac")
Output:
integrate((a*cos(x)^2 + a)^(-5/2), x)
Timed out. \[ \int \frac {1}{\left (a+a \cos ^2(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a\,{\cos \left (x\right )}^2+a\right )}^{5/2}} \,d x \] Input:
int(1/(a + a*cos(x)^2)^(5/2),x)
Output:
int(1/(a + a*cos(x)^2)^(5/2), x)
\[ \int \frac {1}{\left (a+a \cos ^2(x)\right )^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\cos \left (x \right )^{2}+1}}{\cos \left (x \right )^{6}+3 \cos \left (x \right )^{4}+3 \cos \left (x \right )^{2}+1}d x \right )}{a^{3}} \] Input:
int(1/(a+a*cos(x)^2)^(5/2),x)
Output:
(sqrt(a)*int(sqrt(cos(x)**2 + 1)/(cos(x)**6 + 3*cos(x)**4 + 3*cos(x)**2 + 1),x))/a**3