\(\int (a+a \cos ^2(x))^{3/2} \, dx\) [54]

Optimal result

Integrand size = 12, antiderivative size = 94 \[ \int \left (a+a \cos ^2(x)\right )^{3/2} \, dx=\frac {2 a \sqrt {a+a \cos ^2(x)} E\left (\left .\frac {\pi }{2}+x\right |-1\right )}{\sqrt {1+\cos ^2(x)}}-\frac {2 a^2 \sqrt {1+\cos ^2(x)} \operatorname {EllipticF}\left (\frac {\pi }{2}+x,-1\right )}{3 \sqrt {a+a \cos ^2(x)}}+\frac {1}{3} a \cos (x) \sqrt {a+a \cos ^2(x)} \sin (x) \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.61 \[ \int \left (a+a \cos ^2(x)\right )^{3/2} \, dx=\frac {a \sqrt {a \left (1+\cos ^2(x)\right )} \left (24 E\left (x\left |\frac {1}{2}\right .\right )-4 \operatorname {EllipticF}\left (x,\frac {1}{2}\right )+\sqrt {3+\cos (2 x)} \sin (2 x)\right )}{6 \sqrt {3+\cos (2 x)}} \] Input:

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

Output:

(a*Sqrt[a*(1 + Cos[x]^2)]*(24*EllipticE[x, 1/2] - 4*EllipticF[x, 1/2] + Sq 
rt[3 + Cos[2*x]]*Sin[2*x]))/(6*Sqrt[3 + Cos[2*x]])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3659, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3659

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3651

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3657

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3656

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

\(\Big \downarrow \) 3662

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3661

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 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 = 0.37 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.15

method	result	size
default	\(\frac {\sqrt {a \left (1+\cos \left (x \right )^{2}\right ) \sin \left (x \right )^{2}}\, a^{2} \left (-\sin \left (x \right )^{4} \cos \left (x \right )+2 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\sin \left (x \right )^{2}+2}\, \operatorname {EllipticF}\left (\cos \left (x \right ), i\right )-6 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\sin \left (x \right )^{2}+2}\, \operatorname {EllipticE}\left (\cos \left (x \right ), i\right )+2 \sin \left (x \right )^{2} \cos \left (x \right )\right )}{3 \sqrt {-a \left (\cos \left (x \right )^{4}-1\right )}\, \sin \left (x \right ) \sqrt {a \left (1+\cos \left (x \right )^{2}\right )}}\)	\(108\)

Input:

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

Output:

1/3*(a*(1+cos(x)^2)*sin(x)^2)^(1/2)*a^2*(-sin(x)^4*cos(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)+2*sin(x)^2*cos(x))/(-a*(cos(x)^4-1))^(1/2)/sin 
(x)/(a*(1+cos(x)^2))^(1/2)
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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