\(\int \frac {\sqrt {c+d \sin (e+f x)}}{(a+b \sin (e+f x))^{3/2}} \, dx\) [798]

Optimal result

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

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

Mathematica [A] (warning: unable to verify)

Time = 8.81 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+b \sin (e+f x))^{3/2}} \, dx=-\frac {2 \sqrt {2} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {a-b}{a+b}} \cos \left (\frac {1}{4} (2 e+\pi +2 f x)\right )}{\sqrt {\frac {a+b \sin (e+f x)}{a+b}}}\right )|\frac {2 (-b c+a d)}{(a-b) (c+d)}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \sqrt {c+d \sin (e+f x)}}{\sqrt {\frac {a-b}{a+b}} f \sqrt {\frac {(a+b) (1+\sin (e+f x))}{a+b \sin (e+f x)}} (a+b \sin (e+f x))^{3/2} \sqrt {\frac {(a+b) (c+d \sin (e+f x))}{(c+d) (a+b \sin (e+f x))}}} \] Input:

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

Output:

(-2*Sqrt[2]*Cos[(2*e - Pi + 2*f*x)/4]*EllipticE[ArcSin[(Sqrt[(a - b)/(a + 
b)]*Cos[(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[c 
+ d*Sin[e + f*x]])/(Sqrt[(a - b)/(a + b)]*f*Sqrt[((a + b)*(1 + Sin[e + f*x 
]))/(a + b*Sin[e + f*x])]*(a + b*Sin[e + f*x])^(3/2)*Sqrt[((a + b)*(c + d* 
Sin[e + f*x]))/((c + d)*(a + b*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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3274

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3297

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

\(\Big \downarrow \) 3475

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

Input:

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

Output:

(2*(c - d)*Sqrt[c + d]*EllipticE[ArcSin[(Sqrt[a + b]*Sqrt[c + d*Sin[e + f* 
x]])/(Sqrt[c + d]*Sqrt[a + b*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]))/((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]))]*(a + b*Sin[e + f*x]))/((a - b)*Sqrt[a + b]*(b*c - a*d)*f) 
 + (2*Sqrt[a + b]*(c - d)*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]))/((a - b)*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. \(42556\) vs. \(2(379)=758\).

Time = 5.00 (sec) , antiderivative size = 42557, normalized size of antiderivative = 104.05

method	result	size
default	\(\text {Expression too large to display}\)	\(42557\)

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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