\(\int (1+\cos ^2(x))^{3/2} \, dx\) [48]

Optimal result

Integrand size = 10, antiderivative size = 43 \[ \int \left (1+\cos ^2(x)\right )^{3/2} \, dx=2 E\left (\left .\frac {\pi }{2}+x\right |-1\right )-\frac {2}{3} \operatorname {EllipticF}\left (\frac {\pi }{2}+x,-1\right )+\frac {1}{3} \cos (x) \sqrt {1+\cos ^2(x)} \sin (x) \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.91 \[ \int \left (1+\cos ^2(x)\right )^{3/2} \, dx=\frac {24 E\left (x\left |\frac {1}{2}\right .\right )-4 \operatorname {EllipticF}\left (x,\frac {1}{2}\right )+\sqrt {3+\cos (2 x)} \sin (2 x)}{6 \sqrt {2}} \] Input:

Integrate[(1 + Cos[x]^2)^(3/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 3659, 27, 3042, 3651, 3042, 3656, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3659

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3651

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3656

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

\(\Big \downarrow \) 3661

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

Input:

Int[(1 + Cos[x]^2)^(3/2),x]
 

Output:

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (32 ) = 64\).

Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.35

method	result	size
default	\(\frac {\sqrt {\left (1+\cos \left (x \right )^{2}\right ) \sin \left (x \right )^{2}}\, \left (-\sin \left (x \right )^{4} \cos \left (x \right )+2 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\sin \left (x \right )^{2}+2}\, \operatorname {EllipticF}\left (\cos \left (x \right ), i\right )-6 \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \sqrt {-\sin \left (x \right )^{2}+2}\, \operatorname {EllipticE}\left (\cos \left (x \right ), i\right )+2 \sin \left (x \right )^{2} \cos \left (x \right )\right )}{3 \sqrt {1-\cos \left (x \right )^{4}}\, \sin \left (x \right ) \sqrt {1+\cos \left (x \right )^{2}}}\)	\(101\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((cos(x)^2 + 1)^(3/2), x)
 

Sympy [F]

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

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

Output:

Integral((cos(x)**2 + 1)**(3/2), x)
 

Maxima [F]

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

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

Output:

integrate((cos(x)^2 + 1)^(3/2), x)
 

Giac [F]

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

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

Output:

integrate((cos(x)^2 + 1)^(3/2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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