3.289 \(\int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx\)
Optimal. Leaf size=167 \[ \frac {16 \tan ^5(e+f x)}{165 a^3 c^6 f}+\frac {32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac {16 \tan (e+f x)}{33 a^3 c^6 f}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3} \]
[Out]
1/11*sec(f*x+e)^5/a^3/f/(c^2-c^2*sin(f*x+e))^3+8/99*sec(f*x+e)^5/a^3/f/(c^3-c^3*sin(f*x+e))^2+8/99*sec(f*x+e)^
5/a^3/f/(c^6-c^6*sin(f*x+e))+16/33*tan(f*x+e)/a^3/c^6/f+32/99*tan(f*x+e)^3/a^3/c^6/f+16/165*tan(f*x+e)^5/a^3/c
^6/f
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Rubi [A] time = 0.22, antiderivative size = 167, normalized size of antiderivative = 1.00,
number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used =
{2736, 2672, 3767} \[ \frac {16 \tan ^5(e+f x)}{165 a^3 c^6 f}+\frac {32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac {16 \tan (e+f x)}{33 a^3 c^6 f}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x]
[Out]
Sec[e + f*x]^5/(11*a^3*f*(c^2 - c^2*Sin[e + f*x])^3) + (8*Sec[e + f*x]^5)/(99*a^3*f*(c^3 - c^3*Sin[e + f*x])^2
) + (8*Sec[e + f*x]^5)/(99*a^3*f*(c^6 - c^6*Sin[e + f*x])) + (16*Tan[e + f*x])/(33*a^3*c^6*f) + (32*Tan[e + f*
x]^3)/(99*a^3*c^6*f) + (16*Tan[e + f*x]^5)/(165*a^3*c^6*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))^6} \, dx &=\frac {\int \frac {\sec ^6(e+f x)}{(c-c \sin (e+f x))^3} \, dx}{a^3 c^3}\\ &=\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \int \frac {\sec ^6(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{11 a^3 c^4}\\ &=\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {56 \int \frac {\sec ^6(e+f x)}{c-c \sin (e+f x)} \, dx}{99 a^3 c^5}\\ &=\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {16 \int \sec ^6(e+f x) \, dx}{33 a^3 c^6}\\ &=\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}-\frac {16 \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{33 a^3 c^6 f}\\ &=\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {16 \tan (e+f x)}{33 a^3 c^6 f}+\frac {32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac {16 \tan ^5(e+f x)}{165 a^3 c^6 f}\\ \end {align*}
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Mathematica [A] time = 1.63, size = 233, normalized size = 1.40 \[ \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (1802240 \sin (e+f x)+247170 \sin (2 (e+f x))+557056 \sin (3 (e+f x))+187250 \sin (4 (e+f x))-163840 \sin (5 (e+f x))+37450 \sin (6 (e+f x))-98304 \sin (7 (e+f x))-3745 \sin (8 (e+f x))-411950 \cos (e+f x)+1081344 \cos (2 (e+f x))-127330 \cos (3 (e+f x))+819200 \cos (4 (e+f x))+37450 \cos (5 (e+f x))+163840 \cos (6 (e+f x))+22470 \cos (7 (e+f x))-16384 \cos (8 (e+f x)))}{8110080 f (a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^6} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6),x]
[Out]
((Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-411950*Cos[e + f*x] + 1081344*C
os[2*(e + f*x)] - 127330*Cos[3*(e + f*x)] + 819200*Cos[4*(e + f*x)] + 37450*Cos[5*(e + f*x)] + 163840*Cos[6*(e
+ f*x)] + 22470*Cos[7*(e + f*x)] - 16384*Cos[8*(e + f*x)] + 1802240*Sin[e + f*x] + 247170*Sin[2*(e + f*x)] +
557056*Sin[3*(e + f*x)] + 187250*Sin[4*(e + f*x)] - 163840*Sin[5*(e + f*x)] + 37450*Sin[6*(e + f*x)] - 98304*S
in[7*(e + f*x)] - 3745*Sin[8*(e + f*x)]))/(8110080*f*(a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6)
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fricas [A] time = 0.46, size = 163, normalized size = 0.98 \[ \frac {128 \, \cos \left (f x + e\right )^{8} - 576 \, \cos \left (f x + e\right )^{6} + 240 \, \cos \left (f x + e\right )^{4} + 56 \, \cos \left (f x + e\right )^{2} + 8 \, {\left (48 \, \cos \left (f x + e\right )^{6} - 40 \, \cos \left (f x + e\right )^{4} - 14 \, \cos \left (f x + e\right )^{2} - 9\right )} \sin \left (f x + e\right ) + 27}{495 \, {\left (3 \, a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5} - {\left (a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5}\right )} \sin \left (f x + e\right )\right )}} \]
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))^6,x, algorithm="fricas")
[Out]
1/495*(128*cos(f*x + e)^8 - 576*cos(f*x + e)^6 + 240*cos(f*x + e)^4 + 56*cos(f*x + e)^2 + 8*(48*cos(f*x + e)^6
- 40*cos(f*x + e)^4 - 14*cos(f*x + e)^2 - 9)*sin(f*x + e) + 27)/(3*a^3*c^6*f*cos(f*x + e)^7 - 4*a^3*c^6*f*cos
(f*x + e)^5 - (a^3*c^6*f*cos(f*x + e)^7 - 4*a^3*c^6*f*cos(f*x + e)^5)*sin(f*x + e))
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giac [A] time = 0.26, size = 245, normalized size = 1.47 \[ -\frac {\frac {33 \, {\left (555 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1920 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2710 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1760 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 463\right )}}{a^{3} c^{6} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} + \frac {108405 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 784080 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 2901195 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 6652800 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 10407474 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 11435424 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 8949270 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 4899840 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1816265 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 411664 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 47279}{a^{3} c^{6} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{11}}}{63360 \, f} \]
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))^6,x, algorithm="giac")
[Out]
-1/63360*(33*(555*tan(1/2*f*x + 1/2*e)^4 + 1920*tan(1/2*f*x + 1/2*e)^3 + 2710*tan(1/2*f*x + 1/2*e)^2 + 1760*ta
n(1/2*f*x + 1/2*e) + 463)/(a^3*c^6*(tan(1/2*f*x + 1/2*e) + 1)^5) + (108405*tan(1/2*f*x + 1/2*e)^10 - 784080*ta
n(1/2*f*x + 1/2*e)^9 + 2901195*tan(1/2*f*x + 1/2*e)^8 - 6652800*tan(1/2*f*x + 1/2*e)^7 + 10407474*tan(1/2*f*x
+ 1/2*e)^6 - 11435424*tan(1/2*f*x + 1/2*e)^5 + 8949270*tan(1/2*f*x + 1/2*e)^4 - 4899840*tan(1/2*f*x + 1/2*e)^3
+ 1816265*tan(1/2*f*x + 1/2*e)^2 - 411664*tan(1/2*f*x + 1/2*e) + 47279)/(a^3*c^6*(tan(1/2*f*x + 1/2*e) - 1)^1
1))/f
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maple [A] time = 0.32, size = 253, normalized size = 1.51 \[ \frac {-\frac {8}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {106}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {23}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {33}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {217}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {623}{20 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {169}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {365}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {303}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {219}{128 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{40 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {7}{48 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {5}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {37}{128 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{f \,a^{3} c^{6}} \]
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))^6,x)
[Out]
2/f/a^3/c^6*(-4/11/(tan(1/2*f*x+1/2*e)-1)^11-2/(tan(1/2*f*x+1/2*e)-1)^10-53/9/(tan(1/2*f*x+1/2*e)-1)^9-23/2/(t
an(1/2*f*x+1/2*e)-1)^8-33/2/(tan(1/2*f*x+1/2*e)-1)^7-217/12/(tan(1/2*f*x+1/2*e)-1)^6-623/40/(tan(1/2*f*x+1/2*e
)-1)^5-169/16/(tan(1/2*f*x+1/2*e)-1)^4-365/64/(tan(1/2*f*x+1/2*e)-1)^3-303/128/(tan(1/2*f*x+1/2*e)-1)^2-219/25
6/(tan(1/2*f*x+1/2*e)-1)-1/80/(tan(1/2*f*x+1/2*e)+1)^5+1/32/(tan(1/2*f*x+1/2*e)+1)^4-7/96/(tan(1/2*f*x+1/2*e)+
1)^3+5/64/(tan(1/2*f*x+1/2*e)+1)^2-37/256/(tan(1/2*f*x+1/2*e)+1))
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maxima [B] time = 0.73, size = 703, normalized size = 4.21 \[ \text {result too large to display} \]
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))^6,x, algorithm="maxima")
[Out]
-2/495*(255*sin(f*x + e)/(cos(f*x + e) + 1) + 235*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 3065*sin(f*x + e)^3/(c
os(f*x + e) + 1)^3 + 3775*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 667*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 8217
*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 2035*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 8745*sin(f*x + e)^8/(cos(f*x
+ e) + 1)^8 - 11715*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 + 33*sin(f*x + e)^10/(cos(f*x + e) + 1)^10 + 4917*sin
(f*x + e)^11/(cos(f*x + e) + 1)^11 - 2475*sin(f*x + e)^12/(cos(f*x + e) + 1)^12 - 1815*sin(f*x + e)^13/(cos(f*
x + e) + 1)^13 + 1485*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 - 495*sin(f*x + e)^15/(cos(f*x + e) + 1)^15 - 125)
/((a^3*c^6 - 6*a^3*c^6*sin(f*x + e)/(cos(f*x + e) + 1) + 10*a^3*c^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a
^3*c^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 50*a^3*c^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 34*a^3*c^6*sin(f
*x + e)^5/(cos(f*x + e) + 1)^5 + 66*a^3*c^6*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 110*a^3*c^6*sin(f*x + e)^7/(
cos(f*x + e) + 1)^7 + 110*a^3*c^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9 - 66*a^3*c^6*sin(f*x + e)^10/(cos(f*x +
e) + 1)^10 - 34*a^3*c^6*sin(f*x + e)^11/(cos(f*x + e) + 1)^11 + 50*a^3*c^6*sin(f*x + e)^12/(cos(f*x + e) + 1)^
12 - 10*a^3*c^6*sin(f*x + e)^13/(cos(f*x + e) + 1)^13 - 10*a^3*c^6*sin(f*x + e)^14/(cos(f*x + e) + 1)^14 + 6*a
^3*c^6*sin(f*x + e)^15/(cos(f*x + e) + 1)^15 - a^3*c^6*sin(f*x + e)^16/(cos(f*x + e) + 1)^16)*f)
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mupad [B] time = 8.62, size = 185, normalized size = 1.11 \[ -\frac {\frac {2\,\sin \left (e+f\,x\right )}{9}+\frac {2\,\cos \left (2\,e+2\,f\,x\right )}{15}+\frac {10\,\cos \left (4\,e+4\,f\,x\right )}{99}+\frac {2\,\cos \left (6\,e+6\,f\,x\right )}{99}-\frac {\cos \left (8\,e+8\,f\,x\right )}{495}+\frac {34\,\sin \left (3\,e+3\,f\,x\right )}{495}-\frac {2\,\sin \left (5\,e+5\,f\,x\right )}{99}-\frac {2\,\sin \left (7\,e+7\,f\,x\right )}{165}}{a^3\,c^6\,f\,\left (\frac {5\,\cos \left (5\,e+5\,f\,x\right )}{64}-\frac {17\,\cos \left (3\,e+3\,f\,x\right )}{64}-\frac {55\,\cos \left (e+f\,x\right )}{64}+\frac {3\,\cos \left (7\,e+7\,f\,x\right )}{64}+\frac {33\,\sin \left (2\,e+2\,f\,x\right )}{64}+\frac {25\,\sin \left (4\,e+4\,f\,x\right )}{64}+\frac {5\,\sin \left (6\,e+6\,f\,x\right )}{64}-\frac {\sin \left (8\,e+8\,f\,x\right )}{128}\right )} \]
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))^6),x)
[Out]
-((2*sin(e + f*x))/9 + (2*cos(2*e + 2*f*x))/15 + (10*cos(4*e + 4*f*x))/99 + (2*cos(6*e + 6*f*x))/99 - cos(8*e
+ 8*f*x)/495 + (34*sin(3*e + 3*f*x))/495 - (2*sin(5*e + 5*f*x))/99 - (2*sin(7*e + 7*f*x))/165)/(a^3*c^6*f*((5*
cos(5*e + 5*f*x))/64 - (17*cos(3*e + 3*f*x))/64 - (55*cos(e + f*x))/64 + (3*cos(7*e + 7*f*x))/64 + (33*sin(2*e
+ 2*f*x))/64 + (25*sin(4*e + 4*f*x))/64 + (5*sin(6*e + 6*f*x))/64 - sin(8*e + 8*f*x)/128))
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sympy [A] time = 160.15, size = 5661, normalized size = 33.90 \[ \text {result too large to display} \]
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))**6,x)
[Out]
Piecewise((-990*tan(e/2 + f*x/2)**15/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)
**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e
/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3
*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 -
16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x
/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 2970*ta
n(e/2 + f*x/2)**14/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c
**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 +
16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 +
f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*
tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3
*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) - 3630*tan(e/2 + f*x/2)**13
/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*
x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*
tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*
a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**
5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 +
f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) - 4950*tan(e/2 + f*x/2)**12/(495*a**3*c**6*f*
tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a*
*3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**
11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/
2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**
6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a
**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 9834*tan(e/2 + f*x/2)**11/(495*a**3*c**6*f*tan(e/2 + f*x/2)**
16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2
+ f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c*
*6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32
670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/
2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2
+ f*x/2) - 495*a**3*c**6*f) + 66*tan(e/2 + f*x/2)**10/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*
f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750
*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2
)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(
e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c*
*6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3
*c**6*f) - 23430*tan(e/2 + f*x/2)**9/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)
**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e
/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3
*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 -
16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x
/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 17490*t
an(e/2 + f*x/2)**8/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c
**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 +
16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 +
f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*
tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3
*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 4070*tan(e/2 + f*x/2)**7/
(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x
/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*t
an(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a
**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5
+ 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f
*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) - 16434*tan(e/2 + f*x/2)**6/(495*a**3*c**6*f*t
an(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**
3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**1
1 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2
+ f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6
*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a*
*3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 1334*tan(e/2 + f*x/2)**5/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16
- 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 +
f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6
*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 3267
0*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)
**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 +
f*x/2) - 495*a**3*c**6*f) + 7550*tan(e/2 + f*x/2)**4/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f
*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*
a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)
**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e
/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**
6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*
c**6*f) - 6130*tan(e/2 + f*x/2)**3/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**
15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2
+ f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c
**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16
830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2
)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 470*tan(e
/2 + f*x/2)**2/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*
f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 1683
0*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/
2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(
e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**
6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) + 510*tan(e/2 + f*x/2)/(495*a**
3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14
+ 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13 - 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 +
f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2 + f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6
*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750
*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2
+ 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 495*a**3*c**6*f) - 250/(495*a**3*c**6*f*tan(e/2 + f*x/2)**16 - 2970*a**
3*c**6*f*tan(e/2 + f*x/2)**15 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**14 + 4950*a**3*c**6*f*tan(e/2 + f*x/2)**13
- 24750*a**3*c**6*f*tan(e/2 + f*x/2)**12 + 16830*a**3*c**6*f*tan(e/2 + f*x/2)**11 + 32670*a**3*c**6*f*tan(e/2
+ f*x/2)**10 - 54450*a**3*c**6*f*tan(e/2 + f*x/2)**9 + 54450*a**3*c**6*f*tan(e/2 + f*x/2)**7 - 32670*a**3*c**6
*f*tan(e/2 + f*x/2)**6 - 16830*a**3*c**6*f*tan(e/2 + f*x/2)**5 + 24750*a**3*c**6*f*tan(e/2 + f*x/2)**4 - 4950*
a**3*c**6*f*tan(e/2 + f*x/2)**3 - 4950*a**3*c**6*f*tan(e/2 + f*x/2)**2 + 2970*a**3*c**6*f*tan(e/2 + f*x/2) - 4
95*a**3*c**6*f), Ne(f, 0)), (x/((a*sin(e) + a)**3*(-c*sin(e) + c)**6), True))
________________________________________________________________________________________