3.238 \(\int \frac{\sqrt{e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=74 \[ -\frac{2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}-\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{a d \sqrt{\cos (c+d x)}} \]

[Out]

(-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d*x]]) - (2*(e*Cos[c + d*x])^(3/2))/(d*e
*(a + a*Sin[c + d*x]))

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Rubi [A]  time = 0.0679212, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2683, 2640, 2639} \[ -\frac{2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}-\frac{2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{a d \sqrt{\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d*x]]) - (2*(e*Cos[c + d*x])^(3/2))/(d*e
*(a + a*Sin[c + d*x]))

Rule 2683

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(g*Cos[e
+ f*x])^(p + 1))/(a*f*g*(p - 1)*(a + b*Sin[e + f*x])), x] + Dist[p/(a*(p - 1)), Int[(g*Cos[e + f*x])^p, x], x
] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] &&  !GeQ[p, 1] && IntegerQ[2*p]

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx &=-\frac{2 (e \cos (c+d x))^{3/2}}{d e (a+a \sin (c+d x))}-\frac{\int \sqrt{e \cos (c+d x)} \, dx}{a}\\ &=-\frac{2 (e \cos (c+d x))^{3/2}}{d e (a+a \sin (c+d x))}-\frac{\sqrt{e \cos (c+d x)} \int \sqrt{\cos (c+d x)} \, dx}{a \sqrt{\cos (c+d x)}}\\ &=-\frac{2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d \sqrt{\cos (c+d x)}}-\frac{2 (e \cos (c+d x))^{3/2}}{d e (a+a \sin (c+d x))}\\ \end{align*}

Mathematica [C]  time = 0.0419868, size = 66, normalized size = 0.89 \[ -\frac{2^{3/4} (e \cos (c+d x))^{3/2} \, _2F_1\left (\frac{3}{4},\frac{5}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{3 a d e (\sin (c+d x)+1)^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2^(3/4)*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 5/4, 7/4, (1 - Sin[c + d*x])/2])/(3*a*d*e*(1 + Sin[c +
d*x])^(3/4))

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Maple [A]  time = 0.831, size = 115, normalized size = 1.6 \begin{align*} -2\,{\frac{ \left ( \sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) +\sin \left ( 1/2\,dx+c/2 \right ) \right ) e}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}\sin \left ( 1/2\,dx+c/2 \right ) ad}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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