\(\int (4-3 \sin ^2(x))^{3/2} \, dx\) [110]

Optimal result

Integrand size = 12, antiderivative size = 36 \[ \int \left (4-3 \sin ^2(x)\right )^{3/2} \, dx=\frac {20 E\left (x\left |\frac {3}{4}\right .\right )}{3}-\frac {2 \operatorname {EllipticF}\left (x,\frac {3}{4}\right )}{3}+\cos (x) \sin (x) \sqrt {4-3 \sin ^2(x)} \] Output:

20/3*EllipticE(sin(x),1/2*3^(1/2))-2/3*InverseJacobiAM(x,1/2*3^(1/2))+cos( 
x)*sin(x)*(4-3*sin(x)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \left (4-3 \sin ^2(x)\right )^{3/2} \, dx=\frac {1}{12} \left (80 E\left (x\left |\frac {3}{4}\right .\right )-8 \operatorname {EllipticF}\left (x,\frac {3}{4}\right )+3 \sqrt {10+6 \cos (2 x)} \sin (2 x)\right ) \] Input:

Integrate[(4 - 3*Sin[x]^2)^(3/2),x]
 

Output:

(80*EllipticE[x, 3/4] - 8*EllipticF[x, 3/4] + 3*Sqrt[10 + 6*Cos[2*x]]*Sin[ 
2*x])/12
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, 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.

\(\displaystyle \int \left (4-3 \sin ^2(x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (4-3 \sin (x)^2\right )^{3/2}dx\)

\(\Big \downarrow \) 3659

\(\displaystyle \frac {1}{3} \int \frac {6 \left (6-5 \sin ^2(x)\right )}{\sqrt {4-3 \sin ^2(x)}}dx+\sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {6-5 \sin ^2(x)}{\sqrt {4-3 \sin ^2(x)}}dx+\sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int \frac {6-5 \sin (x)^2}{\sqrt {4-3 \sin (x)^2}}dx+\sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\)

\(\Big \downarrow \) 3651

\(\displaystyle 2 \left (\frac {5}{3} \int \sqrt {4-3 \sin ^2(x)}dx-\frac {2}{3} \int \frac {1}{\sqrt {4-3 \sin ^2(x)}}dx\right )+\sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \left (\frac {5}{3} \int \sqrt {4-3 \sin (x)^2}dx-\frac {2}{3} \int \frac {1}{\sqrt {4-3 \sin (x)^2}}dx\right )+\sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\)

\(\Big \downarrow \) 3656

\(\displaystyle 2 \left (\frac {10 E\left (x\left |\frac {3}{4}\right .\right )}{3}-\frac {2}{3} \int \frac {1}{\sqrt {4-3 \sin (x)^2}}dx\right )+\sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\)

\(\Big \downarrow \) 3661

\(\displaystyle \sin (x) \sqrt {4-3 \sin ^2(x)} \cos (x)+2 \left (\frac {10 E\left (x\left |\frac {3}{4}\right .\right )}{3}-\frac {\operatorname {EllipticF}\left (x,\frac {3}{4}\right )}{3}\right )\)

Input:

Int[(4 - 3*Sin[x]^2)^(3/2),x]
 

Output:

2*((10*EllipticE[x, 3/4])/3 - EllipticF[x, 3/4]/3) + Cos[x]*Sin[x]*Sqrt[4 
- 3*Sin[x]^2]
 

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)

Leaf count of result is larger than twice the leaf count of optimal. \(114\) vs. \(2(35)=70\).

Time = 1.70 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.19

method	result	size
default	\(-\frac {\sqrt {-\left (-4+3 \sin \left (x \right )^{2}\right ) \cos \left (x \right )^{2}}\, \left (-9 \cos \left (x \right )^{4} \sin \left (x \right )+2 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {3 \cos \left (x \right )^{2}+1}\, \operatorname {EllipticF}\left (\sin \left (x \right ), \frac {\sqrt {3}}{2}\right )-20 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {3 \cos \left (x \right )^{2}+1}\, \operatorname {EllipticE}\left (\sin \left (x \right ), \frac {\sqrt {3}}{2}\right )-3 \cos \left (x \right )^{2} \sin \left (x \right )\right )}{3 \sqrt {3 \cos \left (x \right )^{4}+\cos \left (x \right )^{2}}\, \cos \left (x \right ) \sqrt {4-3 \sin \left (x \right )^{2}}}\)	\(115\)

Input:

int((4-3*sin(x)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*(-(-4+3*sin(x)^2)*cos(x)^2)^(1/2)*(-9*cos(x)^4*sin(x)+2*(cos(x)^2)^(1 
/2)*(3*cos(x)^2+1)^(1/2)*EllipticF(sin(x),1/2*3^(1/2))-20*(cos(x)^2)^(1/2) 
*(3*cos(x)^2+1)^(1/2)*EllipticE(sin(x),1/2*3^(1/2))-3*cos(x)^2*sin(x))/(3* 
cos(x)^4+cos(x)^2)^(1/2)/cos(x)/(4-3*sin(x)^2)^(1/2)
 

Fricas [F]

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

integrate((4-3*sin(x)^2)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \left (4-3 \sin ^2(x)\right )^{3/2} \, dx=\int \left (4 - 3 \sin ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \] Input:

integrate((4-3*sin(x)**2)**(3/2),x)
 

Output:

Integral((4 - 3*sin(x)**2)**(3/2), x)
 

Maxima [F]

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

integrate((4-3*sin(x)^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((-3*sin(x)^2 + 4)^(3/2), x)
 

Giac [F]

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

integrate((4-3*sin(x)^2)^(3/2),x, algorithm="giac")
 

Output:

integrate((-3*sin(x)^2 + 4)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (4-3 \sin ^2(x)\right )^{3/2} \, dx=\int {\left (4-3\,{\sin \left (x\right )}^2\right )}^{3/2} \,d x \] Input:

int((4 - 3*sin(x)^2)^(3/2),x)
 

Output:

int((4 - 3*sin(x)^2)^(3/2), x)
 

Reduce [F]

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

int((4-3*sin(x)^2)^(3/2),x)
 

Output:

4*int(sqrt( - 3*sin(x)**2 + 4),x) - 3*int(sqrt( - 3*sin(x)**2 + 4)*sin(x)* 
*2,x)