\(\int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}} \, dx\) [141]

Optimal result

Integrand size = 42, antiderivative size = 294 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}} \, dx=-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {10 (g \cos (e+f x))^{5/2}}{9 a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 a c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]

[Out]

-2*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(7/2)+10/9*(g*cos(f*x+e))^(5/2)/a/f/g/(c-c
*sin(f*x+e))^(7/2)/(a+a*sin(f*x+e))^(1/2)+2/3*(g*cos(f*x+e))^(5/2)/a/c/f/g/(c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x
+e))^(1/2)+2/3*(g*cos(f*x+e))^(5/2)/a/c^2/f/g/(c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)-2/3*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/c^3/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2931, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}} \, dx=-\frac {2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{3 a c^3 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c^2 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {10 (g \cos (e+f x))^{5/2}}{9 a f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{7/2}} \]

[In]

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

[Out]

(-2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2)) + (10*(g*Cos[e + f*x])
^(5/2))/(9*a*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) + (2*(g*Cos[e + f*x])^(5/2))/(3*a*c*f*g*
Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (2*(g*Cos[e + f*x])^(5/2))/(3*a*c^2*f*g*Sqrt[a + a*Sin[
e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2]
)/(3*a*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}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {5 \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} \\ & = -\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {10 (g \cos (e+f x))^{5/2}}{9 a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {5 \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 c} \\ & = -\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {10 (g \cos (e+f x))^{5/2}}{9 a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {\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}{3 a c^2} \\ & = -\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {10 (g \cos (e+f x))^{5/2}}{9 a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{3 a c^3} \\ & = -\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {10 (g \cos (e+f x))^{5/2}}{9 a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {(g \cos (e+f x)) \int \sqrt {g \cos (e+f x)} \, dx}{3 a 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}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {10 (g \cos (e+f x))^{5/2}}{9 a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{3 a 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}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}+\frac {10 (g \cos (e+f x))^{5/2}}{9 a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{3 a c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 a c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.84 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.53 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}} \, dx=\frac {\sqrt {\cos (e+f x)} (g \cos (e+f x))^{3/2} \left (3 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) (5 \cos (e+f x)-\cos (3 (e+f x))-4 \sin (2 (e+f x)))+\sqrt {\cos (e+f x)} (4-12 \cos (2 (e+f x))-17 \sin (e+f x)+3 \sin (3 (e+f x)))\right )}{18 c^3 f (-1+\sin (e+f x))^3 (a (1+\sin (e+f x)))^{3/2} \sqrt {c-c \sin (e+f x)}} \]

[In]

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

[Out]

(Sqrt[Cos[e + f*x]]*(g*Cos[e + f*x])^(3/2)*(3*EllipticE[(e + f*x)/2, 2]*(5*Cos[e + f*x] - Cos[3*(e + f*x)] - 4
*Sin[2*(e + f*x)]) + Sqrt[Cos[e + f*x]]*(4 - 12*Cos[2*(e + f*x)] - 17*Sin[e + f*x] + 3*Sin[3*(e + f*x)])))/(18
*c^3*f*(-1 + Sin[e + f*x])^3*(a*(1 + Sin[e + f*x]))^(3/2)*Sqrt[c - c*Sin[e + f*x]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.26 (sec) , antiderivative size = 1131, normalized size of antiderivative = 3.85

method	result	size
default	\(\text {Expression too large to display}\)	\(1131\)

[In]

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

[Out]

-2/9*I/f*(g*cos(f*x+e))^(1/2)*g*(3*(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)^4-3*(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)^4+6*(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)^3+6*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)^2-6*(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)^3-6*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Elli
pticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2-3*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*E
llipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2+12*(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)*sin(f*x+e)+3*(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)^2-12*(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)-12*(c
os(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)+6*s
in(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)+12
*(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)-
6*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)
-6*(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)+6*(1/(1+c
os(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)-2*I*cos(f*x+e)+3*I*
cos(f*x+e)^2*sin(f*x+e)-6*I*cos(f*x+e)^2+I*cos(f*x+e)*sin(f*x+e)-5*I*sin(f*x+e)+4*I)/(cos(f*x+e)^2+2*sin(f*x+e
)-2)/c^3/a/(-c*(sin(f*x+e)-1))^(1/2)/(a*(1+sin(f*x+e)))^(1/2)/(1+cos(f*x+e))

Fricas [C] (verification not implemented)

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

Time = 0.14 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}} \, dx=-\frac {2 \, {\left (6 \, g \cos \left (f x + e\right )^{2} - {\left (3 \, g \cos \left (f x + e\right )^{2} - 5 \, g\right )} \sin \left (f x + e\right ) - 4 \, g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} + 3 \, {\left (-i \, \sqrt {2} g \cos \left (f x + e\right )^{4} - 2 i \, \sqrt {2} g \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) + 2 i \, \sqrt {2} g \cos \left (f x + e\right )^{2}\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, {\left (i \, \sqrt {2} g \cos \left (f x + e\right )^{4} + 2 i \, \sqrt {2} g \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) - 2 i \, \sqrt {2} g \cos \left (f x + e\right )^{2}\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{9 \, {\left (a^{2} c^{4} f \cos \left (f x + e\right )^{4} + 2 \, a^{2} c^{4} f \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) - 2 \, a^{2} c^{4} f \cos \left (f x + e\right )^{2}\right )}} \]

[In]

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

[Out]

-1/9*(2*(6*g*cos(f*x + e)^2 - (3*g*cos(f*x + e)^2 - 5*g)*sin(f*x + e) - 4*g)*sqrt(g*cos(f*x + e))*sqrt(a*sin(f
*x + e) + a)*sqrt(-c*sin(f*x + e) + c) + 3*(-I*sqrt(2)*g*cos(f*x + e)^4 - 2*I*sqrt(2)*g*cos(f*x + e)^2*sin(f*x
+ e) + 2*I*sqrt(2)*g*cos(f*x + e)^2)*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x +
e) + I*sin(f*x + e))) + 3*(I*sqrt(2)*g*cos(f*x + e)^4 + 2*I*sqrt(2)*g*cos(f*x + e)^2*sin(f*x + e) - 2*I*sqrt(2
)*g*cos(f*x + e)^2)*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e
))))/(a^2*c^4*f*cos(f*x + e)^4 + 2*a^2*c^4*f*cos(f*x + e)^2*sin(f*x + e) - 2*a^2*c^4*f*cos(f*x + e)^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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