\(\int (4-3 \sin ^2(x))^{5/2} \, dx\) [109]

Optimal result

Integrand size = 12, antiderivative size = 57 \[ \int \left (4-3 \sin ^2(x)\right )^{5/2} \, dx=\frac {328 E\left (x\left |\frac {3}{4}\right .\right )}{15}-\frac {8 \operatorname {EllipticF}\left (x,\frac {3}{4}\right )}{3}+4 \cos (x) \sin (x) \sqrt {4-3 \sin ^2(x)}+\frac {3}{5} \cos (x) \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \] Output:

328/15*EllipticE(sin(x),1/2*3^(1/2))-8/3*InverseJacobiAM(x,1/2*3^(1/2))+4* 
cos(x)*sin(x)*(4-3*sin(x)^2)^(1/2)+3/5*cos(x)*sin(x)*(4-3*sin(x)^2)^(3/2)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84 \[ \int \left (4-3 \sin ^2(x)\right )^{5/2} \, dx=\frac {328 E\left (x\left |\frac {3}{4}\right .\right )}{15}-\frac {8 \operatorname {EllipticF}\left (x,\frac {3}{4}\right )}{3}+\frac {1}{80} \sqrt {10+6 \cos (2 x)} (110 \sin (2 x)+9 \sin (4 x)) \] Input:

Integrate[(4 - 3*Sin[x]^2)^(5/2),x]
 

Output:

(328*EllipticE[x, 3/4])/15 - (8*EllipticF[x, 3/4])/3 + (Sqrt[10 + 6*Cos[2* 
x]]*(110*Sin[2*x] + 9*Sin[4*x]))/80
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {3042, 3659, 27, 3042, 3649, 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.

\(\displaystyle \int \left (4-3 \sin ^2(x)\right )^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (4-3 \sin (x)^2\right )^{5/2}dx\)

\(\Big \downarrow \) 3659

\(\displaystyle \frac {1}{5} \int 4 \left (17-15 \sin ^2(x)\right ) \sqrt {4-3 \sin ^2(x)}dx+\frac {3}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{5} \int \left (17-15 \sin ^2(x)\right ) \sqrt {4-3 \sin ^2(x)}dx+\frac {3}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4}{5} \int \left (17-15 \sin (x)^2\right ) \sqrt {4-3 \sin (x)^2}dx+\frac {3}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\)

\(\Big \downarrow \) 3649

\(\displaystyle \frac {4}{5} \left (\frac {1}{3} \int \frac {3 \left (48-41 \sin ^2(x)\right )}{\sqrt {4-3 \sin ^2(x)}}dx+5 \sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\right )+\frac {3}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{5} \left (\int \frac {48-41 \sin ^2(x)}{\sqrt {4-3 \sin ^2(x)}}dx+5 \sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\right )+\frac {3}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4}{5} \left (\int \frac {48-41 \sin (x)^2}{\sqrt {4-3 \sin (x)^2}}dx+5 \sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\right )+\frac {3}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\)

\(\Big \downarrow \) 3651

\(\displaystyle \frac {4}{5} \left (-\frac {20}{3} \int \frac {1}{\sqrt {4-3 \sin ^2(x)}}dx+\frac {41}{3} \int \sqrt {4-3 \sin ^2(x)}dx+5 \sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\right )+\frac {3}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4}{5} \left (-\frac {20}{3} \int \frac {1}{\sqrt {4-3 \sin (x)^2}}dx+\frac {41}{3} \int \sqrt {4-3 \sin (x)^2}dx+5 \sqrt {4-3 \sin ^2(x)} \sin (x) \cos (x)\right )+\frac {3}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\)

\(\Big \downarrow \) 3656

\(\displaystyle \frac {4}{5} \left (-\frac {20}{3} \int \frac {1}{\sqrt {4-3 \sin (x)^2}}dx+\frac {82 E\left (x\left |\frac {3}{4}\right .\right )}{3}+5 \sin (x) \sqrt {4-3 \sin ^2(x)} \cos (x)\right )+\frac {3}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)\)

\(\Big \downarrow \) 3661

\(\displaystyle \frac {3}{5} \sin (x) \left (4-3 \sin ^2(x)\right )^{3/2} \cos (x)+\frac {4}{5} \left (-\frac {10 \operatorname {EllipticF}\left (x,\frac {3}{4}\right )}{3}+\frac {82 E\left (x\left |\frac {3}{4}\right .\right )}{3}+5 \sin (x) \sqrt {4-3 \sin ^2(x)} \cos (x)\right )\)

Input:

Int[(4 - 3*Sin[x]^2)^(5/2),x]
 

Output:

(3*Cos[x]*Sin[x]*(4 - 3*Sin[x]^2)^(3/2))/5 + (4*((82*EllipticE[x, 3/4])/3 
- (10*EllipticF[x, 3/4])/3 + 5*Cos[x]*Sin[x]*Sqrt[4 - 3*Sin[x]^2]))/5
 

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 3649

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)]^2), x_Symbol] :> Simp[(-B)*Cos[e + f*x]*Sin[e + f*x]*((a + b* 
Sin[e + f*x]^2)^p/(2*f*(p + 1))), x] + Simp[1/(2*(p + 1))   Int[(a + b*Sin[ 
e + f*x]^2)^(p - 1)*Simp[a*B + 2*a*A*(p + 1) + (2*A*b*(p + 1) + B*(b + 2*a* 
p + 2*b*p))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && G 
tQ[p, 0]
 

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. \(122\) vs. \(2(52)=104\).

Time = 3.92 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.16

method	result	size
default	\(-\frac {\sqrt {-\left (-4+3 \sin \left (x \right )^{2}\right ) \cos \left (x \right )^{2}}\, \left (-81 \cos \left (x \right )^{6} \sin \left (x \right )-234 \cos \left (x \right )^{4} \sin \left (x \right )+40 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {3 \cos \left (x \right )^{2}+1}\, \operatorname {EllipticF}\left (\sin \left (x \right ), \frac {\sqrt {3}}{2}\right )-328 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {3 \cos \left (x \right )^{2}+1}\, \operatorname {EllipticE}\left (\sin \left (x \right ), \frac {\sqrt {3}}{2}\right )-69 \cos \left (x \right )^{2} \sin \left (x \right )\right )}{15 \sqrt {3 \cos \left (x \right )^{4}+\cos \left (x \right )^{2}}\, \cos \left (x \right ) \sqrt {4-3 \sin \left (x \right )^{2}}}\)	\(123\)

Input:

int((4-3*sin(x)^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/15*(-(-4+3*sin(x)^2)*cos(x)^2)^(1/2)*(-81*cos(x)^6*sin(x)-234*cos(x)^4* 
sin(x)+40*(cos(x)^2)^(1/2)*(3*cos(x)^2+1)^(1/2)*EllipticF(sin(x),1/2*3^(1/ 
2))-328*(cos(x)^2)^(1/2)*(3*cos(x)^2+1)^(1/2)*EllipticE(sin(x),1/2*3^(1/2) 
)-69*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]

\[ \int \left (4-3 \sin ^2(x)\right )^{5/2} \, dx=\int { {\left (-3 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((4-3*sin(x)^2)^(5/2),x, algorithm="fricas")
 

Output:

integral((9*cos(x)^4 + 6*cos(x)^2 + 1)*sqrt(3*cos(x)^2 + 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (4-3 \sin ^2(x)\right )^{5/2} \, dx=\text {Timed out} \] Input:

integrate((4-3*sin(x)**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (4-3 \sin ^2(x)\right )^{5/2} \, dx=\int { {\left (-3 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((4-3*sin(x)^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate((-3*sin(x)^2 + 4)^(5/2), x)
 

Giac [F]

\[ \int \left (4-3 \sin ^2(x)\right )^{5/2} \, dx=\int { {\left (-3 \, \sin \left (x\right )^{2} + 4\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((4-3*sin(x)^2)^(5/2),x, algorithm="giac")
 

Output:

integrate((-3*sin(x)^2 + 4)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (4-3 \sin ^2(x)\right )^{5/2} \, dx=\int {\left (4-3\,{\sin \left (x\right )}^2\right )}^{5/2} \,d x \] Input:

int((4 - 3*sin(x)^2)^(5/2),x)
 

Output:

int((4 - 3*sin(x)^2)^(5/2), x)
 

Reduce [F]

\[ \int \left (4-3 \sin ^2(x)\right )^{5/2} \, dx=16 \left (\int \sqrt {-3 \sin \left (x \right )^{2}+4}d x \right )+9 \left (\int \sqrt {-3 \sin \left (x \right )^{2}+4}\, \sin \left (x \right )^{4}d x \right )-24 \left (\int \sqrt {-3 \sin \left (x \right )^{2}+4}\, \sin \left (x \right )^{2}d x \right ) \] Input:

int((4-3*sin(x)^2)^(5/2),x)
 

Output:

16*int(sqrt( - 3*sin(x)**2 + 4),x) + 9*int(sqrt( - 3*sin(x)**2 + 4)*sin(x) 
**4,x) - 24*int(sqrt( - 3*sin(x)**2 + 4)*sin(x)**2,x)