\(\int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx\) [118]

Optimal result

Integrand size = 42, antiderivative size = 343 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {2 a^4 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^4 c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{7 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{77 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{33 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{7/2}}{11 f g \sqrt {c-c \sin (e+f x)}} \]

[Out]

-10/77*a^2*c*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(3/2)/f/g/(c-c*sin(f*x+e))^(1/2)-2/33*a*c*(g*cos(f*x+e))^(5
/2)*(a+a*sin(f*x+e))^(5/2)/f/g/(c-c*sin(f*x+e))^(1/2)+2/11*c*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(7/2)/f/g/(
c-c*sin(f*x+e))^(1/2)-2/3*a^4*c*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)+2*a^4*c
*g*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e)^(1/2)*(g*c
os(f*x+e))^(1/2)/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-2/7*a^3*c*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e
))^(1/2)/f/g/(c-c*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2930, 2921, 2721, 2719} \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {2 a^4 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 a^4 c g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a^3 c \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 c (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{77 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a c (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{33 f g \sqrt {c-c \sin (e+f x)}}+\frac {2 c (a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{5/2}}{11 f g \sqrt {c-c \sin (e+f x)}} \]

[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 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2921

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 2930

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] &&
EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + n + p, 0] &&  !LtQ[0, n, m] && IntegersQ[2*m, 2*n, 2*p]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 5.48 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.05 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \left (\frac {(2+2 i) a^4 e^{\frac {1}{2} i (e+f x)} \left (i+e^{i (e+f x)}\right ) \left (e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right )\right )^{3/2} \left (\sqrt {1+e^{2 i (e+f x)}}+\left (-1+e^{2 i e}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 i (e+f x)}\right )\right )}{\left (-1+e^{2 i e}\right ) \sqrt {-i a e^{-i (e+f x)} \left (i+e^{i (e+f x)}\right )^2} \left (1+e^{2 i (e+f x)}\right )^{3/2} f}-\frac {a^3 \sqrt {\cos (e+f x)} \sqrt {a (1+\sin (e+f x))} (1374 \cos (e+f x)+423 \cos (3 (e+f x))-7 (3 \cos (5 (e+f x))-528 \cot (e)+44 \sin (2 (e+f x))-22 \sin (4 (e+f x))))}{1848 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right )}{\cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[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]

((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]*(((2 + 2*I)*a^4*E^((I/2)*(e + f*x))*(I + E^(I*(e + f*x)))*((1
+ E^((2*I)*(e + f*x)))/E^(I*(e + f*x)))^(3/2)*(Sqrt[1 + E^((2*I)*(e + f*x))] + (-1 + E^((2*I)*e))*Hypergeomet
ric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(e + f*x))]))/((-1 + E^((2*I)*e))*Sqrt[((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*
(e + f*x))]*(1 + E^((2*I)*(e + f*x)))^(3/2)*f) - (a^3*Sqrt[Cos[e + f*x]]*Sqrt[a*(1 + 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*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))))/(Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2
]))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 6.04 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.50

method	result	size
default	\(-\frac {2 \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {g \cos \left (f x +e \right )}\, a^{3} g \left (-21 \left (\cos ^{5}\left (f x +e \right )\right )+231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )-231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )-21 \left (\cos ^{4}\left (f x +e \right )\right )+77 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+462 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sec \left (f x +e \right )-462 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sec \left (f x +e \right )+132 \left (\cos ^{3}\left (f x +e \right )\right )+77 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \left (\sec ^{2}\left (f x +e \right )\right )-231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \left (\sec ^{2}\left (f x +e \right )\right )+132 \left (\cos ^{2}\left (f x +e \right )\right )-77 \cos \left (f x +e \right ) \sin \left (f x +e \right )-77 \sin \left (f x +e \right )-231 \tan \left (f x +e \right )\right )}{231 f \left (1+\cos \left (f x +e \right )\right )}\)	\(516\)

[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,method=_RETURNVERBOSE)

[Out]

-2/231/f*(-c*(sin(f*x+e)-1))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*(g*cos(f*x+e))^(1/2)*a^3*g/(1+cos(f*x+e))*(-21*cos
(f*x+e)^5+231*I*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e))
,I)-231*I*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)-21
*cos(f*x+e)^4+77*cos(f*x+e)^3*sin(f*x+e)+462*I*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Elli
pticF(I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)-462*I*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)
*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)+132*cos(f*x+e)^3+77*cos(f*x+e)^2*sin(f*x+e)+231*I*(1/(1+cos
(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)^2-231*I*(1
/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)^2+1
32*cos(f*x+e)^2-77*cos(f*x+e)*sin(f*x+e)-77*sin(f*x+e)-231*tan(f*x+e))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.50 \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\frac {-231 i \, \sqrt {2} \sqrt {a c g} a^{3} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} a^{3} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (21 \, a^{3} g \cos \left (f x + e\right )^{4} - 132 \, a^{3} g \cos \left (f x + e\right )^{2} - 77 \, {\left (a^{3} g \cos \left (f x + e\right )^{2} - a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{231 \, f} \]

[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]

1/231*(-231*I*sqrt(2)*sqrt(a*c*g)*a^3*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin
(f*x + e))) + 231*I*sqrt(2)*sqrt(a*c*g)*a^3*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) -
I*sin(f*x + e))) + 2*(21*a^3*g*cos(f*x + e)^4 - 132*a^3*g*cos(f*x + e)^2 - 77*(a^3*g*cos(f*x + e)^2 - a^3*g)*
sin(f*x + e))*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/f

Sympy [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\text {Timed out} \]

[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

Maxima [F]

\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \sqrt {-c \sin \left (f x + e\right ) + c} \,d x } \]

[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)

Giac [F]

\[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \sqrt {-c \sin \left (f x + e\right ) + c} \,d x } \]

[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]

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(7/2)*sqrt(-c*sin(f*x + e) + c), x)

Mupad [F(-1)]

Timed out. \[ \int (g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}\,\sqrt {c-c\,\sin \left (e+f\,x\right )} \,d x \]

[In]

int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(1/2),x)

[Out]

int((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2)*(c - c*sin(e + f*x))^(1/2), x)