\(\int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx\) [49]

Optimal result

Integrand size = 35, antiderivative size = 198 \[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \sqrt {c+d} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} c f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {3024} \[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \sqrt {c+d} \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{c f \sqrt {a+b}} \]

[In]

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

[Out]

(-2*Sqrt[c + d]*EllipticPi[(a*(c + d))/((a + b)*c), 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]))/(Sqrt[a + b]*c*f)

Rule 3024

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(sin[(e_.) + (f_.)*(x_)]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x
_)]]), x_Symbol] :> Simp[-2*((a + b*Sin[e + f*x])/(c*f*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*S
in[e + f*x])))]*EllipticPi[a*((c + d)/(c*(a + b))), ArcSin[Rt[(a + b)/(c + d), 2]*(Sqrt[c + d*Sin[e + f*x]]/Sq
rt[a + b*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] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {c+d} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\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))}{\sqrt {a+b} c f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.99 \[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=-\frac {2 \sqrt {c+d} \operatorname {EllipticPi}\left (\frac {a (c+d)}{(a+b) c},\arcsin \left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right ),\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {\frac {(-b c+a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{\sqrt {a+b} c f} \]

[In]

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

[Out]

(-2*Sqrt[c + d]*EllipticPi[(a*(c + d))/((a + b)*c), 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]))/(Sqrt[a + b]*c*f)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(28725\) vs. \(2(183)=366\).

Time = 7.15 (sec) , antiderivative size = 28726, normalized size of antiderivative = 145.08

method	result	size
default	\(\text {Expression too large to display}\)	\(28726\)

[In]

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

[Out]

result too large to display

Fricas [F(-1)]

Timed out. \[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {\sqrt {a + b \sin {\left (e + f x \right )}}}{\sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a}}{\sqrt {d \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {\sqrt {b \sin \left (f x + e\right ) + a}}{\sqrt {d \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {\sqrt {a+b\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]

[In]

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

[Out]

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