\(\int \frac {\sin ^2(e+f x)}{(a+b \sin ^2(e+f x))^{5/2}} \, dx\) [118]

Optimal result

Integrand size = 25, antiderivative size = 223 \[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {\cos (e+f x) \sin (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {(a-b) \cos (e+f x) \sin (e+f x)}{3 a (a+b)^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(a-b) E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{3 a b (a+b)^2 f \sqrt {\frac {a+b \sin ^2(e+f x)}{a}}}+\frac {\operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right ) \sqrt {\frac {a+b \sin ^2(e+f x)}{a}}}{3 b (a+b) f \sqrt {a+b \sin ^2(e+f x)}} \] Output:

-1/3*cos(f*x+e)*sin(f*x+e)/(a+b)/f/(a+b*sin(f*x+e)^2)^(3/2)-1/3*(a-b)*cos( 
f*x+e)*sin(f*x+e)/a/(a+b)^2/f/(a+b*sin(f*x+e)^2)^(1/2)-1/3*(a-b)*EllipticE 
(sin(f*x+e),(-b/a)^(1/2))*(a+b*sin(f*x+e)^2)^(1/2)/a/b/(a+b)^2/f/((a+b*sin 
(f*x+e)^2)/a)^(1/2)+1/3*InverseJacobiAM(f*x+e,(-b/a)^(1/2))*((a+b*sin(f*x+ 
e)^2)/a)^(1/2)/b/(a+b)/f/(a+b*sin(f*x+e)^2)^(1/2)
 

Mathematica [A] (verified)

Time = 2.18 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.78 \[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {-2 a^2 (a-b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} E\left (e+f x\left |-\frac {b}{a}\right .\right )+2 a^2 (a+b) \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )-\sqrt {2} b \left (4 a^2+a b-b^2+b (-a+b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{6 a b (a+b)^2 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \] Input:

Integrate[Sin[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]
 

Output:

(-2*a^2*(a - b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*EllipticE[e + f*x 
, -(b/a)] + 2*a^2*(a + b)*((2*a + b - b*Cos[2*(e + f*x)])/a)^(3/2)*Ellipti 
cF[e + f*x, -(b/a)] - Sqrt[2]*b*(4*a^2 + a*b - b^2 + b*(-a + b)*Cos[2*(e + 
 f*x)])*Sin[2*(e + f*x)])/(6*a*b*(a + b)^2*f*(2*a + b - b*Cos[2*(e + f*x)] 
)^(3/2))
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 3652, 3042, 3652, 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.

\(\displaystyle \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (e+f x)^2}{\left (a+b \sin (e+f x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 3652

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3652

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3651

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3657

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3656

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

\(\Big \downarrow \) 3662

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3661

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

Input:

Int[Sin[e + f*x]^2/(a + b*Sin[e + f*x]^2)^(5/2),x]
 

Output:

-1/3*(Cos[e + f*x]*Sin[e + f*x])/((a + b)*f*(a + b*Sin[e + f*x]^2)^(3/2)) 
+ (-(((a - b)*Cos[e + f*x]*Sin[e + f*x])/((a + b)*f*Sqrt[a + b*Sin[e + f*x 
]^2])) + (-((a*(a - b)*EllipticE[e + f*x, -(b/a)]*Sqrt[a + b*Sin[e + f*x]^ 
2])/(b*f*Sqrt[1 + (b*Sin[e + f*x]^2)/a])) + (a^2*(a + b)*EllipticF[e + f*x 
, -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*f*Sqrt[a + b*Sin[e + f*x]^2]) 
)/(a*(a + b)))/(3*a*(a + b))
 

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 3652

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b - a*B))*Cos[e + f*x]*Sin[e + f*x 
]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(a + b)*(p + 1))), x] - Simp[1/(2* 
a*(a + b)*(p + 1))   Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*( 
p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; 
 FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]
 

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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(482\) vs. \(2(208)=416\).

Time = 3.21 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.17

method	result	size
default	\(\frac {\left (b^{2} a -b^{3}\right ) \cos \left (f x +e \right )^{4} \sin \left (f x +e \right )+\left (-2 a^{2} b -b^{2} a +b^{3}\right ) \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, a b \left (\operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +\operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -\operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +\operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b \right ) \cos \left (f x +e \right )^{2}+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b +\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}}{3 \left (a +b \sin \left (f x +e \right )^{2}\right )^{\frac {3}{2}} \left (a +b \right )^{2} a b \cos \left (f x +e \right ) f}\)	\(483\)

Input:

int(sin(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/3*((a*b^2-b^3)*cos(f*x+e)^4*sin(f*x+e)+(-2*a^2*b-a*b^2+b^3)*cos(f*x+e)^2 
*sin(f*x+e)-(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*a*b*(El 
lipticF(sin(f*x+e),(-b/a)^(1/2))*a+EllipticF(sin(f*x+e),(-b/a)^(1/2))*b-El 
lipticE(sin(f*x+e),(-b/a)^(1/2))*a+EllipticE(sin(f*x+e),(-b/a)^(1/2))*b)*c 
os(f*x+e)^2+(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*Ellipti 
cF(sin(f*x+e),(-b/a)^(1/2))*a^3+2*(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+ 
(a+b)/a)^(1/2)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*a^2*b+(cos(f*x+e)^2)^(1/ 
2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticF(sin(f*x+e),(-b/a)^(1/2))*a* 
b^2-(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^(1/2)*EllipticE(sin(f 
*x+e),(-b/a)^(1/2))*a^3+(cos(f*x+e)^2)^(1/2)*(-b/a*cos(f*x+e)^2+(a+b)/a)^( 
1/2)*EllipticE(sin(f*x+e),(-b/a)^(1/2))*a*b^2)/(a+b*sin(f*x+e)^2)^(3/2)/(a 
+b)^2/a/b/cos(f*x+e)/f
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 1400, normalized size of antiderivative = 6.28 \[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(sin(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="fricas")
 

Output:

1/6*((2*((-I*a*b^3 + I*b^4)*cos(f*x + e)^4 - I*a^3*b - I*a^2*b^2 + I*a*b^3 
 + I*b^4 - 2*(-I*a^2*b^2 + I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b 
)/b^2) - ((2*I*a^2*b^2 - I*a*b^3 - I*b^4)*cos(f*x + e)^4 + 2*I*a^4 + 3*I*a 
^3*b - I*a^2*b^2 - 3*I*a*b^3 - I*b^4 + 2*(-2*I*a^3*b - I*a^2*b^2 + 2*I*a*b 
^3 + I*b^4)*cos(f*x + e)^2)*sqrt(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2* 
a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)* 
(cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*s 
qrt((a^2 + a*b)/b^2))/b^2) + (2*((I*a*b^3 - I*b^4)*cos(f*x + e)^4 + I*a^3* 
b + I*a^2*b^2 - I*a*b^3 - I*b^4 - 2*(I*a^2*b^2 - I*b^4)*cos(f*x + e)^2)*sq 
rt(-b)*sqrt((a^2 + a*b)/b^2) - ((-2*I*a^2*b^2 + I*a*b^3 + I*b^4)*cos(f*x + 
 e)^4 - 2*I*a^4 - 3*I*a^3*b + I*a^2*b^2 + 3*I*a*b^3 + I*b^4 + 2*(2*I*a^3*b 
 + I*a^2*b^2 - 2*I*a*b^3 - I*b^4)*cos(f*x + e)^2)*sqrt(-b))*sqrt((2*b*sqrt 
((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a* 
b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b 
^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) - 4*(((I*a*b^3 + I*b^4)*c 
os(f*x + e)^4 + I*a^3*b + 3*I*a^2*b^2 + 3*I*a*b^3 + I*b^4 + 2*(-I*a^2*b^2 
- 2*I*a*b^3 - I*b^4)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) + ((-2 
*I*a^2*b^2 - I*a*b^3)*cos(f*x + e)^4 - 2*I*a^4 - 5*I*a^3*b - 4*I*a^2*b^2 - 
 I*a*b^3 + 2*(2*I*a^3*b + 3*I*a^2*b^2 + I*a*b^3)*cos(f*x + e)^2)*sqrt(-b)) 
*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_f(arcsin(sqrt((...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(sin(f*x+e)**2/(a+b*sin(f*x+e)**2)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(sin(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate(sin(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(sin(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x, algorithm="giac")
 

Output:

integrate(sin(f*x + e)^2/(b*sin(f*x + e)^2 + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^2}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \] Input:

int(sin(e + f*x)^2/(a + b*sin(e + f*x)^2)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

int(sin(e + f*x)^2/(a + b*sin(e + f*x)^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\sin ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right )^{2} b +a}\, \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{6} b^{3}+3 \sin \left (f x +e \right )^{4} a \,b^{2}+3 \sin \left (f x +e \right )^{2} a^{2} b +a^{3}}d x \] Input:

int(sin(f*x+e)^2/(a+b*sin(f*x+e)^2)^(5/2),x)
 

Output:

int((sqrt(sin(e + f*x)**2*b + a)*sin(e + f*x)**2)/(sin(e + f*x)**6*b**3 + 
3*sin(e + f*x)**4*a*b**2 + 3*sin(e + f*x)**2*a**2*b + a**3),x)