3.785 \(\int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)}} \, dx\)

Optimal. Leaf size=198 \[ \frac {2 \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left (\frac {(a+b) d}{b (c+d)};\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{b f \sqrt {c+d}} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2811} \[ \frac {2 \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left (\frac {(a+b) d}{b (c+d)};\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{b f \sqrt {c+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2811

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[
(2*(a + b*Sin[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])))]*EllipticPi[(b*(c + d))/(d*(a + b)), 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))])/(d*f*
Rt[(a + b)/(c + d), 2]*Cos[e + f*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] && PosQ[(a + b)/(c + d)]

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)}} \, dx &=\frac {2 \sqrt {a+b} \Pi \left (\frac {(a+b) d}{b (c+d)};\sin ^{-1}\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))}{b \sqrt {c+d} f}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 197, normalized size = 0.99 \[ \frac {2 \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {\frac {(a d-b c) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left (\frac {(a+b) d}{b (c+d)};\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{b f \sqrt {c+d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \sin \left (f x + e\right ) + c}}{\sqrt {b \sin \left (f x + e\right ) + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{\sqrt {b \sin \left (f x + e\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 4.64, size = 248841, normalized size = 1256.77 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d \sin \left (f x + e\right ) + c}}{\sqrt {b \sin \left (f x + e\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}}{\sqrt {a+b\,\sin \left (e+f\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c + d \sin {\left (e + f x \right )}}}{\sqrt {a + b \sin {\left (e + f x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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