\(\int (a+b \cos ^2(x))^{3/2} \, dx\) [68]

Optimal result

Integrand size = 12, antiderivative size = 123 \[ \int \left (a+b \cos ^2(x)\right )^{3/2} \, dx=\frac {2 (2 a+b) \sqrt {a+b \cos ^2(x)} E\left (\frac {\pi }{2}+x|-\frac {b}{a}\right )}{3 \sqrt {\frac {a+b \cos ^2(x)}{a}}}-\frac {a (a+b) \sqrt {\frac {a+b \cos ^2(x)}{a}} \operatorname {EllipticF}\left (\frac {\pi }{2}+x,-\frac {b}{a}\right )}{3 \sqrt {a+b \cos ^2(x)}}+\frac {1}{3} b \cos (x) \sqrt {a+b \cos ^2(x)} \sin (x) \] Output:

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

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00 \[ \int \left (a+b \cos ^2(x)\right )^{3/2} \, dx=\frac {8 \left (2 a^2+3 a b+b^2\right ) \sqrt {\frac {2 a+b+b \cos (2 x)}{a+b}} E\left (x\left |\frac {b}{a+b}\right .\right )-4 a (a+b) \sqrt {\frac {2 a+b+b \cos (2 x)}{a+b}} \operatorname {EllipticF}\left (x,\frac {b}{a+b}\right )+\sqrt {2} b (2 a+b+b \cos (2 x)) \sin (2 x)}{12 \sqrt {2 a+b+b \cos (2 x)}} \] Input:

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

Output:

(8*(2*a^2 + 3*a*b + b^2)*Sqrt[(2*a + b + b*Cos[2*x])/(a + b)]*EllipticE[x, 
 b/(a + b)] - 4*a*(a + b)*Sqrt[(2*a + b + b*Cos[2*x])/(a + b)]*EllipticF[x 
, b/(a + b)] + Sqrt[2]*b*(2*a + b + b*Cos[2*x])*Sin[2*x])/(12*Sqrt[2*a + b 
 + b*Cos[2*x]])
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {3042, 3659, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3659

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3651

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3657

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3656

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

\(\Big \downarrow \) 3662

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3661

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

Input:

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

Output:

((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])/3 + (b*Cos[x]*Sqrt[a + b*Cos[x]^2]*Sin[x])/3
 

Defintions of rubi rules used

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 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 3659

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*f*p)), x] + Sim 
p[1/(2*p)   Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(2*a 
+ b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[ 
a + b, 0] && GtQ[p, 1]
 

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]
 

Maple [A] (verified)

Time = 1.75 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.56

method	result	size
default	\(-\frac {-\frac {a^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \cos \left (x \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right )}{3}-\frac {a \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {\frac {a +b \cos \left (x \right )^{2}}{a}}\, \operatorname {EllipticF}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) b}{3}+\frac {4 \sqrt {\frac {a +b \cos \left (x \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, a^{2}}{3}+\frac {2 \sqrt {\frac {a +b \cos \left (x \right )^{2}}{a}}\, \operatorname {EllipticE}\left (\cos \left (x \right ), \sqrt {-\frac {b}{a}}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, a b}{3}+\frac {b^{2} \cos \left (x \right )^{5}}{3}+\frac {a b \cos \left (x \right )^{3}}{3}-\frac {b^{2} \cos \left (x \right )^{3}}{3}-\frac {\cos \left (x \right ) a b}{3}}{\sin \left (x \right ) \sqrt {a +b \cos \left (x \right )^{2}}}\)	\(192\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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