\(\int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx\) [134]

Optimal result

Integrand size = 42, antiderivative size = 294 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {22 c^4 (g \cos (e+f x))^{5/2}}{a f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 c^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {30 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2}} \]

[Out]

-4*c*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(3/2)-30/7*c^2*(g*cos(f*x+e))^(5/2)*(c-c
*sin(f*x+e))^(3/2)/a/f/g/(a+a*sin(f*x+e))^(1/2)-22*c^4*(g*cos(f*x+e))^(5/2)/a/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*
sin(f*x+e))^(1/2)-66*c^4*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*cos(f*x+e))^(1/2)/a/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-66/7*c^3*(g*cos(f*x
+e))^(5/2)*(c-c*sin(f*x+e))^(1/2)/a/f/g/(a+a*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2929, 2930, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {22 c^4 (g \cos (e+f x))^{5/2}}{a f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {66 c^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{a f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {66 c^3 \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{7 a f g \sqrt {a \sin (e+f x)+a}}-\frac {30 c^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 a f g \sqrt {a \sin (e+f x)+a}}-\frac {4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2}} \]

[In]

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

[Out]

(-22*c^4*(g*Cos[e + f*x])^(5/2))/(a*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (66*c^4*g*Sqrt[Co
s[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f
*x]]) - (66*c^3*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(7*a*f*g*Sqrt[a + a*Sin[e + f*x]]) - (30*c^2*
(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*a*f*g*Sqrt[a + a*Sin[e + f*x]]) - (4*c*(g*Cos[e + f*x])^
(5/2)*(c - c*Sin[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2))

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 2929

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*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*(2*n + p + 1))), x] - Dist[b*((2*m + p - 1)/(d*(2*n + p + 1))), Int[(g*Cos[e + f*x])^p*(a + b*
Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 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] && GtQ[m, 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && IntegersQ[2*m, 2*n, 2*p]

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 {4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2}}-\frac {(15 c) \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a} \\ & = -\frac {30 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2}}-\frac {\left (165 c^2\right ) \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{7 a} \\ & = -\frac {66 c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {30 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2}}-\frac {\left (33 c^3\right ) \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a} \\ & = -\frac {22 c^4 (g \cos (e+f x))^{5/2}}{a f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {30 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2}}-\frac {\left (33 c^4\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}{a} \\ & = -\frac {22 c^4 (g \cos (e+f x))^{5/2}}{a f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {30 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2}}-\frac {\left (33 c^4 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{a \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {22 c^4 (g \cos (e+f x))^{5/2}}{a f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {30 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2}}-\frac {\left (33 c^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{a \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {22 c^4 (g \cos (e+f x))^{5/2}}{a f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 c^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 c^3 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {30 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a f g \sqrt {a+a \sin (e+f x)}}-\frac {4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 14.51 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.96 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {66 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c-c \sin (e+f x))^{7/2}}{f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (a (1+\sin (e+f x)))^{3/2}}+\frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (c-c \sin (e+f x))^{7/2} \left (-32-\frac {109}{14} \cos (e+f x)+\frac {1}{14} \cos (3 (e+f x))+\frac {64 \sin \left (\frac {1}{2} (e+f x)\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}+\sin (2 (e+f x))\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (a (1+\sin (e+f x)))^{3/2}} \]

[In]

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

[Out]

(-66*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f
*x])^(7/2))/(f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(3/2)) + ((g*
Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(c - c*Sin[e + f*x])^(7/2)*(-32 - (10
9*Cos[e + f*x])/14 + Cos[3*(e + f*x)]/14 + (64*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]) + Sin[2
*(e + f*x)]))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(3/2))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.26 (sec) , antiderivative size = 1527, normalized size of antiderivative = 5.19

method	result	size
default	\(\text {Expression too large to display}\)	\(1527\)

[In]

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

[Out]

-2/7/f*(g*cos(f*x+e))^(1/2)*(-c*(sin(f*x+e)-1))^(1/2)*g*c^3/(sin(f*x+e)-1)/(1+cos(f*x+e))/(a*(1+sin(f*x+e)))^(
1/2)/a*(-112-28*cos(f*x+e)^2+462*I*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)
/(1+cos(f*x+e)))^(1/2)*cos(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)*cos(f*x+e)^2-462*I*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)
*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)-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)*cos(f*x+e)^2-28*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e
)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*co
s(f*x+e)^3+7*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)*Elliptic
F(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*(c
sc(f*x+e)-cot(f*x+e)),I)-112*cos(f*x+e)+112*sin(f*x+e)-28*cos(f*x+e)^3+28*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)
^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x
+e))^2)^(3/2)*cos(f*x+e)^3-112*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f
*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)^2+112*ln((2*(-co
s(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)
))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)^2-168*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+
e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*c
os(f*x+e)+168*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos
(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)-28*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+
e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+c
os(f*x+e))^2)^(3/2)*sec(f*x+e)+28*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(
f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*sec(f*x+e)+cos(f*x+e)^5+7*
cos(f*x+e)^3*sin(f*x+e)-119*cos(f*x+e)*sin(f*x+e)+cos(f*x+e)^4-112*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2
)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^
2)^(3/2)+112*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(
f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2))

Fricas [C] (verification not implemented)

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

Time = 0.13 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.73 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {2 \, {\left (6 \, c^{3} g \cos \left (f x + e\right )^{2} + 133 \, c^{3} g - {\left (c^{3} g \cos \left (f x + e\right )^{2} - 21 \, c^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} + 231 \, {\left (-i \, \sqrt {2} c^{3} g \sin \left (f x + e\right ) - i \, \sqrt {2} c^{3} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 \, {\left (i \, \sqrt {2} c^{3} g \sin \left (f x + e\right ) + i \, \sqrt {2} c^{3} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{7 \, {\left (a^{2} f \sin \left (f x + e\right ) + a^{2} f\right )}} \]

[In]

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

[Out]

-1/7*(2*(6*c^3*g*cos(f*x + e)^2 + 133*c^3*g - (c^3*g*cos(f*x + e)^2 - 21*c^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) + 231*(-I*sqrt(2)*c^3*g*sin(f*x + e) - I*sqrt(2)*c^3*g
)*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 231*(I*sqrt(
2)*c^3*g*sin(f*x + e) + I*sqrt(2)*c^3*g)*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x
+ e) - I*sin(f*x + e))))/(a^2*f*sin(f*x + e) + a^2*f)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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