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

Optimal. Leaf size=235 \[ -\frac{2 a^2 c (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 a^2 c g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{6 a c \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{c-c \sin (e+f x)}} \]

[Out]

(-2*a^2*c*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a^2*c*g*Sqrt[
Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e +
f*x]]) - (6*a*c*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(35*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*
Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[c - c*Sin[e + f*x]])

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Rubi [A]  time = 1.13886, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ -\frac{2 a^2 c (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 a^2 c g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{6 a c \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*c*(g*Cos[e + f*x])^(5/2))/(5*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (6*a^2*c*g*Sqrt[
Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e +
f*x]]) - (6*a*c*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(35*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*
Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(7*f*g*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 (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)} \, dx &=\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{7} (3 c) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{6 a c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{35 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{5} (3 a 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 a^2 c (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{6 a c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{35 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{5} \left (3 a^2 c\right ) \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 a^2 c (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{6 a c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{35 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 f g \sqrt{c-c \sin (e+f x)}}+\frac{\left (3 a^2 c g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{5 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{2 a^2 c (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{6 a c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{35 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 f g \sqrt{c-c \sin (e+f x)}}+\frac{\left (3 a^2 c g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{5 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{2 a^2 c (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{6 a^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 )}{5 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{6 a c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{35 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 f g \sqrt{c-c \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 7.75171, size = 257, normalized size = 1.09 \[ \frac{i a^2 g e^{-3 i (e+f x)} \left (e^{i (e+f x)}+i\right ) \left (112 i e^{5 i (e+f x)} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right )+\sqrt{1+e^{2 i (e+f x)}} \left (-14 i e^{i (e+f x)}+15 e^{2 i (e+f x)}-168 i e^{3 i (e+f x)}+15 e^{4 i (e+f x)}+14 i e^{5 i (e+f x)}+5 e^{6 i (e+f x)}+5\right )\right ) \sqrt{c-c \sin (e+f x)} \sqrt{g \cos (e+f x)}}{140 f \left (e^{i (e+f x)}-i\right ) \sqrt{1+e^{2 i (e+f x)}} \sqrt{a (\sin (e+f x)+1)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/140)*a^2*(I + E^(I*(e + f*x)))*g*Sqrt[g*Cos[e + f*x]]*(Sqrt[1 + E^((2*I)*(e + f*x))]*(5 - (14*I)*E^(I*(e +
f*x)) + 15*E^((2*I)*(e + f*x)) - (168*I)*E^((3*I)*(e + f*x)) + 15*E^((4*I)*(e + f*x)) + (14*I)*E^((5*I)*(e +
f*x)) + 5*E^((6*I)*(e + f*x))) + (112*I)*E^((5*I)*(e + f*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f
*x))])*Sqrt[c - c*Sin[e + f*x]])/(E^((3*I)*(e + f*x))*(-I + E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]*f*S
qrt[a*(1 + Sin[e + f*x])])

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Maple [C]  time = 0.327, size = 372, normalized size = 1.6 \begin{align*} -{\frac{2}{35\,f \left ( 1+\sin \left ( fx+e \right ) \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}}\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( 21\,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 ) -21\,i\sin \left ( fx+e \right ) \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}}}+21\,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 ) -21\,i\sin \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}}}+5\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+7\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}+14\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-21\,\cos \left ( fx+e \right ) \right ) \left ( g\cos \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/35/f*(-c*(-1+sin(f*x+e)))^(1/2)*(21*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)
*cos(f*x+e)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)-21*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1)
)^(1/2)*sin(f*x+e)*cos(f*x+e)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)+21*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x
+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)-21*I*(1/(cos(f*x+e)+1))^(1/2)*(
cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)+5*sin(f*x+e)*cos(f*x+e)^
4+7*cos(f*x+e)^4+14*cos(f*x+e)^2-21*cos(f*x+e))*(g*cos(f*x+e))^(3/2)*(a*(1+sin(f*x+e)))^(3/2)/(1+sin(f*x+e))/s
in(f*x+e)/cos(f*x+e)^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError