\(\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx\) [120]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 1.08 (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} \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {22 a^4 (g \cos (e+f x))^{5/2}}{c f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {66 a^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {66 a^3 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{7 c f g \sqrt {c-c \sin (e+f x)}}+\frac {30 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 c f g \sqrt {c-c \sin (e+f x)}}+\frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}} \]

[In]

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

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(f*g*(c - c*Sin[e + f*x])^(3/2)) + (22*a^4*(g*Cos[e +
f*x])^(5/2))/(c*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (66*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*C
os[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (66*a^3*(g*C
os[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(7*c*f*g*Sqrt[c - c*Sin[e + f*x]]) + (30*a^2*(g*Cos[e + f*x])^(5/
2)*(a + a*Sin[e + f*x])^(3/2))/(7*c*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 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 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {(15 a) \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}{c} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}+\frac {30 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 c f g \sqrt {c-c \sin (e+f x)}}-\frac {\left (165 a^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}{7 c} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}+\frac {66 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{7 c f g \sqrt {c-c \sin (e+f x)}}+\frac {30 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 c f g \sqrt {c-c \sin (e+f x)}}-\frac {\left (33 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}{c} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{c f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {66 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{7 c f g \sqrt {c-c \sin (e+f x)}}+\frac {30 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 c f g \sqrt {c-c \sin (e+f x)}}-\frac {\left (33 a^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}{c} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{c f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {66 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{7 c f g \sqrt {c-c \sin (e+f x)}}+\frac {30 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 c f g \sqrt {c-c \sin (e+f x)}}-\frac {\left (33 a^4 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{c \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{c f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {66 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{7 c f g \sqrt {c-c \sin (e+f x)}}+\frac {30 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 c f g \sqrt {c-c \sin (e+f x)}}-\frac {\left (33 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{c \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{c f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {66 a^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 )}{c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {66 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{7 c f g \sqrt {c-c \sin (e+f x)}}+\frac {30 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 c f g \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 14.35 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.97 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {66 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{7/2}}{f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{3/2}}+\frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{7/2} \left (32+\frac {109}{14} \cos (e+f x)-\frac {1}{14} \cos (3 (e+f x))+\frac {64 \sin \left (\frac {1}{2} (e+f x)\right )}{\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )}+\sin (2 (e+f x))\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{3/2}} \]

[In]

Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*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*(a*(1 + Sin[e +
f*x]))^(7/2))/(f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*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*(a*(1 + Sin[e + f*x]))^(7/2)*(32 + (1
09*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*(c - c*Sin[e + f*x])^(3/2))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 3.91 (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)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/7/f*(g*cos(f*x+e))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*g*a^3/(1+sin(f*x+e))/(1+cos(f*x+e))/(-c*(sin(f*x+e)-1))^(
1/2)/c*(-112-231*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+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)-c
ot(f*x+e)),I)*cos(f*x+e)^2-462*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)+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)*cos(f*x+e)-28*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)-112*cos(f*x+e)-112*sin(f*x+e)-28*cos(f*x+e)^3+28*ln((2*(-cos(f*x+e)/(1+co
s(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+11
2*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)^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)*cos(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+cos(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)+231*I*(cos(f*x+e)/(1+cos(f*x+e)))^(
1/2)*(1/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)-231*I*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(
1/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I))

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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