\(\int \frac {1}{(a+a \cos ^2(x))^{3/2}} \, dx\) [57]

Optimal result

Integrand size = 12, antiderivative size = 62 \[ \int \frac {1}{\left (a+a \cos ^2(x)\right )^{3/2}} \, dx=\frac {\sqrt {a+a \cos ^2(x)} E\left (\left .\frac {\pi }{2}+x\right |-1\right )}{2 a^2 \sqrt {1+\cos ^2(x)}}-\frac {\cos (x) \sin (x)}{2 a \sqrt {a+a \cos ^2(x)}} \] Output:

1/2*(a+a*cos(x)^2)^(1/2)*EllipticE(cos(x),I)/a^2/(1+cos(x)^2)^(1/2)-1/2*co 
s(x)*sin(x)/a/(a+a*cos(x)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a+a \cos ^2(x)\right )^{3/2}} \, dx=\frac {2 \sqrt {3+\cos (2 x)} E\left (x\left |\frac {1}{2}\right .\right )-\sin (2 x)}{2 \sqrt {2} a \sqrt {a (3+\cos (2 x))}} \] Input:

Integrate[(a + a*Cos[x]^2)^(-3/2),x]
 

Output:

(2*Sqrt[3 + Cos[2*x]]*EllipticE[x, 1/2] - Sin[2*x])/(2*Sqrt[2]*a*Sqrt[a*(3 
 + Cos[2*x])])
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {3042, 3663, 25, 3042, 3657, 3042, 3656}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3663

\(\displaystyle -\frac {\int -\sqrt {a \cos ^2(x)+a}dx}{2 a^2}-\frac {\sin (x) \cos (x)}{2 a \sqrt {a \cos ^2(x)+a}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \sqrt {a \cos ^2(x)+a}dx}{2 a^2}-\frac {\sin (x) \cos (x)}{2 a \sqrt {a \cos ^2(x)+a}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3657

\(\displaystyle \frac {\sqrt {a \cos ^2(x)+a} \int \sqrt {\cos ^2(x)+1}dx}{2 a^2 \sqrt {\cos ^2(x)+1}}-\frac {\sin (x) \cos (x)}{2 a \sqrt {a \cos ^2(x)+a}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3656

\(\displaystyle \frac {E\left (\left .x+\frac {\pi }{2}\right |-1\right ) \sqrt {a \cos ^2(x)+a}}{2 a^2 \sqrt {\cos ^2(x)+1}}-\frac {\sin (x) \cos (x)}{2 a \sqrt {a \cos ^2(x)+a}}\)

Input:

Int[(a + a*Cos[x]^2)^(-3/2),x]
 

Output:

(Sqrt[a + a*Cos[x]^2]*EllipticE[Pi/2 + x, -1])/(2*a^2*Sqrt[1 + Cos[x]^2]) 
- (Cos[x]*Sin[x])/(2*a*Sqrt[a + a*Cos[x]^2])
 

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 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 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 [A] (verified)

Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.26

method	result	size
default	\(-\frac {\sqrt {-a \sin \left (x \right )^{4}+2 a \sin \left (x \right )^{2}}\, \left (\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\sin \left (x \right )^{2}+2}\, \operatorname {EllipticE}\left (\cos \left (x \right ), i\right )+\sin \left (x \right )^{2} \cos \left (x \right )\right )}{2 a \sqrt {-a \left (\cos \left (x \right )^{4}-1\right )}\, \sin \left (x \right ) \sqrt {a \left (1+\cos \left (x \right )^{2}\right )}}\)	\(78\)

Input:

int(1/(a+a*cos(x)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2/a*(-a*sin(x)^4+2*a*sin(x)^2)^(1/2)*((sin(x)^2)^(1/2)*(-sin(x)^2+2)^(1 
/2)*EllipticE(cos(x),I)+sin(x)^2*cos(x))/(-a*(cos(x)^4-1))^(1/2)/sin(x)/(a 
*(1+cos(x)^2))^(1/2)
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (47) = 94\).

Time = 0.11 (sec) , antiderivative size = 267, normalized size of antiderivative = 4.31 \[ \int \frac {1}{\left (a+a \cos ^2(x)\right )^{3/2}} \, dx=\frac {{\left ({\left (2 i \, \sqrt {2} - 3 i\right )} \cos \left (x\right )^{2} + 2 i \, \sqrt {2} - 3 i\right )} \sqrt {a} \sqrt {2 \, \sqrt {2} - 3} E(\arcsin \left (\sqrt {2 \, \sqrt {2} - 3} {\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right )}\right )\,|\,12 \, \sqrt {2} + 17) + {\left ({\left (-2 i \, \sqrt {2} + 3 i\right )} \cos \left (x\right )^{2} - 2 i \, \sqrt {2} + 3 i\right )} \sqrt {a} \sqrt {2 \, \sqrt {2} - 3} E(\arcsin \left (\sqrt {2 \, \sqrt {2} - 3} {\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right )}\right )\,|\,12 \, \sqrt {2} + 17) - 4 \, {\left ({\left (-i \, \sqrt {2} - 3 i\right )} \cos \left (x\right )^{2} - i \, \sqrt {2} - 3 i\right )} \sqrt {a} \sqrt {2 \, \sqrt {2} - 3} F(\arcsin \left (\sqrt {2 \, \sqrt {2} - 3} {\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right )}\right )\,|\,12 \, \sqrt {2} + 17) - 4 \, {\left ({\left (i \, \sqrt {2} + 3 i\right )} \cos \left (x\right )^{2} + i \, \sqrt {2} + 3 i\right )} \sqrt {a} \sqrt {2 \, \sqrt {2} - 3} F(\arcsin \left (\sqrt {2 \, \sqrt {2} - 3} {\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right )}\right )\,|\,12 \, \sqrt {2} + 17) - 2 \, \sqrt {a \cos \left (x\right )^{2} + a} \cos \left (x\right ) \sin \left (x\right )}{4 \, {\left (a^{2} \cos \left (x\right )^{2} + a^{2}\right )}} \] Input:

integrate(1/(a+a*cos(x)^2)^(3/2),x, algorithm="fricas")
 

Output:

1/4*(((2*I*sqrt(2) - 3*I)*cos(x)^2 + 2*I*sqrt(2) - 3*I)*sqrt(a)*sqrt(2*sqr 
t(2) - 3)*elliptic_e(arcsin(sqrt(2*sqrt(2) - 3)*(cos(x) + I*sin(x))), 12*s 
qrt(2) + 17) + ((-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) - 4*((-I*sqrt(2) - 3*I)*cos(x)^2 - I*sqrt(2) - 3*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) - 4*((I*sqrt(2) + 3*I)*cos(x)^2 + I*sqrt( 
2) + 3*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)*cos(x)*s 
in(x))/(a^2*cos(x)^2 + a^2)
 

Sympy [F]

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

integrate(1/(a+a*cos(x)**2)**(3/2),x)
 

Output:

Integral((a*cos(x)**2 + a)**(-3/2), x)
 

Maxima [F]

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

integrate(1/(a+a*cos(x)^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((a*cos(x)^2 + a)^(-3/2), x)
 

Giac [F]

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

integrate(1/(a+a*cos(x)^2)^(3/2),x, algorithm="giac")
 

Output:

integrate((a*cos(x)^2 + a)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+a \cos ^2(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (a\,{\cos \left (x\right )}^2+a\right )}^{3/2}} \,d x \] Input:

int(1/(a + a*cos(x)^2)^(3/2),x)
 

Output:

int(1/(a + a*cos(x)^2)^(3/2), x)
 

Reduce [F]

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

int(1/(a+a*cos(x)^2)^(3/2),x)
 

Output:

(sqrt(a)*int(sqrt(cos(x)**2 + 1)/(cos(x)**4 + 2*cos(x)**2 + 1),x))/a**2