3.118 \(\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx\)

Optimal. Leaf size=343 \[ -\frac{2 a^4 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 c \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{77 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 a^4 c g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt{c-c \sin (e+f x)}} \]

[Out]

(-2*a^4*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*a^4*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]]) - (2*a^3*c*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(7*f*g*Sqrt[c - c*Sin[e + f*x]]) - (10*a^2*c
*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(77*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*c*(g*Cos[e + f*x]
)^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(33*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin
[e + f*x])^(7/2))/(11*f*g*Sqrt[c - c*Sin[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 1.71989, antiderivative size = 343, 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} \[ -\frac{2 a^4 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a^3 c \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{77 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 a^4 c g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt{c-c \sin (e+f x)}}+\frac{2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt{c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^4*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*a^4*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]]) - (2*a^3*c*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(7*f*g*Sqrt[c - c*Sin[e + f*x]]) - (10*a^2*c
*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(77*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*c*(g*Cos[e + f*x]
)^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(33*f*g*Sqrt[c - c*Sin[e + f*x]]) + (2*c*(g*Cos[e + f*x])^(5/2)*(a + a*Sin
[e + f*x])^(7/2))/(11*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))^{7/2} \sqrt{c-c \sin (e+f x)} \, dx &=\frac{2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{11 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{11} (3 c) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{11} (5 a c) \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{10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{7} \left (5 a^2 c\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^3 c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt{c-c \sin (e+f x)}}+\left (a^3 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^4 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 a^3 c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt{c-c \sin (e+f x)}}+\left (a^4 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^4 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 a^3 c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt{c-c \sin (e+f x)}}+\frac{\left (a^4 c g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{2 a^4 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 a^3 c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt{c-c \sin (e+f x)}}+\frac{\left (a^4 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 a^4 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 a^4 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)}}-\frac{2 a^3 c (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{7 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 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))^{7/2}}{11 f g \sqrt{c-c \sin (e+f x)}}\\ \end{align*}

Mathematica [C]  time = 5.50583, size = 333, normalized size = 0.97 \[ -\frac{a^3 g^2 \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (1374 \cos (e+f x)+423 \cos (3 (e+f x))-7 (44 \sin (2 (e+f x))-22 \sin (4 (e+f x))+3 \cos (5 (e+f x))-528 \cot (e)))}{1848 f \sqrt{g \cos (e+f x)}}+\frac{4 a^4 \left (e^{i (e+f x)}+i\right ) \sqrt{e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right )} \left (\left (-1+e^{2 i e}\right ) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (e+f x)}\right )+\sqrt{1+e^{2 i (e+f x)}}\right ) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2}}{\left (-1+e^{2 i e}\right ) f \left (e^{i (e+f x)}-i\right ) \sqrt{1+e^{2 i (e+f x)}} \sqrt{-i a e^{-i (e+f x)} \left (e^{i (e+f x)}+i\right )^2} \cos ^{\frac{3}{2}}(e+f x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^4*(I + 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 + E^((2*I)*e))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(e + f*x))])*Sqrt[c - c*
Sin[e + f*x]])/((-1 + E^((2*I)*e))*(-I + E^(I*(e + f*x)))*Sqrt[((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x)
)]*Sqrt[1 + E^((2*I)*(e + f*x))]*f*Cos[e + f*x]^(3/2)) - (a^3*g^2*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e
+ f*x]]*(1374*Cos[e + f*x] + 423*Cos[3*(e + f*x)] - 7*(3*Cos[5*(e + f*x)] - 528*Cot[e] + 44*Sin[2*(e + f*x)] -
22*Sin[4*(e + f*x)])))/(1848*f*Sqrt[g*Cos[e + f*x]])

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Maple [C]  time = 0.394, size = 425, normalized size = 1.2 \begin{align*}{\frac{2}{231\,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 ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}}\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( -21\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}\sin \left ( fx+e \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}}}-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 ) -77\, \left ( \cos \left ( fx+e \right ) \right ) ^{6}+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 ) +132\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+154\, \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{7}{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))^(7/2)*(c-c*sin(f*x+e))^(1/2),x)

[Out]

2/231/f*(-c*(-1+sin(f*x+e)))^(1/2)*(-21*cos(f*x+e)^6*sin(f*x+e)+231*I*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),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)-231*I*EllipticF(I*(-1+cos(f
*x+e))/sin(f*x+e),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)-77*cos(f
*x+e)^6+231*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)-231*I*EllipticF(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)+132*sin(f*x+e)*cos(f*x+e)^4+154*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)))^(7/2)/(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)^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{7}{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))^(7/2)*(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)^(7/2)*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 (3 \, a^{3} g \cos \left (f x + e\right )^{3} - 4 \, a^{3} g \cos \left (f x + e\right ) +{\left (a^{3} g \cos \left (f x + e\right )^{3} - 4 \, a^{3} g \cos \left (f x + e\right )\right )} \sin \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))^(7/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(-(3*a^3*g*cos(f*x + e)^3 - 4*a^3*g*cos(f*x + e) + (a^3*g*cos(f*x + e)^3 - 4*a^3*g*cos(f*x + e))*sin(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))**(7/2)*(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))^(7/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError