\(\int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx\) [150]

Optimal result

Integrand size = 42, antiderivative size = 350 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}-\frac {14 (g \cos (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{9 a^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {14 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{15 a^2 c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]

[Out]

-2/5*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(7/2)-14/5*(g*cos(f*x+e))^(5/2)/a/f/g/(a
+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(7/2)+14/9*(g*cos(f*x+e))^(5/2)/a^2/f/g/(c-c*sin(f*x+e))^(7/2)/(a+a*sin(
f*x+e))^(1/2)+14/15*(g*cos(f*x+e))^(5/2)/a^2/c/f/g/(c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2)+14/15*(g*cos(
f*x+e))^(5/2)/a^2/c^2/f/g/(c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)-14/15*g*(cos(1/2*f*x+1/2*e)^2)^(1/2)/c
os(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^2/c^3/f/(a+a*s
in(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 350, 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 = {2931, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=-\frac {14 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{15 a^2 c^3 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c^2 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{9 a^2 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {14 (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{7/2}} \]

[In]

Int[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2)),x]

[Out]

(-2*(g*Cos[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2)) - (14*(g*Cos[e + f*x
])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2)) + (14*(g*Cos[e + f*x])^(5/2))/(9*a^2
*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (14*(g*Cos[e + f*x])^(5/2))/(15*a^2*c*f*g*Sqrt[a +
a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (14*(g*Cos[e + f*x])^(5/2))/(15*a^2*c^2*f*g*Sqrt[a + a*Sin[e +
f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (14*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(
15*a^2*c^3*f*Sqrt[a + a*Sin[e + f*x]]*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 2931

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*((c + d*Sin[e + f*x])^
n/(a*f*g*(2*m + p + 1))), x] + Dist[(m + n + p + 1)/(a*(2*m + p + 1)), 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] && E
qQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] &&  !LtQ[m, n, -1] && IntegersQ[2*m, 2*n, 2*p]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}+\frac {7 \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}} \, dx}{5 a} \\ & = -\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}-\frac {14 (g \cos (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {7 \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \, dx}{a^2} \\ & = -\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}-\frac {14 (g \cos (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{9 a^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {7 \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx}{3 a^2 c} \\ & = -\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}-\frac {14 (g \cos (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{9 a^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {7 \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{15 a^2 c^2} \\ & = -\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}-\frac {14 (g \cos (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{9 a^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {7 \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{15 a^2 c^3} \\ & = -\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}-\frac {14 (g \cos (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{9 a^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {(7 g \cos (e+f x)) \int \sqrt {g \cos (e+f x)} \, dx}{15 a^2 c^3 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}-\frac {14 (g \cos (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{9 a^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (7 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{15 a^2 c^3 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}-\frac {14 (g \cos (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{9 a^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 (g \cos (e+f x))^{5/2}}{15 a^2 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {14 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 a^2 c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.30 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.49 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=-\frac {\sqrt {\cos (e+f x)} (g \cos (e+f x))^{3/2} \left (42 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (-3 \cos (e+f x)-\cos (3 (e+f x))+4 \cos ^3(e+f x) \sin (e+f x)\right )+\sqrt {\cos (e+f x)} (-9+28 \cos (2 (e+f x))+21 \cos (4 (e+f x))+98 \sin (e+f x)+42 \sin (3 (e+f x)))\right )}{180 c^3 f (-1+\sin (e+f x))^3 (a (1+\sin (e+f x)))^{5/2} \sqrt {c-c \sin (e+f x)}} \]

[In]

Integrate[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2)),x]

[Out]

-1/180*(Sqrt[Cos[e + f*x]]*(g*Cos[e + f*x])^(3/2)*(42*EllipticE[(e + f*x)/2, 2]*(-3*Cos[e + f*x] - Cos[3*(e +
f*x)] + 4*Cos[e + f*x]^3*Sin[e + f*x]) + Sqrt[Cos[e + f*x]]*(-9 + 28*Cos[2*(e + f*x)] + 21*Cos[4*(e + f*x)] +
98*Sin[e + f*x] + 42*Sin[3*(e + f*x)])))/(c^3*f*(-1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^(5/2)*Sqrt[c - c*
Sin[e + f*x]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.05 (sec) , antiderivative size = 885, normalized size of antiderivative = 2.53

method	result	size
default	\(\text {Expression too large to display}\)	\(885\)

[In]

int((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/45*I/f*(g*cos(f*x+e))^(1/2)*g/(sin(f*x+e)-1)/(a*(1+sin(f*x+e)))^(1/2)/(-c*(sin(f*x+e)-1))^(1/2)/a^2/c^3/(1+c
os(f*x+e))*(-21*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(
f*x+e)))^(1/2)*cos(f*x+e)^2+21*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)
),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2+21*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*
x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2-42*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-co
t(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)*sin(f*x+e)-21*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(
csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2+42*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF
(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)*sin(f*x+e)+42*(cos(f*x+e)/(1+cos(f*x+e)))^(1
/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)-21*sin(f*x+e)*(cos(f*x+e)/(1+co
s(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)-42*(cos(f*x+e)/(1+cos(f*x+e))
)^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)+21*sin(f*x+e)*(cos(f*x+e)/(
1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)-2*I*sec(f*x+e)^2+21*(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*(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)+7*I*cos(f*x+e)-2*I*sec(f*x
+e)+7*I*tan(f*x+e)+7*I*tan(f*x+e)*sec(f*x+e)+21*I*cos(f*x+e)^2-14*I+21*I*sin(f*x+e))

Fricas [C] (verification not implemented)

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

Time = 0.15 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.73 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=-\frac {2 \, {\left (21 \, g \cos \left (f x + e\right )^{4} - 14 \, g \cos \left (f x + e\right )^{2} + 7 \, {\left (3 \, g \cos \left (f x + e\right )^{2} + g\right )} \sin \left (f x + e\right ) - 2 \, g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} + 21 \, {\left (-i \, \sqrt {2} g \cos \left (f x + e\right )^{4} \sin \left (f x + e\right ) + i \, \sqrt {2} g \cos \left (f x + e\right )^{4}\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, {\left (i \, \sqrt {2} g \cos \left (f x + e\right )^{4} \sin \left (f x + e\right ) - i \, \sqrt {2} g \cos \left (f x + e\right )^{4}\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{45 \, {\left (a^{3} c^{4} f \cos \left (f x + e\right )^{4} \sin \left (f x + e\right ) - a^{3} c^{4} f \cos \left (f x + e\right )^{4}\right )}} \]

[In]

integrate((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

-1/45*(2*(21*g*cos(f*x + e)^4 - 14*g*cos(f*x + e)^2 + 7*(3*g*cos(f*x + e)^2 + g)*sin(f*x + e) - 2*g)*sqrt(g*co
s(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c) + 21*(-I*sqrt(2)*g*cos(f*x + e)^4*sin(f*x + e)
+ I*sqrt(2)*g*cos(f*x + e)^4)*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*s
in(f*x + e))) + 21*(I*sqrt(2)*g*cos(f*x + e)^4*sin(f*x + e) - I*sqrt(2)*g*cos(f*x + e)^4)*sqrt(a*c*g)*weierstr
assZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))))/(a^3*c^4*f*cos(f*x + e)^4*sin(f*x +
e) - a^3*c^4*f*cos(f*x + e)^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))**(3/2)/(a+a*sin(f*x+e))**(5/2)/(c-c*sin(f*x+e))**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(7/2),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)^(7/2)), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(7/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \]

[In]

int((g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(7/2)),x)

[Out]

int((g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(7/2)), x)