\(\int \frac {\sqrt {a+b \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx\) [776]

Optimal result

Integrand size = 29, antiderivative size = 409 \[ \int \frac {\sqrt {a+b \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 (a-b) \sqrt {a+b} E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} (b c-a d) f}+\frac {2 (a-b) \sqrt {a+b} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(c-d) \sqrt {c+d} (b c-a d) f} \] Output:

-2*(a-b)*(a+b)^(1/2)*EllipticE((c+d)^(1/2)*(a+b*sin(f*x+e))^(1/2)/(a+b)^(1 
/2)/(c+d*sin(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*sec(f*x+e)*((- 
a*d+b*c)*(1-sin(f*x+e))/(a+b)/(c+d*sin(f*x+e)))^(1/2)*(-(-a*d+b*c)*(1+sin( 
f*x+e))/(a-b)/(c+d*sin(f*x+e)))^(1/2)*(c+d*sin(f*x+e))/(c-d)/(c+d)^(1/2)/( 
-a*d+b*c)/f+2*(a-b)*(a+b)^(1/2)*EllipticF((c+d)^(1/2)*(a+b*sin(f*x+e))^(1/ 
2)/(a+b)^(1/2)/(c+d*sin(f*x+e))^(1/2),((a+b)*(c-d)/(a-b)/(c+d))^(1/2))*sec 
(f*x+e)*((-a*d+b*c)*(1-sin(f*x+e))/(a+b)/(c+d*sin(f*x+e)))^(1/2)*(-(-a*d+b 
*c)*(1+sin(f*x+e))/(a-b)/(c+d*sin(f*x+e)))^(1/2)*(c+d*sin(f*x+e))/(c-d)/(c 
+d)^(1/2)/(-a*d+b*c)/f
 

Mathematica [A] (warning: unable to verify)

Time = 11.36 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {a+b \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 \left (-((b c-a d) \cos (e+f x))-\frac {\sqrt {2} \sqrt {\frac {a-b}{a+b}} (a+b) (c+d) \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {a-b}{a+b}} \cos \left (\frac {1}{4} (2 e+\pi +2 f x)\right )}{\sqrt {\frac {a+b \sin (e+f x)}{a+b}}}\right )|\frac {2 (-b c+a d)}{(a-b) (c+d)}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \sqrt {\frac {(a+b) (c+d \sin (e+f x))}{(c+d) (a+b \sin (e+f x))}}}{\sqrt {\frac {(a+b) (1+\sin (e+f x))}{a+b \sin (e+f x)}}}\right )}{(c-d) (c+d) f \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \] Input:

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

Output:

(2*(-((b*c - a*d)*Cos[e + f*x]) - (Sqrt[2]*Sqrt[(a - b)/(a + b)]*(a + b)*( 
c + d)*Cos[(2*e - Pi + 2*f*x)/4]*EllipticE[ArcSin[(Sqrt[(a - b)/(a + b)]*C 
os[(2*e + Pi + 2*f*x)/4])/Sqrt[(a + b*Sin[e + f*x])/(a + b)]], (2*(-(b*c) 
+ a*d))/((a - b)*(c + d))]*Sqrt[(a + b*Sin[e + f*x])/(a + b)]*Sqrt[((a + b 
)*(c + d*Sin[e + f*x]))/((c + d)*(a + b*Sin[e + f*x]))])/Sqrt[((a + b)*(1 
+ Sin[e + f*x]))/(a + b*Sin[e + f*x])]))/((c - d)*(c + d)*f*Sqrt[a + b*Sin 
[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3274, 3042, 3297, 3475}

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.

\(\displaystyle \int \frac {\sqrt {a+b \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {a+b \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}}dx\)

\(\Big \downarrow \) 3274

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3297

\(\displaystyle \frac {(b c-a d) \int \frac {\sin (e+f x)+1}{\sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}dx}{c-d}+\frac {2 (a-b) \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}\)

\(\Big \downarrow \) 3475

\(\displaystyle \frac {2 (a-b) \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right ),\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}-\frac {2 (a-b) \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} E\left (\arcsin \left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{f (c-d) \sqrt {c+d} (b c-a d)}\)

Input:

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

Output:

(-2*(a - b)*Sqrt[a + b]*EllipticE[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e + f 
*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - b)* 
(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*(c + 
 d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c + d 
*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d)*f 
) + (2*(a - b)*Sqrt[a + b]*EllipticF[ArcSin[(Sqrt[c + d]*Sqrt[a + b*Sin[e 
+ f*x]])/(Sqrt[a + b]*Sqrt[c + d*Sin[e + f*x]])], ((a + b)*(c - d))/((a - 
b)*(c + d))]*Sec[e + f*x]*Sqrt[((b*c - a*d)*(1 - Sin[e + f*x]))/((a + b)*( 
c + d*Sin[e + f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Sin[e + f*x]))/((a - b)*(c 
+ d*Sin[e + f*x])))]*(c + d*Sin[e + f*x]))/((c - d)*Sqrt[c + d]*(b*c - a*d 
)*f)
 

Defintions of rubi rules used

rule 3042

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

rule 3274

Int[Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]/((a_.) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)])^(3/2), x_Symbol] :> Simp[(c - d)/(a - b)   Int[1/(Sqrt[a + b*Si 
n[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x], x] - Simp[(b*c - a*d)/(a - b) 
Int[(1 + Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]] 
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - 
 b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3297

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_ 
.) + (f_.)*(x_)]]), x_Symbol] :> Simp[2*((c + d*Sin[e + f*x])/(f*(b*c - a*d 
)*Rt[(c + d)/(a + b), 2]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 - Sin[e + f*x] 
)/((a + b)*(c + d*Sin[e + f*x])))]*Sqrt[(-(b*c - a*d))*((1 + Sin[e + f*x])/ 
((a - b)*(c + d*Sin[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)/(a + b), 2]*(S 
qrt[a + b*Sin[e + f*x]]/Sqrt[c + d*Sin[e + f*x]])], (a + b)*((c - d)/((a - 
b)*(c + d)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && N 
eQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/(a + b)]
 

rule 3475

Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_) + (b_.)*sin[(e_.) + (f_.) 
*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim 
p[-2*A*(c - d)*((a + b*Sin[e + f*x])/(f*(b*c - a*d)^2*Rt[(a + b)/(c + d), 2 
]*Cos[e + f*x]))*Sqrt[(b*c - a*d)*((1 + Sin[e + f*x])/((c - d)*(a + b*Sin[e 
 + f*x])))]*Sqrt[(-(b*c - a*d))*((1 - Sin[e + f*x])/((c + d)*(a + b*Sin[e + 
 f*x])))]*EllipticE[ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]] 
/Sqrt[a + b*Sin[e + f*x]])], (a - b)*((c + d)/((a + b)*(c - d)))], x] /; Fr 
eeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] 
&& NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(a + b)/(c + d)]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(41872\) vs. \(2(379)=758\).

Time = 4.90 (sec) , antiderivative size = 41873, normalized size of antiderivative = 102.38

method	result	size
default	\(\text {Expression too large to display}\)	\(41873\)

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Output:

integral(-sqrt(b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)/(d^2*cos(f*x + 
 e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2), x)
 

Sympy [F]

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

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

Output:

Integral(sqrt(a + b*sin(e + f*x))/(c + d*sin(e + f*x))**(3/2), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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