\(\int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}} \, dx\) [149]

Optimal result

Integrand size = 42, antiderivative size = 291 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}} \, dx=-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}-\frac {2 (g \cos (e+f x))^{5/2}}{a f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {6 (g \cos (e+f x))^{5/2}}{5 a^2 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {6 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 a^2 c^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:

-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))^(5/2 
)-2*(g*cos(f*x+e))^(5/2)/a/f/g/(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(5/ 
2)+6/5*(g*cos(f*x+e))^(5/2)/a^2/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e) 
)^(5/2)+6/5*(g*cos(f*x+e))^(5/2)/a^2/c/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin 
(f*x+e))^(3/2)-6/5*g*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)*EllipticE(sin(1 
/2*f*x+1/2*e),2^(1/2))/a^2/c^2/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^( 
1/2)
 

Mathematica [A] (verified)

Time = 5.66 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.36 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}} \, dx=\frac {(g \cos (e+f x))^{3/2} \sec ^3(e+f x) \left (-12 \cos ^{\frac {5}{2}}(e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+7 \sin (e+f x)+3 \sin (3 (e+f x))\right )}{10 a^2 c^2 f \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)}} \] Input:

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

Output:

((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]^3*(-12*Cos[e + f*x]^(5/2)*EllipticE[( 
e + f*x)/2, 2] + 7*Sin[e + f*x] + 3*Sin[3*(e + f*x)]))/(10*a^2*c^2*f*Sqrt[ 
a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])
 

Rubi [A] (verified)

Time = 2.46 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3331, 3042, 3331, 3042, 3331, 3042, 3331, 3042, 3321, 3042, 3121, 3042, 3119}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(g \cos (e+f x))^{3/2}}{(a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}dx\)

\(\Big \downarrow \) 3331

\(\displaystyle \frac {\int \frac {(g \cos (e+f x))^{3/2}}{(\sin (e+f x) a+a)^{3/2} (c-c \sin (e+f x))^{5/2}}dx}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(g \cos (e+f x))^{3/2}}{(\sin (e+f x) a+a)^{3/2} (c-c \sin (e+f x))^{5/2}}dx}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3331

\(\displaystyle \frac {\frac {3 \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{5/2}}dx}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3331

\(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} (c-c \sin (e+f x))^{3/2}}dx}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3331

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx}{c}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3321

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3121

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {3 \left (\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{c \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}+\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {3 \left (\frac {2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {\frac {2 (g \cos (e+f x))^{5/2}}{f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}}{5 c}\right )}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{5/2}}}{a}-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{5/2}}\)

Input:

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

Output:

(-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])^(5/2)) + ((-2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^( 
3/2)*(c - c*Sin[e + f*x])^(5/2)) + (3*((2*(g*Cos[e + f*x])^(5/2))/(5*f*g*S 
qrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + ((2*(g*Cos[e + f*x]) 
^(5/2))/(f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (2*g*S 
qrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(c*f*Sqr 
t[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]))/(5*c)))/a)/a
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

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 3121

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) 
^n/Sin[c + d*x]^n   Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt 
Q[-1, n, 1] && IntegerQ[2*n]
 

rule 3321

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[g* 
(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))   Int[(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 3331

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] + Simp[(m + n + p + 1)/(a*(2*m + p + 1))   Int[(g*C 
os[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] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] &&  !LtQ[m, n, -1] && Integers 
Q[2*m, 2*n, 2*p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(777\) vs. \(2(253)=506\).

Time = 18.36 (sec) , antiderivative size = 778, normalized size of antiderivative = 2.67

method	result	size
default	\(\text {Expression too large to display}\)	\(778\)

Input:

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

Output:

1/5*g/a^2/c^2/f/(2*2^(1/2)-3)/(1+2^(1/2))*(2*sin(1/2*f*x+1/2*e)*(3+(-3-24* 
cos(1/2*f*x+1/2*e)^5-12*cos(1/2*f*x+1/2*e)^4+24*cos(1/2*f*x+1/2*e)^3+10*co 
s(1/2*f*x+1/2*e)^2-8*cos(1/2*f*x+1/2*e))*2^(1/2)+24*cos(1/2*f*x+1/2*e)^5+1 
2*cos(1/2*f*x+1/2*e)^4-24*cos(1/2*f*x+1/2*e)^3-10*cos(1/2*f*x+1/2*e)^2+8*c 
os(1/2*f*x+1/2*e))+6*(-2+(4*cos(1/2*f*x+1/2*e)^6+8*cos(1/2*f*x+1/2*e)^5-8* 
cos(1/2*f*x+1/2*e)^3-3*cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2*e)+1)*2^(1/2 
)-8*cos(1/2*f*x+1/2*e)^6-16*cos(1/2*f*x+1/2*e)^5+16*cos(1/2*f*x+1/2*e)^3+6 
*cos(1/2*f*x+1/2*e)^2-4*cos(1/2*f*x+1/2*e))*(-2*(2^(1/2)*cos(1/2*f*x+1/2*e 
)-2^(1/2)-2*cos(1/2*f*x+1/2*e)+1)/(cos(1/2*f*x+1/2*e)+1))^(1/2)*((2^(1/2)* 
cos(1/2*f*x+1/2*e)-2^(1/2)+2*cos(1/2*f*x+1/2*e)-1)/(cos(1/2*f*x+1/2*e)+1)) 
^(1/2)*EllipticF((1+2^(1/2))*(csc(1/2*f*x+1/2*e)-cot(1/2*f*x+1/2*e)),-2*2^ 
(1/2)+3)+3*(4*cos(1/2*f*x+1/2*e)^6+8*cos(1/2*f*x+1/2*e)^5-8*cos(1/2*f*x+1/ 
2*e)^3-3*cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2*e)+1)*2^(1/2)*(-2*(2^(1/2) 
*cos(1/2*f*x+1/2*e)-2^(1/2)-2*cos(1/2*f*x+1/2*e)+1)/(cos(1/2*f*x+1/2*e)+1) 
)^(1/2)*((2^(1/2)*cos(1/2*f*x+1/2*e)-2^(1/2)+2*cos(1/2*f*x+1/2*e)-1)/(cos( 
1/2*f*x+1/2*e)+1))^(1/2)*EllipticE((1+2^(1/2))*(csc(1/2*f*x+1/2*e)-cot(1/2 
*f*x+1/2*e)),-2*2^(1/2)+3))*(g*(-1+2*cos(1/2*f*x+1/2*e)^2))^(1/2)/((2*cos( 
1/2*f*x+1/2*e)*sin(1/2*f*x+1/2*e)+1)*a)^(1/2)/(-(2*cos(1/2*f*x+1/2*e)*sin( 
1/2*f*x+1/2*e)-1)*c)^(1/2)/(4*cos(1/2*f*x+1/2*e)^5+4*cos(1/2*f*x+1/2*e)^4- 
4*cos(1/2*f*x+1/2*e)^3-4*cos(1/2*f*x+1/2*e)^2+cos(1/2*f*x+1/2*e)+1)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.54 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}} \, dx=-\frac {2 \, {\left (-3 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} g \cos \left (f x + e\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} g \cos \left (f x + e\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - {\left (3 \, g \cos \left (f x + e\right )^{2} + g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )\right )}}{5 \, a^{3} c^{3} f \cos \left (f x + e\right )^{4}} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}} \, dx=-\frac {\sqrt {g}\, \sqrt {c}\, \sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right )}{\sin \left (f x +e \right )^{6}-3 \sin \left (f x +e \right )^{4}+3 \sin \left (f x +e \right )^{2}-1}d x \right ) g}{a^{3} c^{3}} \] Input:

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

Output:

( - sqrt(g)*sqrt(c)*sqrt(a)*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f* 
x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x))/(sin(e + f*x)**6 - 3*sin(e + f*x) 
**4 + 3*sin(e + f*x)**2 - 1),x)*g)/(a**3*c**3)