\(\int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx\) [147]

Optimal result

Integrand size = 42, antiderivative size = 179 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx=-\frac {2 (g \cos (e+f x))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}-\frac {2 (g \cos (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 a^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] 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))^(1/2 
)-2/5*(g*cos(f*x+e))^(5/2)/a/f/g/(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^( 
1/2)-2/5*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/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 4.76 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.06 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx=-\frac {(g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (2 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+\sqrt {\cos (e+f x)} \left (3 \cos \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {3}{2} (e+f x)\right )-4 \sin ^3\left (\frac {1}{2} (e+f x)\right )\right )\right )}{5 f \cos ^{\frac {3}{2}}(e+f x) (a (1+\sin (e+f x)))^{5/2} \sqrt {c-c \sin (e+f x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3331

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3331

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3321

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3121

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

Input:

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

Output:

(-2*(g*Cos[e + f*x])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*S 
in[e + f*x]]) + ((-2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/ 
2)*Sqrt[c - c*Sin[e + f*x]]) - (2*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x] 
]*EllipticE[(e + f*x)/2, 2])/(a*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[ 
e + f*x]]))/(5*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. \(1365\) vs. \(2(155)=310\).

Time = 39.29 (sec) , antiderivative size = 1366, normalized size of antiderivative = 7.63

method	result	size
default	\(\text {Expression too large to display}\)	\(1366\)

Input:

int((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(1/2),x,m 
ethod=_RETURNVERBOSE)
 

Output:

g/a^2*(-2/7/f/(1+2^(1/2))*((cos(1/2*f*x+1/2*e)+1)*((2*cos(1/2*f*x+1/2*e)^2 
+1)*sin(1/2*f*x+1/2*e)-cos(1/2*f*x+1/2*e)*(2*cos(1/2*f*x+1/2*e)^2-3))*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)*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)*((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-cos(1/2*f*x+1/2*e)^2)*sin 
(1/2*f*x+1/2*e)-cos(1/2*f*x+1/2*e)^3)*2^(1/2)+2*(1-cos(1/2*f*x+1/2*e)^2)*s 
in(1/2*f*x+1/2*e)-2*cos(1/2*f*x+1/2*e)^3)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2))^ 
(1/2)/(-2*cos(1/2*f*x+1/2*e)^3+2*cos(1/2*f*x+1/2*e)^2*sin(1/2*f*x+1/2*e)+3 
*cos(1/2*f*x+1/2*e)+sin(1/2*f*x+1/2*e))/((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) 
+1/35/f/(2*2^(1/2)-3)/(1+2^(1/2))*(-14+7*(cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f 
*x+1/2*e)+1)*(-(2*cos(1/2*f*x+1/2*e)^2+1)*sin(1/2*f*x+1/2*e)+cos(1/2*f*x+1 
/2*e)*(2*cos(1/2*f*x+1/2*e)^2-3))*2^(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)*(-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)+4*(cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2*e)+1)*(4*((2*cos(1/2*f*x+ 
1/2*e)^2+1)*sin(1/2*f*x+1/2*e)-cos(1/2*f*x+1/2*e)*(2*cos(1/2*f*x+1/2*e)^2- 
3))*2^(1/2)-3*(2*cos(1/2*f*x+1/2*e)^2+1)*sin(1/2*f*x+1/2*e)+3*cos(1/2*f...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.11 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx=\frac {2 \, {\left (\sqrt {\frac {1}{2}} \sqrt {a c g} {\left (i \, g \cos \left (f x + e\right )^{2} - 2 i \, g \sin \left (f x + e\right ) - 2 i \, g\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {\frac {1}{2}} \sqrt {a c g} {\left (-i \, g \cos \left (f x + e\right )^{2} + 2 i \, g \sin \left (f x + e\right ) + 2 i \, g\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (g \sin \left (f x + e\right ) + 2 \, g\right )}\right )}}{5 \, {\left (a^{3} c f \cos \left (f x + e\right )^{2} - 2 \, a^{3} c f \sin \left (f x + e\right ) - 2 \, a^{3} c f\right )}} \] Input:

integrate((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(1/ 
2),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(1/ 
2),x, algorithm="maxima")
 

Output:

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

Giac [F(-1)]

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

integrate((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(1/ 
2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

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