3.507 \(\int \frac{1}{(a+a \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx\)

Optimal. Leaf size=181 \[ -\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}+\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{a f \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{a f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]

[Out]

-((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((c - d)*f*(a + a*Sin[e + f*x]))) - (EllipticE[(e - Pi/2 + f*x)/2, (
2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(a*(c - d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (EllipticF[(e - Pi/
2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]])

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Rubi [A]  time = 0.214433, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2768, 2752, 2663, 2661, 2655, 2653} \[ -\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{f (c-d) (a \sin (e+f x)+a)}+\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{a f \sqrt{c+d \sin (e+f x)}}-\frac{\sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{a f (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + a*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x]

[Out]

-((Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/((c - d)*f*(a + a*Sin[e + f*x]))) - (EllipticE[(e - Pi/2 + f*x)/2, (
2*d)/(c + d)]*Sqrt[c + d*Sin[e + f*x]])/(a*(c - d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (EllipticF[(e - Pi/
2 + f*x)/2, (2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(a*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2768

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

Rule 2752

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*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]

Rule 2663

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2661

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, (2*b)
/(a + b)])/(d*Sqrt[a + b]), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2655

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+a \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx &=-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{(c-d) f (a+a \sin (e+f x))}+\frac{d \int \frac{-\frac{a}{2}-\frac{1}{2} a \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{a^2 (c-d)}\\ &=-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{(c-d) f (a+a \sin (e+f x))}+\frac{\int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{2 a}-\frac{\int \sqrt{c+d \sin (e+f x)} \, dx}{2 a (c-d)}\\ &=-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{(c-d) f (a+a \sin (e+f x))}-\frac{\sqrt{c+d \sin (e+f x)} \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{2 a (c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\sqrt{\frac{c+d \sin (e+f x)}{c+d}} \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{2 a \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{\cos (e+f x) \sqrt{c+d \sin (e+f x)}}{(c-d) f (a+a \sin (e+f x))}-\frac{E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{a (c-d) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{a f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 1.09524, size = 210, normalized size = 1.16 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (2 \sin \left (\frac{1}{2} (e+f x)\right ) (c+d \sin (e+f x))-\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left ((c-d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-(c+d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+c+d \sin (e+f x)\right )\right )}{a f (c-d) (\sin (e+f x)+1) \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + a*Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]),x]

[Out]

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

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Maple [A]  time = 3.408, size = 443, normalized size = 2.5 \begin{align*}{\frac{1}{af\cos \left ( fx+e \right ) }\sqrt{- \left ( -d\sin \left ( fx+e \right ) -c \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}} \left ( -{\frac{- \left ( \sin \left ( fx+e \right ) \right ) ^{2}d-c\sin \left ( fx+e \right ) +d\sin \left ( fx+e \right ) +c}{c-d}{\frac{1}{\sqrt{ \left ( -d\sin \left ( fx+e \right ) -c \right ) \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( 1+\sin \left ( fx+e \right ) \right ) }}}}-2\,{\frac{d}{ \left ( 2\,c-2\,d \right ) \sqrt{- \left ( -d\sin \left ( fx+e \right ) -c \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}} \left ({\frac{c}{d}}-1 \right ) \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{{\frac{d \left ( 1-\sin \left ( fx+e \right ) \right ) }{c+d}}}\sqrt{{\frac{ \left ( -\sin \left ( fx+e \right ) -1 \right ) d}{c-d}}}{\it EllipticF} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ) }-{\frac{d}{c-d} \left ({\frac{c}{d}}-1 \right ) \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}}\sqrt{{\frac{d \left ( 1-\sin \left ( fx+e \right ) \right ) }{c+d}}}\sqrt{{\frac{ \left ( -\sin \left ( fx+e \right ) -1 \right ) d}{c-d}}} \left ( \left ( -{\frac{c}{d}}-1 \right ){\it EllipticE} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ) +{\it EllipticF} \left ( \sqrt{{\frac{c+d\sin \left ( fx+e \right ) }{c-d}}},\sqrt{{\frac{c-d}{c+d}}} \right ) \right ){\frac{1}{\sqrt{- \left ( -d\sin \left ( fx+e \right ) -c \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}}}} \right ){\frac{1}{\sqrt{c+d\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sin \left (f x + e\right ) + a\right )} \sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{d \sin \left (f x + e\right ) + c}}{a d \cos \left (f x + e\right )^{2} - a c - a d -{\left (a c + a d\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{\sqrt{c + d \sin{\left (e + f x \right )}} \sin{\left (e + f x \right )} + \sqrt{c + d \sin{\left (e + f x \right )}}}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(c + d*sin(e + f*x))*sin(e + f*x) + sqrt(c + d*sin(e + f*x))), x)/a

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sin \left (f x + e\right ) + a\right )} \sqrt{d \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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