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

Optimal. Leaf size=352 \[ -\frac{14 a^3 c^2 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{15 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 a^3 c^2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a c^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{99 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g} \]

[Out]

(-14*a^3*c^2*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (14*a^3*c^2*
g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c
*Sin[e + f*x]]) - (2*a^2*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(15*f*g*Sqrt[c - c*Sin[e + f*x]]
) - (2*a*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(33*f*g*Sqrt[c - c*Sin[e + f*x]]) + (14*c^2*(g
*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(99*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/
2)*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(11*f*g)

________________________________________________________________________________________

Rubi [A]  time = 1.75728, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ -\frac{14 a^3 c^2 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^2 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{15 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 a^3 c^2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{15 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a c^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{99 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{5/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{11 f g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-14*a^3*c^2*(g*Cos[e + f*x])^(5/2))/(45*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (14*a^3*c^2*
g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(15*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c
*Sin[e + f*x]]) - (2*a^2*c^2*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(15*f*g*Sqrt[c - c*Sin[e + f*x]]
) - (2*a*c^2*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(33*f*g*Sqrt[c - c*Sin[e + f*x]]) + (14*c^2*(g
*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(99*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/
2)*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(11*f*g)

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))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx &=\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{11 f g}+\frac{1}{11} (7 c) \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)} \, dx\\ &=\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{99 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))^{5/2} \sqrt{c-c \sin (e+f x)}}{11 f g}+\frac{1}{33} \left (7 c^2\right ) \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 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{99 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))^{5/2} \sqrt{c-c \sin (e+f x)}}{11 f g}+\frac{1}{3} \left (a c^2\right ) \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{2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{15 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{99 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))^{5/2} \sqrt{c-c \sin (e+f x)}}{11 f g}+\frac{1}{15} \left (7 a^2 c^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{14 a^3 c^2 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{15 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{99 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))^{5/2} \sqrt{c-c \sin (e+f x)}}{11 f g}+\frac{1}{15} \left (7 a^3 c^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{14 a^3 c^2 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{15 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{99 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))^{5/2} \sqrt{c-c \sin (e+f x)}}{11 f g}+\frac{\left (7 a^3 c^2 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{15 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{14 a^3 c^2 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{15 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{99 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))^{5/2} \sqrt{c-c \sin (e+f x)}}{11 f g}+\frac{\left (7 a^3 c^2 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{15 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{14 a^3 c^2 (g \cos (e+f x))^{5/2}}{45 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{14 a^3 c^2 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{15 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^2 c^2 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{15 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{99 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))^{5/2} \sqrt{c-c \sin (e+f x)}}{11 f g}\\ \end{align*}

Mathematica [A]  time = 1.36524, size = 189, normalized size = 0.54 \[ \frac{c (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{5/2} \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left (\sqrt{\cos (e+f x)} (-836 \sin (2 (e+f x))-110 \sin (4 (e+f x))+450 \cos (e+f x)+225 \cos (3 (e+f x))+45 \cos (5 (e+f x)))-3696 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{3960 f \cos ^{\frac{3}{2}}(e+f x) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(g*Cos[e + f*x])^(3/2)*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*Sin[e + f*x]]*(-3696*Ell
ipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(450*Cos[e + f*x] + 225*Cos[3*(e + f*x)] + 45*Cos[5*(e + f*x)] - 8
36*Sin[2*(e + f*x)] - 110*Sin[4*(e + f*x)])))/(3960*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])
^3*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)

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Maple [C]  time = 0.394, size = 382, normalized size = 1.1 \begin{align*}{\frac{2}{495\,f \left ( 1+\sin \left ( fx+e \right ) \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{5}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}} \left ( -45\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}\sin \left ( fx+e \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 ) -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 ) \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}}}-55\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+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 ) -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}}}-22\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-154\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+231\,\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{5}{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))^(5/2)*(c-c*sin(f*x+e))^(3/2),x)

[Out]

2/495/f*(-c*(-1+sin(f*x+e)))^(3/2)*(-45*cos(f*x+e)^6*sin(f*x+e)+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)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-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)-55*cos(f
*x+e)^6+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)-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+co
s(f*x+e))/sin(f*x+e),I)-22*cos(f*x+e)^4-154*cos(f*x+e)^2+231*cos(f*x+e))*(g*cos(f*x+e))^(3/2)*(a*(1+sin(f*x+e)
))^(5/2)/(1+sin(f*x+e))/sin(f*x+e)/cos(f*x+e)^5

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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{5}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}\,{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))^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c g \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + a^{2} c g \cos \left (f x + e\right )^{3}\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))^(5/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

integral((a^2*c*g*cos(f*x + e)^3*sin(f*x + e) + a^2*c*g*cos(f*x + e)^3)*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))**(5/2)*(c-c*sin(f*x+e))**(3/2),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))^(5/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

Exception raised: TypeError