3.287 \(\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx\)
Optimal. Leaf size=97 \[ \frac {6 \tan ^5(e+f x)}{35 a^3 c^4 f}+\frac {4 \tan ^3(e+f x)}{7 a^3 c^4 f}+\frac {6 \tan (e+f x)}{7 a^3 c^4 f}+\frac {\sec ^5(e+f x)}{7 a^3 f \left (c^4-c^4 \sin (e+f x)\right )} \]
[Out]
1/7*sec(f*x+e)^5/a^3/f/(c^4-c^4*sin(f*x+e))+6/7*tan(f*x+e)/a^3/c^4/f+4/7*tan(f*x+e)^3/a^3/c^4/f+6/35*tan(f*x+e
)^5/a^3/c^4/f
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Rubi [A] time = 0.12, antiderivative size = 97, 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, 3767} \[ \frac {6 \tan ^5(e+f x)}{35 a^3 c^4 f}+\frac {4 \tan ^3(e+f x)}{7 a^3 c^4 f}+\frac {6 \tan (e+f x)}{7 a^3 c^4 f}+\frac {\sec ^5(e+f x)}{7 a^3 f \left (c^4-c^4 \sin (e+f x)\right )} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4),x]
[Out]
Sec[e + f*x]^5/(7*a^3*f*(c^4 - c^4*Sin[e + f*x])) + (6*Tan[e + f*x])/(7*a^3*c^4*f) + (4*Tan[e + f*x]^3)/(7*a^3
*c^4*f) + (6*Tan[e + f*x]^5)/(35*a^3*c^4*f)
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]))
Rule 3767
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^4} \, dx &=\frac {\int \frac {\sec ^6(e+f x)}{c-c \sin (e+f x)} \, dx}{a^3 c^3}\\ &=\frac {\sec ^5(e+f x)}{7 a^3 f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {6 \int \sec ^6(e+f x) \, dx}{7 a^3 c^4}\\ &=\frac {\sec ^5(e+f x)}{7 a^3 f \left (c^4-c^4 \sin (e+f x)\right )}-\frac {6 \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{7 a^3 c^4 f}\\ &=\frac {\sec ^5(e+f x)}{7 a^3 f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {6 \tan (e+f x)}{7 a^3 c^4 f}+\frac {4 \tan ^3(e+f x)}{7 a^3 c^4 f}+\frac {6 \tan ^5(e+f x)}{35 a^3 c^4 f}\\ \end {align*}
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Mathematica [A] time = 1.19, size = 193, normalized size = 1.99 \[ \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (5120 \sin (e+f x)+125 \sin (2 (e+f x))+2560 \sin (3 (e+f x))+100 \sin (4 (e+f x))+512 \sin (5 (e+f x))+25 \sin (6 (e+f x))-500 \cos (e+f x)+1280 \cos (2 (e+f x))-250 \cos (3 (e+f x))+1024 \cos (4 (e+f x))-50 \cos (5 (e+f x))+256 \cos (6 (e+f x)))}{17920 f (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^4} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4),x]
[Out]
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-500*Cos[e + f*x] + 1280*Cos[2*(
e + f*x)] - 250*Cos[3*(e + f*x)] + 1024*Cos[4*(e + f*x)] - 50*Cos[5*(e + f*x)] + 256*Cos[6*(e + f*x)] + 5120*S
in[e + f*x] + 125*Sin[2*(e + f*x)] + 2560*Sin[3*(e + f*x)] + 100*Sin[4*(e + f*x)] + 512*Sin[5*(e + f*x)] + 25*
Sin[6*(e + f*x)]))/(17920*f*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^4)
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fricas [A] time = 0.47, size = 106, normalized size = 1.09 \[ -\frac {16 \, \cos \left (f x + e\right )^{6} - 8 \, \cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 2 \, {\left (8 \, \cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{2} + 3\right )} \sin \left (f x + e\right ) - 1}{35 \, {\left (a^{3} c^{4} f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right ) - a^{3} c^{4} f \cos \left (f x + e\right )^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algorithm="fricas")
[Out]
-1/35*(16*cos(f*x + e)^6 - 8*cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 2*(8*cos(f*x + e)^4 + 4*cos(f*x + e)^2 + 3)*s
in(f*x + e) - 1)/(a^3*c^4*f*cos(f*x + e)^5*sin(f*x + e) - a^3*c^4*f*cos(f*x + e)^5)
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giac [B] time = 1.43, size = 189, normalized size = 1.95 \[ -\frac {\frac {7 \, {\left (55 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 180 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 250 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 160 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 43\right )}}{a^{3} c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} + \frac {735 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 3360 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 7315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 8820 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6321 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2492 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 461}{a^{3} c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}}}{560 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algorithm="giac")
[Out]
-1/560*(7*(55*tan(1/2*f*x + 1/2*e)^4 + 180*tan(1/2*f*x + 1/2*e)^3 + 250*tan(1/2*f*x + 1/2*e)^2 + 160*tan(1/2*f
*x + 1/2*e) + 43)/(a^3*c^4*(tan(1/2*f*x + 1/2*e) + 1)^5) + (735*tan(1/2*f*x + 1/2*e)^6 - 3360*tan(1/2*f*x + 1/
2*e)^5 + 7315*tan(1/2*f*x + 1/2*e)^4 - 8820*tan(1/2*f*x + 1/2*e)^3 + 6321*tan(1/2*f*x + 1/2*e)^2 - 2492*tan(1/
2*f*x + 1/2*e) + 461)/(a^3*c^4*(tan(1/2*f*x + 1/2*e) - 1)^7))/f
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maple [B] time = 0.29, size = 193, normalized size = 1.99 \[ \frac {-\frac {2}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {1}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {21}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {11}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {11}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {15}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {21}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{10 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {1}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {11}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{f \,c^{4} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x)
[Out]
2/f/c^4/a^3*(-1/7/(tan(1/2*f*x+1/2*e)-1)^7-1/2/(tan(1/2*f*x+1/2*e)-1)^6-21/20/(tan(1/2*f*x+1/2*e)-1)^5-11/8/(t
an(1/2*f*x+1/2*e)-1)^4-11/8/(tan(1/2*f*x+1/2*e)-1)^3-15/16/(tan(1/2*f*x+1/2*e)-1)^2-21/32/(tan(1/2*f*x+1/2*e)-
1)-1/20/(tan(1/2*f*x+1/2*e)+1)^5+1/8/(tan(1/2*f*x+1/2*e)+1)^4-1/4/(tan(1/2*f*x+1/2*e)+1)^3+1/4/(tan(1/2*f*x+1/
2*e)+1)^2-11/32/(tan(1/2*f*x+1/2*e)+1))
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maxima [B] time = 1.02, size = 519, normalized size = 5.35 \[ \frac {2 \, {\left (\frac {25 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {55 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {15 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {130 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {26 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {182 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {126 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {105 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {35 \, \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} - \frac {35 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {35 \, \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} + 5\right )}}{35 \, {\left (a^{3} c^{4} - \frac {2 \, a^{3} c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {4 \, a^{3} c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {20 \, a^{3} c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {20 \, a^{3} c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {5 \, a^{3} c^{4} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {10 \, a^{3} c^{4} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {4 \, a^{3} c^{4} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {2 \, a^{3} c^{4} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - \frac {a^{3} c^{4} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algorithm="maxima")
[Out]
2/35*(25*sin(f*x + e)/(cos(f*x + e) + 1) - 55*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 15*sin(f*x + e)^3/(cos(f*x
+ e) + 1)^3 + 130*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 26*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 182*sin(f*x
+ e)^6/(cos(f*x + e) + 1)^6 + 126*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 105*sin(f*x + e)^8/(cos(f*x + e) + 1)^
8 - 35*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 35*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 35*sin(f*x + e)^11/(co
s(f*x + e) + 1)^11 + 5)/((a^3*c^4 - 2*a^3*c^4*sin(f*x + e)/(cos(f*x + e) + 1) - 4*a^3*c^4*sin(f*x + e)^2/(cos(
f*x + e) + 1)^2 + 10*a^3*c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 5*a^3*c^4*sin(f*x + e)^4/(cos(f*x + e) + 1)
^4 - 20*a^3*c^4*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 20*a^3*c^4*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 5*a^3*c
^4*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - 10*a^3*c^4*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 4*a^3*c^4*sin(f*x +
e)^10/(cos(f*x + e) + 1)^10 + 2*a^3*c^4*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 - a^3*c^4*sin(f*x + e)^12/(cos(f
*x + e) + 1)^12)*f)
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mupad [B] time = 9.42, size = 180, normalized size = 1.86 \[ -\frac {2\,\left (35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}-35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+105\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+126\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7-182\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+26\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+130\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3-55\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+25\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+5\right )}{35\,a^3\,c^4\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-1\right )}^7\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^4),x)
[Out]
-(2*(25*tan(e/2 + (f*x)/2) - 55*tan(e/2 + (f*x)/2)^2 + 15*tan(e/2 + (f*x)/2)^3 + 130*tan(e/2 + (f*x)/2)^4 + 26
*tan(e/2 + (f*x)/2)^5 - 182*tan(e/2 + (f*x)/2)^6 + 126*tan(e/2 + (f*x)/2)^7 + 105*tan(e/2 + (f*x)/2)^8 - 35*ta
n(e/2 + (f*x)/2)^9 - 35*tan(e/2 + (f*x)/2)^10 + 35*tan(e/2 + (f*x)/2)^11 + 5))/(35*a^3*c^4*f*(tan(e/2 + (f*x)/
2) - 1)^7*(tan(e/2 + (f*x)/2) + 1)^5)
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sympy [A] time = 55.96, size = 3186, normalized size = 32.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**4,x)
[Out]
Piecewise((-70*tan(e/2 + f*x/2)**11/(35*a**3*c**4*f*tan(e/2 + f*x/2)**12 - 70*a**3*c**4*f*tan(e/2 + f*x/2)**11
- 140*a**3*c**4*f*tan(e/2 + f*x/2)**10 + 350*a**3*c**4*f*tan(e/2 + f*x/2)**9 + 175*a**3*c**4*f*tan(e/2 + f*x/
2)**8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 700*a**3*c**4*f*tan(e/2 + f*x/2)**5 - 175*a**3*c**4*f*tan(e/2 +
f*x/2)**4 - 350*a**3*c**4*f*tan(e/2 + f*x/2)**3 + 140*a**3*c**4*f*tan(e/2 + f*x/2)**2 + 70*a**3*c**4*f*tan(e/2
+ f*x/2) - 35*a**3*c**4*f) + 70*tan(e/2 + f*x/2)**10/(35*a**3*c**4*f*tan(e/2 + f*x/2)**12 - 70*a**3*c**4*f*ta
n(e/2 + f*x/2)**11 - 140*a**3*c**4*f*tan(e/2 + f*x/2)**10 + 350*a**3*c**4*f*tan(e/2 + f*x/2)**9 + 175*a**3*c**
4*f*tan(e/2 + f*x/2)**8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 700*a**3*c**4*f*tan(e/2 + f*x/2)**5 - 175*a**3
*c**4*f*tan(e/2 + f*x/2)**4 - 350*a**3*c**4*f*tan(e/2 + f*x/2)**3 + 140*a**3*c**4*f*tan(e/2 + f*x/2)**2 + 70*a
**3*c**4*f*tan(e/2 + f*x/2) - 35*a**3*c**4*f) + 70*tan(e/2 + f*x/2)**9/(35*a**3*c**4*f*tan(e/2 + f*x/2)**12 -
70*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 140*a**3*c**4*f*tan(e/2 + f*x/2)**10 + 350*a**3*c**4*f*tan(e/2 + f*x/2)*
*9 + 175*a**3*c**4*f*tan(e/2 + f*x/2)**8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 700*a**3*c**4*f*tan(e/2 + f*x
/2)**5 - 175*a**3*c**4*f*tan(e/2 + f*x/2)**4 - 350*a**3*c**4*f*tan(e/2 + f*x/2)**3 + 140*a**3*c**4*f*tan(e/2 +
f*x/2)**2 + 70*a**3*c**4*f*tan(e/2 + f*x/2) - 35*a**3*c**4*f) - 210*tan(e/2 + f*x/2)**8/(35*a**3*c**4*f*tan(e
/2 + f*x/2)**12 - 70*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 140*a**3*c**4*f*tan(e/2 + f*x/2)**10 + 350*a**3*c**4*f
*tan(e/2 + f*x/2)**9 + 175*a**3*c**4*f*tan(e/2 + f*x/2)**8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 700*a**3*c*
*4*f*tan(e/2 + f*x/2)**5 - 175*a**3*c**4*f*tan(e/2 + f*x/2)**4 - 350*a**3*c**4*f*tan(e/2 + f*x/2)**3 + 140*a**
3*c**4*f*tan(e/2 + f*x/2)**2 + 70*a**3*c**4*f*tan(e/2 + f*x/2) - 35*a**3*c**4*f) - 252*tan(e/2 + f*x/2)**7/(35
*a**3*c**4*f*tan(e/2 + f*x/2)**12 - 70*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 140*a**3*c**4*f*tan(e/2 + f*x/2)**10
+ 350*a**3*c**4*f*tan(e/2 + f*x/2)**9 + 175*a**3*c**4*f*tan(e/2 + f*x/2)**8 - 700*a**3*c**4*f*tan(e/2 + f*x/2
)**7 + 700*a**3*c**4*f*tan(e/2 + f*x/2)**5 - 175*a**3*c**4*f*tan(e/2 + f*x/2)**4 - 350*a**3*c**4*f*tan(e/2 + f
*x/2)**3 + 140*a**3*c**4*f*tan(e/2 + f*x/2)**2 + 70*a**3*c**4*f*tan(e/2 + f*x/2) - 35*a**3*c**4*f) + 364*tan(e
/2 + f*x/2)**6/(35*a**3*c**4*f*tan(e/2 + f*x/2)**12 - 70*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 140*a**3*c**4*f*ta
n(e/2 + f*x/2)**10 + 350*a**3*c**4*f*tan(e/2 + f*x/2)**9 + 175*a**3*c**4*f*tan(e/2 + f*x/2)**8 - 700*a**3*c**4
*f*tan(e/2 + f*x/2)**7 + 700*a**3*c**4*f*tan(e/2 + f*x/2)**5 - 175*a**3*c**4*f*tan(e/2 + f*x/2)**4 - 350*a**3*
c**4*f*tan(e/2 + f*x/2)**3 + 140*a**3*c**4*f*tan(e/2 + f*x/2)**2 + 70*a**3*c**4*f*tan(e/2 + f*x/2) - 35*a**3*c
**4*f) - 52*tan(e/2 + f*x/2)**5/(35*a**3*c**4*f*tan(e/2 + f*x/2)**12 - 70*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 1
40*a**3*c**4*f*tan(e/2 + f*x/2)**10 + 350*a**3*c**4*f*tan(e/2 + f*x/2)**9 + 175*a**3*c**4*f*tan(e/2 + f*x/2)**
8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 700*a**3*c**4*f*tan(e/2 + f*x/2)**5 - 175*a**3*c**4*f*tan(e/2 + f*x/
2)**4 - 350*a**3*c**4*f*tan(e/2 + f*x/2)**3 + 140*a**3*c**4*f*tan(e/2 + f*x/2)**2 + 70*a**3*c**4*f*tan(e/2 + f
*x/2) - 35*a**3*c**4*f) - 260*tan(e/2 + f*x/2)**4/(35*a**3*c**4*f*tan(e/2 + f*x/2)**12 - 70*a**3*c**4*f*tan(e/
2 + f*x/2)**11 - 140*a**3*c**4*f*tan(e/2 + f*x/2)**10 + 350*a**3*c**4*f*tan(e/2 + f*x/2)**9 + 175*a**3*c**4*f*
tan(e/2 + f*x/2)**8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 700*a**3*c**4*f*tan(e/2 + f*x/2)**5 - 175*a**3*c**
4*f*tan(e/2 + f*x/2)**4 - 350*a**3*c**4*f*tan(e/2 + f*x/2)**3 + 140*a**3*c**4*f*tan(e/2 + f*x/2)**2 + 70*a**3*
c**4*f*tan(e/2 + f*x/2) - 35*a**3*c**4*f) - 30*tan(e/2 + f*x/2)**3/(35*a**3*c**4*f*tan(e/2 + f*x/2)**12 - 70*a
**3*c**4*f*tan(e/2 + f*x/2)**11 - 140*a**3*c**4*f*tan(e/2 + f*x/2)**10 + 350*a**3*c**4*f*tan(e/2 + f*x/2)**9 +
175*a**3*c**4*f*tan(e/2 + f*x/2)**8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 700*a**3*c**4*f*tan(e/2 + f*x/2)*
*5 - 175*a**3*c**4*f*tan(e/2 + f*x/2)**4 - 350*a**3*c**4*f*tan(e/2 + f*x/2)**3 + 140*a**3*c**4*f*tan(e/2 + f*x
/2)**2 + 70*a**3*c**4*f*tan(e/2 + f*x/2) - 35*a**3*c**4*f) + 110*tan(e/2 + f*x/2)**2/(35*a**3*c**4*f*tan(e/2 +
f*x/2)**12 - 70*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 140*a**3*c**4*f*tan(e/2 + f*x/2)**10 + 350*a**3*c**4*f*tan
(e/2 + f*x/2)**9 + 175*a**3*c**4*f*tan(e/2 + f*x/2)**8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 700*a**3*c**4*f
*tan(e/2 + f*x/2)**5 - 175*a**3*c**4*f*tan(e/2 + f*x/2)**4 - 350*a**3*c**4*f*tan(e/2 + f*x/2)**3 + 140*a**3*c*
*4*f*tan(e/2 + f*x/2)**2 + 70*a**3*c**4*f*tan(e/2 + f*x/2) - 35*a**3*c**4*f) - 50*tan(e/2 + f*x/2)/(35*a**3*c*
*4*f*tan(e/2 + f*x/2)**12 - 70*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 140*a**3*c**4*f*tan(e/2 + f*x/2)**10 + 350*a
**3*c**4*f*tan(e/2 + f*x/2)**9 + 175*a**3*c**4*f*tan(e/2 + f*x/2)**8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 7
00*a**3*c**4*f*tan(e/2 + f*x/2)**5 - 175*a**3*c**4*f*tan(e/2 + f*x/2)**4 - 350*a**3*c**4*f*tan(e/2 + f*x/2)**3
+ 140*a**3*c**4*f*tan(e/2 + f*x/2)**2 + 70*a**3*c**4*f*tan(e/2 + f*x/2) - 35*a**3*c**4*f) - 10/(35*a**3*c**4*
f*tan(e/2 + f*x/2)**12 - 70*a**3*c**4*f*tan(e/2 + f*x/2)**11 - 140*a**3*c**4*f*tan(e/2 + f*x/2)**10 + 350*a**3
*c**4*f*tan(e/2 + f*x/2)**9 + 175*a**3*c**4*f*tan(e/2 + f*x/2)**8 - 700*a**3*c**4*f*tan(e/2 + f*x/2)**7 + 700*
a**3*c**4*f*tan(e/2 + f*x/2)**5 - 175*a**3*c**4*f*tan(e/2 + f*x/2)**4 - 350*a**3*c**4*f*tan(e/2 + f*x/2)**3 +
140*a**3*c**4*f*tan(e/2 + f*x/2)**2 + 70*a**3*c**4*f*tan(e/2 + f*x/2) - 35*a**3*c**4*f), Ne(f, 0)), (x/((a*sin
(e) + a)**3*(-c*sin(e) + c)**4), True))
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