\(\int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx\) [139]

Optimal result

Integrand size = 42, antiderivative size = 176 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx=-\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}+\frac {2 (g \cos (e+f x))^{5/2}}{a f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:

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

Mathematica [A] (verified)

Time = 4.58 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.52 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx=-\frac {2 (g \cos (e+f x))^{5/2} \left (-\sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sin (e+f x)\right )}{c f g (-1+\sin (e+f x)) (a (1+\sin (e+f x)))^{3/2} \sqrt {c-c \sin (e+f x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.48 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.99, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3331

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3331

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3321

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3121

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

Input:

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

Output:

(-2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*(c - c*Sin[e + 
 f*x])^(3/2)) + ((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*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]] 
*EllipticE[(e + f*x)/2, 2])/(c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e 
 + f*x]]))/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. \(498\) vs. \(2(158)=316\).

Time = 12.43 (sec) , antiderivative size = 499, normalized size of antiderivative = 2.84

method	result	size
default	\(\frac {g \left (\sin \left (\frac {f x}{2}+\frac {e}{2}\right ) \left (\left (-4 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-2\right ) \sqrt {2}+4 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+2\right )+\left (\left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+4 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+2\right ) \sqrt {2}-4 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-8 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-4\right ) \operatorname {EllipticF}\left (\left (1+\sqrt {2}\right ) \left (\csc \left (\frac {f x}{2}+\frac {e}{2}\right )-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right ), -2 \sqrt {2}+3\right ) \sqrt {\frac {\sqrt {2}\, \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\sqrt {2}+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}\, \sqrt {-\frac {2 \left (\sqrt {2}\, \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\sqrt {2}-2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}+\left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \sqrt {2}\, \operatorname {EllipticE}\left (\left (1+\sqrt {2}\right ) \left (\csc \left (\frac {f x}{2}+\frac {e}{2}\right )-\cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right ), -2 \sqrt {2}+3\right ) \sqrt {\frac {\sqrt {2}\, \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\sqrt {2}+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-1}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}\, \sqrt {-\frac {2 \left (\sqrt {2}\, \cos \left (\frac {f x}{2}+\frac {e}{2}\right )-\sqrt {2}-2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}\right ) \sqrt {g \left (-1+2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )}}{c a f \left (1+\sqrt {2}\right ) \left (2 \sqrt {2}-3\right ) \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \sqrt {\left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) a}\, \sqrt {-\left (2 \cos \left (\frac {f x}{2}+\frac {e}{2}\right ) \sin \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) c}}\)	\(499\)
risch	\(\text {Expression too large to display}\)	\(1138\)

Input:

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

Output:

g/c/a/f/(1+2^(1/2))/(2*2^(1/2)-3)*(sin(1/2*f*x+1/2*e)*((-4*cos(1/2*f*x+1/2 
*e)-2)*2^(1/2)+4*cos(1/2*f*x+1/2*e)+2)+((2*cos(1/2*f*x+1/2*e)^2+4*cos(1/2* 
f*x+1/2*e)+2)*2^(1/2)-4*cos(1/2*f*x+1/2*e)^2-8*cos(1/2*f*x+1/2*e)-4)*Ellip 
ticF((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*(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)+(cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2* 
e)+1)*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^(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*(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))*(g*(-1+2*cos(1/2*f*x 
+1/2*e)^2))^(1/2)/(cos(1/2*f*x+1/2*e)+1)/((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 
)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.83 \[ \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}} \, dx=-\frac {2 \, {\left (-i \, \sqrt {\frac {1}{2}} \sqrt {a c g} g \cos \left (f x + e\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + i \, \sqrt {\frac {1}{2}} \sqrt {a c g} g \cos \left (f x + e\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} g \sin \left (f x + e\right )\right )}}{a^{2} c^{2} f \cos \left (f x + e\right )^{2}} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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