\(\int \frac {1}{(a+b \sin ^2(e+f x))^{3/2}} \, dx\) [460]

Optimal result

Integrand size = 16, antiderivative size = 102 \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {b \cos (e+f x) \sin (e+f x)}{a (a+b) f \sqrt {a+b \sin ^2(e+f x)}}+\frac {E\left (e+f x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(e+f x)}}{a (a+b) f \sqrt {\frac {a+b \sin ^2(e+f x)}{a}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {2 a \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )+\sqrt {2} b \sin (2 (e+f x))}{2 a (a+b) f \sqrt {2 a+b-b \cos (2 (e+f x))}} \] Input:

Integrate[(a + b*Sin[e + f*x]^2)^(-3/2),x]
 

Output:

(2*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a)] + S 
qrt[2]*b*Sin[2*(e + f*x)])/(2*a*(a + b)*f*Sqrt[2*a + b - b*Cos[2*(e + f*x) 
]])
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3042, 3663, 25, 3042, 3657, 3042, 3656}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3663

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3657

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3656

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

Input:

Int[(a + b*Sin[e + f*x]^2)^(-3/2),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, 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 3663

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*a*f*(p + 1)*(a 
+ b))), x] + Simp[1/(2*a*(p + 1)*(a + b))   Int[(a + b*Sin[e + f*x]^2)^(p + 
 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] 
 /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {a \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (f x +e \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )+\cos \left (f x +e \right )^{2} b \sin \left (f x +e \right )}{a \left (a +b \right ) \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )^{2}}\, f}\)	\(103\)

Input:

int(1/(a+b*sin(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

(a*(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))+cos(f*x+e)^2*b*sin(f*x+e))/a/(a+b)/cos(f*x+e)/(a+b*sin( 
f*x+e)^2)^(1/2)/f
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 938, normalized size of antiderivative = 9.20 \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="fricas")
 

Output:

-1/2*(2*sqrt(-b*cos(f*x + e)^2 + a + b)*b^3*cos(f*x + e)*sin(f*x + e) - (2 
*(I*b^3*cos(f*x + e)^2 - I*a*b^2 - I*b^3)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - 
 (2*I*a^2*b + 3*I*a*b^2 + I*b^3 + (-2*I*a*b^2 - I*b^3)*cos(f*x + e)^2)*sqr 
t(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqr 
t((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) - (2 
*(-I*b^3*cos(f*x + e)^2 + I*a*b^2 + I*b^3)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) 
- (-2*I*a^2*b - 3*I*a*b^2 - I*b^3 + (2*I*a*b^2 + I*b^3)*cos(f*x + e)^2)*sq 
rt(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sq 
rt((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) + 2 
*(2*(-I*a^2*b - 2*I*a*b^2 - I*b^3 + (I*a*b^2 + I*b^3)*cos(f*x + e)^2)*sqrt 
(-b)*sqrt((a^2 + a*b)/b^2) + (2*I*a^3 + 3*I*a^2*b + I*a*b^2 + (-2*I*a^2*b 
- I*a*b^2)*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((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)*sq 
rt((a^2 + a*b)/b^2))/b^2) + 2*(2*(I*a^2*b + 2*I*a*b^2 + I*b^3 + (-I*a*b^2 
- I*b^3)*cos(f*x + e)^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) + (-2*I*a^3 - 3*I* 
a^2*b - I*a*b^2 + (2*I*a^2*b + I*a*b^2)*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((2*b*sqrt(...
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(a+b*sin(f*x+e)**2)**(3/2),x)
 

Output:

Integral((a + b*sin(e + f*x)**2)**(-3/2), x)
 

Maxima [F]

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

integrate(1/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*sin(f*x + e)^2 + a)^(-3/2), x)
 

Giac [F]

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

integrate(1/(a+b*sin(f*x+e)^2)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*sin(f*x + e)^2 + a)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \] Input:

int(1/(a + b*sin(e + f*x)^2)^(3/2),x)
 

Output:

int(1/(a + b*sin(e + f*x)^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\sqrt {\sin \left (f x +e \right )^{2} b +a}}{\sin \left (f x +e \right )^{4} b^{2}+2 \sin \left (f x +e \right )^{2} a b +a^{2}}d x \] Input:

int(1/(a+b*sin(f*x+e)^2)^(3/2),x)
 

Output:

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