\(\int \frac {1}{(a+b \cos ^2(x))^{3/2}} \, dx\) [71]

Optimal result

Integrand size = 12, antiderivative size = 79 \[ \int \frac {1}{\left (a+b \cos ^2(x)\right )^{3/2}} \, dx=\frac {\sqrt {a+b \cos ^2(x)} E\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{a (a+b) \sqrt {\frac {a+b \cos ^2(x)}{a}}}-\frac {b \cos (x) \sin (x)}{a (a+b) \sqrt {a+b \cos ^2(x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+b \cos ^2(x)\right )^{3/2}} \, dx=\frac {2 (a+b) \sqrt {\frac {2 a+b+b \cos (2 x)}{a+b}} E\left (x\left |\frac {b}{a+b}\right .\right )-\sqrt {2} b \sin (2 x)}{2 a (a+b) \sqrt {2 a+b+b \cos (2 x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3663

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3657

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3656

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

Input:

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

Output:

(Sqrt[a + b*Cos[x]^2]*EllipticE[Pi/2 + x, -(b/a)])/(a*(a + b)*Sqrt[1 + (b* 
Cos[x]^2)/a]) - (b*Cos[x]*Sin[x])/(a*(a + b)*Sqrt[a + b*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 = 0.46 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\frac {b \sin \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, a \operatorname {EllipticE}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right )+b \cos \left (x \right ) \sin \left (x \right )^{2}}{a \left (a +b \right ) \sin \left (x \right ) \sqrt {a +b \cos \left (x \right )^{2}}}\)	\(73\)

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 775, normalized size of antiderivative = 9.81 \[ \int \frac {1}{\left (a+b \cos ^2(x)\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(2*sqrt(b*cos(x)^2 + a)*b^3*cos(x)*sin(x) + (2*I*a^2*b + I*a*b^2 + (2 
*I*a*b^2 + I*b^3)*cos(x)^2 - 2*(I*b^3*cos(x)^2 + I*a*b^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(ar 
csin(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) + (-2*I* 
a^2*b - I*a*b^2 + (-2*I*a*b^2 - I*b^3)*cos(x)^2 - 2*(-I*b^3*cos(x)^2 - I*a 
*b^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) + 2*(-2*I*a^3 - 3*I*a^2*b - I*a*b^2 + (-2*I*a^2*b - 3*I*a*b^ 
2 - I*b^3)*cos(x)^2 + 2*(-I*a*b^2*cos(x)^2 - I*a^2*b)*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) + 2*(2*I*a^3 
 + 3*I*a^2*b + I*a*b^2 + (2*I*a^2*b + 3*I*a*b^2 + I*b^3)*cos(x)^2 + 2*(I*a 
*b^2*cos(x)^2 + I*a^2*b)*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))/(a^3*b^2 + a^2*b^3 + (a^2*b^3 + a*b^4)*c 
os(x)^2)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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