\(\int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx\) [42]

Optimal result

Integrand size = 38, antiderivative size = 97 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{12 a c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2}}{120 a c^2 f (c-c \sin (e+f x))^{11/2}} \] Output:

1/12*cos(f*x+e)*(a+a*sin(f*x+e))^(9/2)/a/c/f/(c-c*sin(f*x+e))^(13/2)+1/120 
*cos(f*x+e)*(a+a*sin(f*x+e))^(9/2)/a/c^2/f/(c-c*sin(f*x+e))^(11/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(419\) vs. \(2(97)=194\).

Time = 12.88 (sec) , antiderivative size = 419, normalized size of antiderivative = 4.32 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {8 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{7/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}-\frac {32 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (a (1+\sin (e+f x)))^{7/2}}{5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}+\frac {6 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}-\frac {8 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{7/2}}{3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}}+\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11} (a (1+\sin (e+f x)))^{7/2}}{2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{15/2}} \] Input:

Integrate[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x]) 
^(15/2),x]
 

Output:

(8*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3*(a*(1 + Sin[e + f*x]))^(7/2))/( 
3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) - 
 (32*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*(a*(1 + Sin[e + f*x]))^(7/2)) 
/(5*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) 
 + (6*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2) 
)/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) 
- (8*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(7/2)) 
/(3*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2)) 
 + ((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(a*(1 + Sin[e + f*x]))^(7/2)) 
/(2*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(15/2))
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3042, 3320, 3042, 3222, 3042, 3221}

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.

\(\displaystyle \int \frac {\cos ^2(e+f x) (a \sin (e+f x)+a)^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (e+f x)^2 (a \sin (e+f x)+a)^{7/2}}{(c-c \sin (e+f x))^{15/2}}dx\)

\(\Big \downarrow \) 3320

\(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{9/2}}{(c-c \sin (e+f x))^{13/2}}dx}{a c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(\sin (e+f x) a+a)^{9/2}}{(c-c \sin (e+f x))^{13/2}}dx}{a c}\)

\(\Big \downarrow \) 3222

\(\displaystyle \frac {\frac {\int \frac {(\sin (e+f x) a+a)^{9/2}}{(c-c \sin (e+f x))^{11/2}}dx}{12 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{12 f (c-c \sin (e+f x))^{13/2}}}{a c}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {(\sin (e+f x) a+a)^{9/2}}{(c-c \sin (e+f x))^{11/2}}dx}{12 c}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{12 f (c-c \sin (e+f x))^{13/2}}}{a c}\)

\(\Big \downarrow \) 3221

\(\displaystyle \frac {\frac {\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{120 c f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{12 f (c-c \sin (e+f x))^{13/2}}}{a c}\)

Input:

Int[(Cos[e + f*x]^2*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(15/2 
),x]
 

Output:

((Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(12*f*(c - c*Sin[e + f*x])^(13/ 
2)) + (Cos[e + f*x]*(a + a*Sin[e + f*x])^(9/2))/(120*c*f*(c - c*Sin[e + f* 
x])^(11/2)))/(a*c)
 

Defintions of rubi rules used

rule 3042

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

rule 3221

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*( 
(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))), x] /; FreeQ[{a, b, c, d, e, f, m, 
n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[m + n + 1, 0] && Ne 
Q[m, -2^(-1)]
 

rule 3222

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*( 
(c + d*Sin[e + f*x])^n/(a*f*(2*m + 1))), x] + Simp[(m + n + 1)/(a*(2*m + 1) 
)   Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n, x], x] /; Free 
Q[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && 
 ILtQ[Simplify[m + n + 1], 0] && NeQ[m, -2^(-1)] && (SumSimplerQ[m, 1] || 
!SumSimplerQ[n, 1])
 

rule 3320

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ 
.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(a^(p/ 
2)*c^(p/2))   Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(n + 
p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && 
EqQ[a^2 - b^2, 0] && IntegerQ[p/2]
 

Maple [F]

\[\int \frac {\cos \left (f x +e \right )^{2} \left (a +a \sin \left (f x +e \right )\right )^{\frac {7}{2}}}{\left (c -c \sin \left (f x +e \right )\right )^{\frac {15}{2}}}d x\]

Input:

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x)
 

Output:

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (85) = 170\).

Time = 0.10 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.97 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=-\frac {{\left (15 \, a^{3} \cos \left (f x + e\right )^{4} - 60 \, a^{3} \cos \left (f x + e\right )^{2} + 48 \, a^{3} - 4 \, {\left (5 \, a^{3} \cos \left (f x + e\right )^{2} - 8 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{30 \, {\left (c^{8} f \cos \left (f x + e\right )^{7} - 18 \, c^{8} f \cos \left (f x + e\right )^{5} + 48 \, c^{8} f \cos \left (f x + e\right )^{3} - 32 \, c^{8} f \cos \left (f x + e\right ) + 2 \, {\left (3 \, c^{8} f \cos \left (f x + e\right )^{5} - 16 \, c^{8} f \cos \left (f x + e\right )^{3} + 16 \, c^{8} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \] Input:

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x, a 
lgorithm="fricas")
 

Output:

-1/30*(15*a^3*cos(f*x + e)^4 - 60*a^3*cos(f*x + e)^2 + 48*a^3 - 4*(5*a^3*c 
os(f*x + e)^2 - 8*a^3)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin( 
f*x + e) + c)/(c^8*f*cos(f*x + e)^7 - 18*c^8*f*cos(f*x + e)^5 + 48*c^8*f*c 
os(f*x + e)^3 - 32*c^8*f*cos(f*x + e) + 2*(3*c^8*f*cos(f*x + e)^5 - 16*c^8 
*f*cos(f*x + e)^3 + 16*c^8*f*cos(f*x + e))*sin(f*x + e))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(f*x+e)**2*(a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))**(15/2),x 
)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {15}{2}}} \,d x } \] Input:

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x, a 
lgorithm="maxima")
 

Output:

integrate((a*sin(f*x + e) + a)^(7/2)*cos(f*x + e)^2/(-c*sin(f*x + e) + c)^ 
(15/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x, a 
lgorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 21.93 (sec) , antiderivative size = 373, normalized size of antiderivative = 3.85 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=-\frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {504\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{5\,c^8\,f}+\frac {576\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{5\,c^8\,f}-\frac {96\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{c^8\,f}+\frac {8\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{c^8\,f}-\frac {64\,a^3\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (3\,e+3\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,c^8\,f}\right )}{-858\,\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}+858\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )-130\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )+2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\cos \left (7\,e+7\,f\,x\right )+1144\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )-416\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )+24\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )} \] Input:

int((cos(e + f*x)^2*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(15/2 
),x)
 

Output:

-((c - c*sin(e + f*x))^(1/2)*((504*a^3*exp(e*7i + f*x*7i)*(a + a*sin(e + f 
*x))^(1/2))/(5*c^8*f) + (576*a^3*exp(e*7i + f*x*7i)*sin(e + f*x)*(a + a*si 
n(e + f*x))^(1/2))/(5*c^8*f) - (96*a^3*exp(e*7i + f*x*7i)*cos(2*e + 2*f*x) 
*(a + a*sin(e + f*x))^(1/2))/(c^8*f) + (8*a^3*exp(e*7i + f*x*7i)*cos(4*e + 
 4*f*x)*(a + a*sin(e + f*x))^(1/2))/(c^8*f) - (64*a^3*exp(e*7i + f*x*7i)*s 
in(3*e + 3*f*x)*(a + a*sin(e + f*x))^(1/2))/(3*c^8*f)))/(858*exp(e*7i + f* 
x*7i)*cos(3*e + 3*f*x) - 858*cos(e + f*x)*exp(e*7i + f*x*7i) - 130*exp(e*7 
i + f*x*7i)*cos(5*e + 5*f*x) + 2*exp(e*7i + f*x*7i)*cos(7*e + 7*f*x) + 114 
4*exp(e*7i + f*x*7i)*sin(2*e + 2*f*x) - 416*exp(e*7i + f*x*7i)*sin(4*e + 4 
*f*x) + 24*exp(e*7i + f*x*7i)*sin(6*e + 6*f*x))
 

Reduce [F]

\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, a^{3} \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{3}}{\sin \left (f x +e \right )^{8}-8 \sin \left (f x +e \right )^{7}+28 \sin \left (f x +e \right )^{6}-56 \sin \left (f x +e \right )^{5}+70 \sin \left (f x +e \right )^{4}-56 \sin \left (f x +e \right )^{3}+28 \sin \left (f x +e \right )^{2}-8 \sin \left (f x +e \right )+1}d x +3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{8}-8 \sin \left (f x +e \right )^{7}+28 \sin \left (f x +e \right )^{6}-56 \sin \left (f x +e \right )^{5}+70 \sin \left (f x +e \right )^{4}-56 \sin \left (f x +e \right )^{3}+28 \sin \left (f x +e \right )^{2}-8 \sin \left (f x +e \right )+1}d x \right )+3 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )}{\sin \left (f x +e \right )^{8}-8 \sin \left (f x +e \right )^{7}+28 \sin \left (f x +e \right )^{6}-56 \sin \left (f x +e \right )^{5}+70 \sin \left (f x +e \right )^{4}-56 \sin \left (f x +e \right )^{3}+28 \sin \left (f x +e \right )^{2}-8 \sin \left (f x +e \right )+1}d x \right )+\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \cos \left (f x +e \right )^{2}}{\sin \left (f x +e \right )^{8}-8 \sin \left (f x +e \right )^{7}+28 \sin \left (f x +e \right )^{6}-56 \sin \left (f x +e \right )^{5}+70 \sin \left (f x +e \right )^{4}-56 \sin \left (f x +e \right )^{3}+28 \sin \left (f x +e \right )^{2}-8 \sin \left (f x +e \right )+1}d x \right )}{c^{8}} \] Input:

int(cos(f*x+e)^2*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(15/2),x)
 

Output:

(sqrt(c)*sqrt(a)*a**3*(int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 
1)*cos(e + f*x)**2*sin(e + f*x)**3)/(sin(e + f*x)**8 - 8*sin(e + f*x)**7 + 
 28*sin(e + f*x)**6 - 56*sin(e + f*x)**5 + 70*sin(e + f*x)**4 - 56*sin(e + 
 f*x)**3 + 28*sin(e + f*x)**2 - 8*sin(e + f*x) + 1),x) + 3*int((sqrt(sin(e 
 + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*cos(e + f*x)**2*sin(e + f*x)**2)/(s 
in(e + f*x)**8 - 8*sin(e + f*x)**7 + 28*sin(e + f*x)**6 - 56*sin(e + f*x)* 
*5 + 70*sin(e + f*x)**4 - 56*sin(e + f*x)**3 + 28*sin(e + f*x)**2 - 8*sin( 
e + f*x) + 1),x) + 3*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1) 
*cos(e + f*x)**2*sin(e + f*x))/(sin(e + f*x)**8 - 8*sin(e + f*x)**7 + 28*s 
in(e + f*x)**6 - 56*sin(e + f*x)**5 + 70*sin(e + f*x)**4 - 56*sin(e + f*x) 
**3 + 28*sin(e + f*x)**2 - 8*sin(e + f*x) + 1),x) + int((sqrt(sin(e + f*x) 
 + 1)*sqrt( - sin(e + f*x) + 1)*cos(e + f*x)**2)/(sin(e + f*x)**8 - 8*sin( 
e + f*x)**7 + 28*sin(e + f*x)**6 - 56*sin(e + f*x)**5 + 70*sin(e + f*x)**4 
 - 56*sin(e + f*x)**3 + 28*sin(e + f*x)**2 - 8*sin(e + f*x) + 1),x)))/c**8