3.129 \(\int \frac{(g \cos (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx\)

Optimal. Leaf size=122 \[ \frac{2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}} \]

[Out]

(2*c*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c*g*Sqrt[Cos[e + f
*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

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Rubi [A]  time = 0.539677, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ \frac{2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/Sqrt[a + a*Sin[e + f*x]],x]

[Out]

(2*c*(g*Cos[e + f*x])^(5/2))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*c*g*Sqrt[Cos[e + f
*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

Rule 2851

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e +
f*x])^n)/(f*g*(m + n + p)), x] + Dist[(a*(2*m + p - 1))/(m + n + p), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*
x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && Eq
Q[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + n + p, 0] &&  !LtQ[0, n, m] && IntegersQ[2*m, 2*n, 2*p]

Rule 2842

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(g*Cos[e + f*x])/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), In
t[(g*Cos[e + f*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2
, 0]

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{(g \cos (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx &=\frac{2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+c \int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx\\ &=\frac{2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{(c g \cos (e+f x)) \int \sqrt{g \cos (e+f x)} \, dx}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{\left (c g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{2 c g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 2.56402, size = 215, normalized size = 1.76 \[ \frac{\sqrt{e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right )} \sqrt{-i a e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2} \left (12 i e^{i (e+f x)} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (e+f x)}\right )+\sqrt{1+e^{2 i (e+f x)}} \left (-6 i e^{i (e+f x)}+e^{2 i (e+f x)}+1\right )\right ) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2}}{3 a f \left (1+e^{2 i (e+f x)}\right )^{3/2} \cos ^{\frac{3}{2}}(e+f x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/Sqrt[a + a*Sin[e + f*x]],x]

[Out]

(Sqrt[((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*Sqrt[(1 + E^((2*I)*(e + f*x)))/E^(I*(e + f*x))]*(g*Cos
[e + f*x])^(3/2)*(Sqrt[1 + E^((2*I)*(e + f*x))]*(1 - (6*I)*E^(I*(e + f*x)) + E^((2*I)*(e + f*x))) + (12*I)*E^(
I*(e + f*x))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(e + f*x))])*Sqrt[c - c*Sin[e + f*x]])/(3*a*(1 + E^((
2*I)*(e + f*x)))^(3/2)*f*Cos[e + f*x]^(3/2))

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Maple [C]  time = 0.343, size = 361, normalized size = 3. \begin{align*} -{\frac{2}{3\,f \left ( -1+\sin \left ( fx+e \right ) \right ) \sin \left ( fx+e \right ) \cos \left ( fx+e \right ) } \left ( g\cos \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( 3\,i\cos \left ( fx+e \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ) -3\,i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ) \cos \left ( fx+e \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) +3\,i{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ) -3\,i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) + \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+3\,\cos \left ( fx+e \right ) \right ){\frac{1}{\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2),x)

[Out]

-2/3/f*(g*cos(f*x+e))^(3/2)*(-c*(-1+sin(f*x+e)))^(1/2)*(3*I*cos(f*x+e)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),
I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)-3*I*EllipticE(I*(-1+cos(f*x+e))/sin(f
*x+e),I)*cos(f*x+e)*sin(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+3*I*EllipticF(I*(-1+
cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)-3*I*EllipticE(
I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+cos(f*x+
e)^2*sin(f*x+e)-3*cos(f*x+e)^2+3*cos(f*x+e))/(-1+sin(f*x+e))/(a*(1+sin(f*x+e)))^(1/2)/sin(f*x+e)/cos(f*x+e)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}} \sqrt{-c \sin \left (f x + e\right ) + c}}{\sqrt{a \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(3/2)*sqrt(-c*sin(f*x + e) + c)/sqrt(a*sin(f*x + e) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(g*cos(f*x + e))*sqrt(-c*sin(f*x + e) + c)*g*cos(f*x + e)/sqrt(a*sin(f*x + e) + 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((g*cos(f*x+e))**(3/2)*(c-c*sin(f*x+e))**(1/2)/(a+a*sin(f*x+e))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}} \sqrt{-c \sin \left (f x + e\right ) + c}}{\sqrt{a \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((g*cos(f*x + e))^(3/2)*sqrt(-c*sin(f*x + e) + c)/sqrt(a*sin(f*x + e) + a), x)