3.109 \(\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt{c-c \sin (e+f x)}} \, dx\)

Optimal. Leaf size=234 \[ -\frac{22 a^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 a^3 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{c-c \sin (e+f x)}} \]

[Out]

(-22*a^3*(g*Cos[e + f*x])^(5/2))/(15*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a^3*g*Sqrt[C
os[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]]) - (22*a^2*(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*a*(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 = 1.14248, antiderivative size = 234, 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} \[ -\frac{22 a^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 a^3 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-22*a^3*(g*Cos[e + f*x])^(5/2))/(15*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a^3*g*Sqrt[C
os[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]]) - (22*a^2*(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*a*(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 \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt{c-c \sin (e+f x)}} \, dx &=-\frac{2 a (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} (11 a) \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{22 a^2 (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 a (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 (11 a^2\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{22 a^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^2 (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 a (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 (11 a^3\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{22 a^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^2 (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 a (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 (11 a^3 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{22 a^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^2 (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 a (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 (11 a^3 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{22 a^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{22 a^3 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{22 a^2 (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 a (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 [A]  time = 1.86945, size = 158, normalized size = 0.68 \[ -\frac{(a (\sin (e+f x)+1))^{5/2} (g \cos (e+f x))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\sqrt{\cos (e+f x)} (126 \sin (2 (e+f x))+515 \cos (e+f x)-15 \cos (3 (e+f x)))-924 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{210 f \cos ^{\frac{3}{2}}(e+f x) \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2)*(-924*EllipticE[(e
+ f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(515*Cos[e + f*x] - 15*Cos[3*(e + f*x)] + 126*Sin[2*(e + f*x)])))/(210*f*Co
s[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sqrt[c - c*Sin[e + f*x]])

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Maple [C]  time = 0.338, size = 415, normalized size = 1.8 \begin{align*} -{\frac{2}{105\,f \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +3\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-4\,\sin \left ( fx+e \right ) -4 \right ) \sin \left ( fx+e \right ) \cos \left ( fx+e \right ) } \left ( g\cos \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( a \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{5}{2}}} \left ( -231\,i\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) +231\,i\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) +15\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}-231\,i{\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}+231\,i\sin \left ( fx+e \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}{\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) +63\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-140\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) -294\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+231\,\cos \left ( fx+e \right ) \right ){\frac{1}{\sqrt{-c \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)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(1/2),x)

[Out]

-2/105/f*(g*cos(f*x+e))^(3/2)*(a*(1+sin(f*x+e)))^(5/2)*(-231*I*sin(f*x+e)*cos(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*
(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+231*I*sin(f*x+e)*cos(f*x+e)*(1/(co
s(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)+15*sin(f*x+e)*c
os(f*x+e)^4-231*I*sin(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*
x+e))/sin(f*x+e),I)+231*I*sin(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-
1+cos(f*x+e))/sin(f*x+e),I)+63*cos(f*x+e)^4-140*cos(f*x+e)^2*sin(f*x+e)-294*cos(f*x+e)^2+231*cos(f*x+e))/(cos(
f*x+e)^2*sin(f*x+e)+3*cos(f*x+e)^2-4*sin(f*x+e)-4)/sin(f*x+e)/cos(f*x+e)/(-c*(-1+sin(f*x+e)))^(1/2)

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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}}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{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))^(5/2)/(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)^(5/2)/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 (\frac{{\left (a^{2} g \cos \left (f x + e\right )^{3} - 2 \, a^{2} g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{2} 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}}{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))^(5/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral((a^2*g*cos(f*x + e)^3 - 2*a^2*g*cos(f*x + e)*sin(f*x + e) - 2*a^2*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)/(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))**(5/2)/(c-c*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}}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{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))^(5/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(5/2)/sqrt(-c*sin(f*x + e) + c), x)