3.235 \(\int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^5} \, dx\)
Optimal. Leaf size=126 \[ \frac {2 a c \cos ^3(e+f x)}{315 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^4}+\frac {a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6} \]
[Out]
1/9*a*c*cos(f*x+e)^3/f/(c-c*sin(f*x+e))^6+1/21*a*cos(f*x+e)^3/f/(c-c*sin(f*x+e))^5+2/105*a*cos(f*x+e)^3/c/f/(c
-c*sin(f*x+e))^4+2/315*a*c*cos(f*x+e)^3/f/(c^2-c^2*sin(f*x+e))^3
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Rubi [A] time = 0.22, antiderivative size = 126, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used =
{2736, 2672, 2671} \[ \frac {2 a c \cos ^3(e+f x)}{315 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^4}+\frac {a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^5,x]
[Out]
(a*c*Cos[e + f*x]^3)/(9*f*(c - c*Sin[e + f*x])^6) + (a*Cos[e + f*x]^3)/(21*f*(c - c*Sin[e + f*x])^5) + (2*a*Co
s[e + f*x]^3)/(105*c*f*(c - c*Sin[e + f*x])^4) + (2*a*c*Cos[e + f*x]^3)/(315*f*(c^2 - c^2*Sin[e + f*x])^3)
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)}{(c-c \sin (e+f x))^5} \, dx &=(a c) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^6} \, dx\\ &=\frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}+\frac {1}{3} a \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^5} \, dx\\ &=\frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}+\frac {a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac {(2 a) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^4} \, dx}{21 c}\\ &=\frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}+\frac {a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^4}+\frac {(2 a) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^3} \, dx}{105 c^2}\\ &=\frac {a c \cos ^3(e+f x)}{9 f (c-c \sin (e+f x))^6}+\frac {a \cos ^3(e+f x)}{21 f (c-c \sin (e+f x))^5}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^4}+\frac {2 a \cos ^3(e+f x)}{315 c^2 f (c-c \sin (e+f x))^3}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 124, normalized size = 0.98 \[ \frac {a \left (36 \sin \left (2 e+\frac {5 f x}{2}\right )-\sin \left (4 e+\frac {9 f x}{2}\right )+315 \cos \left (e+\frac {f x}{2}\right )-84 \cos \left (e+\frac {3 f x}{2}\right )+9 \cos \left (3 e+\frac {7 f x}{2}\right )+189 \sin \left (\frac {f x}{2}\right )\right )}{1260 c^5 f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])/(c - c*Sin[e + f*x])^5,x]
[Out]
(a*(315*Cos[e + (f*x)/2] - 84*Cos[e + (3*f*x)/2] + 9*Cos[3*e + (7*f*x)/2] + 189*Sin[(f*x)/2] + 36*Sin[2*e + (5
*f*x)/2] - Sin[4*e + (9*f*x)/2]))/(1260*c^5*f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9)
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fricas [B] time = 0.47, size = 256, normalized size = 2.03 \[ -\frac {2 \, a \cos \left (f x + e\right )^{5} - 8 \, a \cos \left (f x + e\right )^{4} - 25 \, a \cos \left (f x + e\right )^{3} + 20 \, a \cos \left (f x + e\right )^{2} - 35 \, a \cos \left (f x + e\right ) + {\left (2 \, a \cos \left (f x + e\right )^{4} + 10 \, a \cos \left (f x + e\right )^{3} - 15 \, a \cos \left (f x + e\right )^{2} - 35 \, a \cos \left (f x + e\right ) - 70 \, a\right )} \sin \left (f x + e\right ) - 70 \, a}{315 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="fricas")
[Out]
-1/315*(2*a*cos(f*x + e)^5 - 8*a*cos(f*x + e)^4 - 25*a*cos(f*x + e)^3 + 20*a*cos(f*x + e)^2 - 35*a*cos(f*x + e
) + (2*a*cos(f*x + e)^4 + 10*a*cos(f*x + e)^3 - 15*a*cos(f*x + e)^2 - 35*a*cos(f*x + e) - 70*a)*sin(f*x + e) -
70*a)/(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 = 0.60, size = 144, normalized size = 1.14 \[ -\frac {2 \, {\left (315 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 945 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 2625 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 3465 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3843 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2247 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1143 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 207 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 58 \, a\right )}}{315 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="giac")
[Out]
-2/315*(315*a*tan(1/2*f*x + 1/2*e)^8 - 945*a*tan(1/2*f*x + 1/2*e)^7 + 2625*a*tan(1/2*f*x + 1/2*e)^6 - 3465*a*t
an(1/2*f*x + 1/2*e)^5 + 3843*a*tan(1/2*f*x + 1/2*e)^4 - 2247*a*tan(1/2*f*x + 1/2*e)^3 + 1143*a*tan(1/2*f*x + 1
/2*e)^2 - 207*a*tan(1/2*f*x + 1/2*e) + 58*a)/(c^5*f*(tan(1/2*f*x + 1/2*e) - 1)^9)
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maple [A] time = 0.27, size = 146, normalized size = 1.16 \[ \frac {2 a \left (-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {46}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {32}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {236}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {248}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {148}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}\right )}{f \,c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x)
[Out]
2/f*a/c^5*(-5/(tan(1/2*f*x+1/2*e)-1)^2-32/(tan(1/2*f*x+1/2*e)-1)^4-1/(tan(1/2*f*x+1/2*e)-1)-46/3/(tan(1/2*f*x+
1/2*e)-1)^3-32/9/(tan(1/2*f*x+1/2*e)-1)^9-16/(tan(1/2*f*x+1/2*e)-1)^8-236/5/(tan(1/2*f*x+1/2*e)-1)^5-248/7/(ta
n(1/2*f*x+1/2*e)-1)^7-148/3/(tan(1/2*f*x+1/2*e)-1)^6)
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maxima [B] time = 0.61, size = 733, normalized size = 5.82 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="maxima")
[Out]
-2/315*(a*(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/(cos
(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) - 5*a*(45*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/(c
os(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))/f
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mupad [B] time = 8.77, size = 119, normalized size = 0.94 \[ \frac {\sqrt {2}\,a\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {121\,\cos \left (3\,e+3\,f\,x\right )}{4}-\frac {1575\,\sin \left (e+f\,x\right )}{4}-\frac {625\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {635\,\cos \left (e+f\,x\right )}{4}+\frac {7\,\cos \left (4\,e+4\,f\,x\right )}{2}+\frac {399\,\sin \left (2\,e+2\,f\,x\right )}{4}+\frac {141\,\sin \left (3\,e+3\,f\,x\right )}{4}-\frac {15\,\sin \left (4\,e+4\,f\,x\right )}{4}+\frac {1357}{4}\right )}{5040\,c^5\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + a*sin(e + f*x))/(c - c*sin(e + f*x))^5,x)
[Out]
(2^(1/2)*a*cos(e/2 + (f*x)/2)*((121*cos(3*e + 3*f*x))/4 - (1575*sin(e + f*x))/4 - (625*cos(2*e + 2*f*x))/4 - (
635*cos(e + f*x))/4 + (7*cos(4*e + 4*f*x))/2 + (399*sin(2*e + 2*f*x))/4 + (141*sin(3*e + 3*f*x))/4 - (15*sin(4
*e + 4*f*x))/4 + 1357/4))/(5040*c^5*f*cos(e/2 + pi/4 + (f*x)/2)^9)
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sympy [A] time = 33.03, size = 1700, normalized size = 13.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))/(c-c*sin(f*x+e))**5,x)
[Out]
Piecewise((-630*a*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 + 1134
0*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*ta
n(e/2 + f*x/2) - 315*c**5*f) + 1890*a*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) - 5250*a*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 + 3969
0*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) + 6930*a*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) - 7686*a*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 + 2646
0*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) +
4494*a*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*t
an(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) - 2286*a*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) + 414*a*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) - 116*a/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*t
an(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)/(-c*sin(e) + c)**5, True))
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