3.245 \(\int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx\)
Optimal. Leaf size=98 \[ \frac {a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac {2 a^2 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}+\frac {2 a^2 c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6} \]
[Out]
1/9*a^2*c^2*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^7+2/63*a^2*c*cos(f*x+e)^5/f/(c-c*sin(f*x+e))^6+2/315*a^2*cos(f*x+e
)^5/f/(c-c*sin(f*x+e))^5
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Rubi [A] time = 0.18, antiderivative size = 98, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used =
{2736, 2672, 2671} \[ \frac {a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac {2 a^2 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}+\frac {2 a^2 c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^5,x]
[Out]
(a^2*c^2*Cos[e + f*x]^5)/(9*f*(c - c*Sin[e + f*x])^7) + (2*a^2*c*Cos[e + f*x]^5)/(63*f*(c - c*Sin[e + f*x])^6)
+ (2*a^2*Cos[e + f*x]^5)/(315*f*(c - c*Sin[e + f*x])^5)
Rule 2671
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*m), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] && !ILtQ[p, 0]
Rule 2672
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*Simplify[2*m + p + 1]), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] && !IGtQ[m, 0]
Rule 2736
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0,
n, m] || LtQ[m, n, 0]))
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^5} \, dx &=\left (a^2 c^2\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^7} \, dx\\ &=\frac {a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac {1}{9} \left (2 a^2 c\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^6} \, dx\\ &=\frac {a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac {2 a^2 c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6}+\frac {1}{63} \left (2 a^2\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^5} \, dx\\ &=\frac {a^2 c^2 \cos ^5(e+f x)}{9 f (c-c \sin (e+f x))^7}+\frac {2 a^2 c \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^6}+\frac {2 a^2 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^5}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 121, normalized size = 1.23 \[ \frac {a^2 \left (441 \sin \left (\frac {1}{2} (e+f x)\right )+210 \sin \left (\frac {3}{2} (e+f x)\right )-36 \sin \left (\frac {5}{2} (e+f x)\right )+\sin \left (\frac {9}{2} (e+f x)\right )+315 \cos \left (\frac {1}{2} (e+f x)\right )-126 \cos \left (\frac {3}{2} (e+f x)\right )-9 \cos \left (\frac {7}{2} (e+f x)\right )\right )}{1260 c^5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])^2/(c - c*Sin[e + f*x])^5,x]
[Out]
(a^2*(315*Cos[(e + f*x)/2] - 126*Cos[(3*(e + f*x))/2] - 9*Cos[(7*(e + f*x))/2] + 441*Sin[(e + f*x)/2] + 210*Si
n[(3*(e + f*x))/2] - 36*Sin[(5*(e + f*x))/2] + Sin[(9*(e + f*x))/2]))/(1260*c^5*f*(Cos[(e + f*x)/2] - Sin[(e +
f*x)/2])^9)
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fricas [B] time = 0.44, size = 278, normalized size = 2.84 \[ \frac {2 \, a^{2} \cos \left (f x + e\right )^{5} - 8 \, a^{2} \cos \left (f x + e\right )^{4} - 25 \, a^{2} \cos \left (f x + e\right )^{3} - 85 \, a^{2} \cos \left (f x + e\right )^{2} + 70 \, a^{2} \cos \left (f x + e\right ) + 140 \, a^{2} + {\left (2 \, a^{2} \cos \left (f x + e\right )^{4} + 10 \, a^{2} \cos \left (f x + e\right )^{3} - 15 \, a^{2} \cos \left (f x + e\right )^{2} + 70 \, a^{2} \cos \left (f x + e\right ) + 140 \, a^{2}\right )} \sin \left (f x + e\right )}{315 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm="fricas")
[Out]
1/315*(2*a^2*cos(f*x + e)^5 - 8*a^2*cos(f*x + e)^4 - 25*a^2*cos(f*x + e)^3 - 85*a^2*cos(f*x + e)^2 + 70*a^2*co
s(f*x + e) + 140*a^2 + (2*a^2*cos(f*x + e)^4 + 10*a^2*cos(f*x + e)^3 - 15*a^2*cos(f*x + e)^2 + 70*a^2*cos(f*x
+ e) + 140*a^2)*sin(f*x + e))/(c^5*f*cos(f*x + e)^5 + 5*c^5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^3 - 20*c^5
*f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f - (c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*
f*cos(f*x + e)^2 + 8*c^5*f*cos(f*x + e) + 16*c^5*f)*sin(f*x + e))
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giac [A] time = 0.27, size = 162, normalized size = 1.65 \[ -\frac {2 \, {\left (315 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 630 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 2310 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 2520 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3402 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1638 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1062 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 108 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 47 \, a^{2}\right )}}{315 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm="giac")
[Out]
-2/315*(315*a^2*tan(1/2*f*x + 1/2*e)^8 - 630*a^2*tan(1/2*f*x + 1/2*e)^7 + 2310*a^2*tan(1/2*f*x + 1/2*e)^6 - 25
20*a^2*tan(1/2*f*x + 1/2*e)^5 + 3402*a^2*tan(1/2*f*x + 1/2*e)^4 - 1638*a^2*tan(1/2*f*x + 1/2*e)^3 + 1062*a^2*t
an(1/2*f*x + 1/2*e)^2 - 108*a^2*tan(1/2*f*x + 1/2*e) + 47*a^2)/(c^5*f*(tan(1/2*f*x + 1/2*e) - 1)^9)
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maple [A] time = 0.30, size = 148, normalized size = 1.51 \[ \frac {2 a^{2} \left (-\frac {50}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {64}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {272}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {6}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {64}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {480}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {404}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}\right )}{f \,c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x)
[Out]
2/f*a^2/c^5*(-50/(tan(1/2*f*x+1/2*e)-1)^4-64/9/(tan(1/2*f*x+1/2*e)-1)^9-272/3/(tan(1/2*f*x+1/2*e)-1)^6-6/(tan(
1/2*f*x+1/2*e)-1)^2-1/(tan(1/2*f*x+1/2*e)-1)-64/3/(tan(1/2*f*x+1/2*e)-1)^3-480/7/(tan(1/2*f*x+1/2*e)-1)^7-404/
5/(tan(1/2*f*x+1/2*e)-1)^5-32/(tan(1/2*f*x+1/2*e)-1)^8)
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maxima [B] time = 1.19, size = 1073, normalized size = 10.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^2/(c-c*sin(f*x+e))^5,x, algorithm="maxima")
[Out]
-2/315*(a^2*(432*sin(f*x + e)/(cos(f*x + e) + 1) - 1728*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3612*sin(f*x + e
)^3/(cos(f*x + e) + 1)^3 - 5418*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5040*sin(f*x + e)^5/(cos(f*x + e) + 1)^5
- 3360*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1260*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 315*sin(f*x + e)^8/(c
os(f*x + e) + 1)^8 - 83)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) +
1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(
f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x
+ e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) - 10*a^2*(4
5*sin(f*x + e)/(cos(f*x + e) + 1) - 117*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 273*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3 - 315*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 315*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 147*sin(f*x + e)
^6/(cos(f*x + e) + 1)^6 + 63*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 5)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e)
+ 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f
*x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x
+ e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin
(f*x + e)^9/(cos(f*x + e) + 1)^9) + 14*a^2*(9*sin(f*x + e)/(cos(f*x + e) + 1) - 36*sin(f*x + e)^2/(cos(f*x + e
) + 1)^2 + 54*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 81*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 45*sin(f*x + e)^5
/(cos(f*x + e) + 1)^5 - 30*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) +
1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x
+ e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x +
e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(f
*x + e)^9/(cos(f*x + e) + 1)^9))/f
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mupad [B] time = 8.81, size = 121, normalized size = 1.23 \[ \frac {\sqrt {2}\,a^2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {89\,\cos \left (3\,e+3\,f\,x\right )}{4}-\frac {2205\,\sin \left (e+f\,x\right )}{8}-\frac {265\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {625\,\cos \left (e+f\,x\right )}{4}+\frac {49\,\cos \left (4\,e+4\,f\,x\right )}{16}+\frac {567\,\sin \left (2\,e+2\,f\,x\right )}{8}+\frac {243\,\sin \left (3\,e+3\,f\,x\right )}{8}-\frac {45\,\sin \left (4\,e+4\,f\,x\right )}{16}+\frac {4967}{16}\right )}{5040\,c^5\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + a*sin(e + f*x))^2/(c - c*sin(e + f*x))^5,x)
[Out]
(2^(1/2)*a^2*cos(e/2 + (f*x)/2)*((89*cos(3*e + 3*f*x))/4 - (2205*sin(e + f*x))/8 - (265*cos(2*e + 2*f*x))/2 -
(625*cos(e + f*x))/4 + (49*cos(4*e + 4*f*x))/16 + (567*sin(2*e + 2*f*x))/8 + (243*sin(3*e + 3*f*x))/8 - (45*si
n(4*e + 4*f*x))/16 + 4967/16))/(5040*c^5*f*cos(e/2 + pi/4 + (f*x)/2)^9)
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sympy [A] time = 50.69, size = 1717, normalized size = 17.52 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))**2/(c-c*sin(f*x+e))**5,x)
[Out]
Piecewise((-630*a**2*tan(e/2 + f*x/2)**8/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 1
1340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*
c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f
*tan(e/2 + f*x/2) - 315*c**5*f) + 1260*a**2*tan(e/2 + f*x/2)**7/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*
tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e
/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 +
f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 4620*a**2*tan(e/2 + f*x/2)**6/(315*c**5*f*tan(e/2 + f
*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)*
*6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 -
11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) + 5040*a**2*tan(e/2 + f*x/2)**5/
(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c
**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f
*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 6804*a*
*2*tan(e/2 + f*x/2)**4/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/
2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f
*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2)
- 315*c**5*f) + 3276*a**2*tan(e/2 + f*x/2)**3/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**
8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 3
9690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c
**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 2124*a**2*tan(e/2 + f*x/2)**2/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c*
*5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*
tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e
/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) + 216*a**2*tan(e/2 + f*x/2)/(315*c**5*f*tan(e/2 +
f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)
**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 -
11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 94*a**2/(315*c**5*f*tan(e/2 +
f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2
)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3
- 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f), Ne(f, 0)), (x*(a*sin(e) + a)*
*2/(-c*sin(e) + c)**5, True))
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