\(\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\) [119]

Optimal result

Integrand size = 42, antiderivative size = 288 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}} \]

[Out]

-10/21*a^2*(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)-2/9*a*(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)-22/9*a^4*(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)+22/3*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)/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-22/21*a^3*(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 = 1.07 (sec) , antiderivative size = 288, 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 = {2930, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{3 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}} \]

[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]

(-22*a^4*(g*Cos[e + f*x])^(5/2))/(9*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (22*a^4*g*Sqrt[Co
s[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(3*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f
*x]]) - (22*a^3*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(21*f*g*Sqrt[c - c*Sin[e + f*x]]) - (10*a^2*(
g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(21*f*g*Sqrt[c - c*Sin[e + f*x]]) - (2*a*(g*Cos[e + f*x])^(5
/2)*(a + a*Sin[e + f*x])^(5/2))/(9*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 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}+\frac {1}{3} (5 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 \\ & = -\frac {10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}+\frac {1}{21} \left (55 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 \\ & = -\frac {22 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}+\frac {1}{3} \left (11 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 \\ & = -\frac {22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}+\frac {1}{3} \left (11 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 \\ & = -\frac {22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}+\frac {\left (11 a^4 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{3 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}}+\frac {\left (11 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{3 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 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 )}{3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.51 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.63 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {a^3 (g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3 \sqrt {a (1+\sin (e+f x))} \left (-1848 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (1128 \cos (e+f x)-72 \cos (3 (e+f x))+350 \sin (2 (e+f x))-7 \sin (4 (e+f x)))\right )}{252 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \sqrt {c-c \sin (e+f x)}} \]

[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]

-1/252*(a^3*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^3*Sqrt[a*(1 + Sin[
e + f*x])]*(-1848*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x]]*(1128*Cos[e + f*x] - 72*Cos[3*(e + f*x)] + 35
0*Sin[2*(e + f*x)] - 7*Sin[4*(e + f*x)])))/(f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*Sqrt[
c - c*Sin[e + f*x]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.47 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.88

method	result	size
default	\(-\frac {2 \left (231 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{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 (\cos ^{2}\left (f x +e \right )\right )-231 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{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 (\cos ^{2}\left (f x +e \right )\right )-7 \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )+462 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{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 ) \cos \left (f x +e \right )-462 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{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 ) \cos \left (f x +e \right )-36 \left (\cos ^{5}\left (f x +e \right )\right )-7 \left (\cos ^{4}\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 )-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 )-36 \left (\cos ^{4}\left (f x +e \right )\right )+91 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+168 \left (\cos ^{3}\left (f x +e \right )\right )+91 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+168 \left (\cos ^{2}\left (f x +e \right )\right )-231 \cos \left (f x +e \right ) \sin \left (f x +e \right )\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {g \cos \left (f x +e \right )}\, g \,a^{3}}{63 f \left (1+\sin \left (f x +e \right )\right ) \left (1+\cos \left (f x +e \right )\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\)	\(542\)

[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]

-2/63/f*(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)-cot(f
*x+e)),I)*cos(f*x+e)^2-7*cos(f*x+e)^5*sin(f*x+e)+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)-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)-36*cos(f*x+e)^5-7*cos(f*x+e)^4*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)-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)-36*cos(f*x+e)^4+91*co
s(f*x+e)^3*sin(f*x+e)+168*cos(f*x+e)^3+91*cos(f*x+e)^2*sin(f*x+e)+168*cos(f*x+e)^2-231*cos(f*x+e)*sin(f*x+e))*
(a*(1+sin(f*x+e)))^(1/2)*(g*cos(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)

Fricas [C] (verification not implemented)

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

Time = 0.13 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.58 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {-231 i \, \sqrt {2} \sqrt {a c g} a^{3} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} a^{3} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (36 \, a^{3} g \cos \left (f x + e\right )^{2} - 168 \, a^{3} g + 7 \, {\left (a^{3} g \cos \left (f x + e\right )^{2} - 13 \, a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{63 \, c f} \]

[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]

1/63*(-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*(36*a^3*g*cos(f*x + e)^2 - 168*a^3*g + 7*(a^3*g*cos(f*x + e)^2 - 13*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))/(c*f)

Sympy [F(-1)]

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

[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]

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

[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(-1)]

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

[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, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

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

[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)