3.321 \(\int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx\)

Optimal. Leaf size=144 \[ \frac {4 \cos (e+f x)}{315 f \left (a^6 \sin (e+f x)+a^6\right )}+\frac {4 \cos (e+f x)}{315 f \left (a^3 \sin (e+f x)+a^3\right )^2}+\frac {2 \cos (e+f x)}{105 f \left (a^2 \sin (e+f x)+a^2\right )^3}-\frac {19 \cos (e+f x)}{63 a^2 f (a \sin (e+f x)+a)^4}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5} \]

[Out]

2/9*cos(f*x+e)/a/f/(a+a*sin(f*x+e))^5-19/63*cos(f*x+e)/a^2/f/(a+a*sin(f*x+e))^4+2/105*cos(f*x+e)/f/(a^2+a^2*si
n(f*x+e))^3+4/315*cos(f*x+e)/f/(a^3+a^3*sin(f*x+e))^2+4/315*cos(f*x+e)/f/(a^6+a^6*sin(f*x+e))

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Rubi [A]  time = 0.16, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2857, 2750, 2650, 2648} \[ \frac {4 \cos (e+f x)}{315 f \left (a^6 \sin (e+f x)+a^6\right )}+\frac {4 \cos (e+f x)}{315 f \left (a^3 \sin (e+f x)+a^3\right )^2}+\frac {2 \cos (e+f x)}{105 f \left (a^2 \sin (e+f x)+a^2\right )^3}-\frac {19 \cos (e+f x)}{63 a^2 f (a \sin (e+f x)+a)^4}+\frac {2 \cos (e+f x)}{9 a f (a \sin (e+f x)+a)^5} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[e + f*x]^2*Sin[e + f*x])/(a + a*Sin[e + f*x])^6,x]

[Out]

(2*Cos[e + f*x])/(9*a*f*(a + a*Sin[e + f*x])^5) - (19*Cos[e + f*x])/(63*a^2*f*(a + a*Sin[e + f*x])^4) + (2*Cos
[e + f*x])/(105*f*(a^2 + a^2*Sin[e + f*x])^3) + (4*Cos[e + f*x])/(315*f*(a^3 + a^3*Sin[e + f*x])^2) + (4*Cos[e
+ f*x])/(315*f*(a^6 + a^6*Sin[e + f*x]))

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2650

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(a*
d*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 2750

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

Rule 2857

Int[cos[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_
)]), x_Symbol] :> Simp[(2*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(2*m + 3)), x] + Dist[
1/(b^3*(2*m + 3)), Int[(a + b*Sin[e + f*x])^(m + 2)*(b*c + 2*a*d*(m + 1) - b*d*(2*m + 3)*Sin[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -3/2]

Rubi steps

\begin {align*} \int \frac {\cos ^2(e+f x) \sin (e+f x)}{(a+a \sin (e+f x))^6} \, dx &=\frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {\int \frac {-10 a+9 a \sin (e+f x)}{(a+a \sin (e+f x))^4} \, dx}{9 a^3}\\ &=\frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {19 \cos (e+f x)}{63 a^2 f (a+a \sin (e+f x))^4}-\frac {2 \int \frac {1}{(a+a \sin (e+f x))^3} \, dx}{21 a^3}\\ &=\frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {19 \cos (e+f x)}{63 a^2 f (a+a \sin (e+f x))^4}+\frac {2 \cos (e+f x)}{105 f \left (a^2+a^2 \sin (e+f x)\right )^3}-\frac {4 \int \frac {1}{(a+a \sin (e+f x))^2} \, dx}{105 a^4}\\ &=\frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {19 \cos (e+f x)}{63 a^2 f (a+a \sin (e+f x))^4}+\frac {2 \cos (e+f x)}{105 f \left (a^2+a^2 \sin (e+f x)\right )^3}+\frac {4 \cos (e+f x)}{315 f \left (a^3+a^3 \sin (e+f x)\right )^2}-\frac {4 \int \frac {1}{a+a \sin (e+f x)} \, dx}{315 a^5}\\ &=\frac {2 \cos (e+f x)}{9 a f (a+a \sin (e+f x))^5}-\frac {19 \cos (e+f x)}{63 a^2 f (a+a \sin (e+f x))^4}+\frac {2 \cos (e+f x)}{105 f \left (a^2+a^2 \sin (e+f x)\right )^3}+\frac {4 \cos (e+f x)}{315 f \left (a^3+a^3 \sin (e+f x)\right )^2}+\frac {4 \cos (e+f x)}{315 f \left (a^6+a^6 \sin (e+f x)\right )}\\ \end {align*}

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Mathematica [A]  time = 0.96, size = 171, normalized size = 1.19 \[ -\frac {2562 \sin \left (2 e+\frac {3 f x}{2}\right )-900 \sin \left (2 e+\frac {5 f x}{2}\right )-27 \sin \left (4 e+\frac {7 f x}{2}\right )+25 \sin \left (4 e+\frac {9 f x}{2}\right )+378 \cos \left (e+\frac {f x}{2}\right )+210 \cos \left (e+\frac {3 f x}{2}\right )-108 \cos \left (3 e+\frac {5 f x}{2}\right )+225 \cos \left (3 e+\frac {7 f x}{2}\right )+3 \cos \left (5 e+\frac {9 f x}{2}\right )+3150 \sin \left (\frac {f x}{2}\right )}{13860 a^6 f \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[e + f*x]^2*Sin[e + f*x])/(a + a*Sin[e + f*x])^6,x]

[Out]

-1/13860*(378*Cos[e + (f*x)/2] + 210*Cos[e + (3*f*x)/2] - 108*Cos[3*e + (5*f*x)/2] + 225*Cos[3*e + (7*f*x)/2]
+ 3*Cos[5*e + (9*f*x)/2] + 3150*Sin[(f*x)/2] + 2562*Sin[2*e + (3*f*x)/2] - 900*Sin[2*e + (5*f*x)/2] - 27*Sin[4
*e + (7*f*x)/2] + 25*Sin[4*e + (9*f*x)/2])/(a^6*f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^
9)

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fricas [A]  time = 0.50, size = 243, normalized size = 1.69 \[ \frac {4 \, \cos \left (f x + e\right )^{5} - 16 \, \cos \left (f x + e\right )^{4} - 50 \, \cos \left (f x + e\right )^{3} - 65 \, \cos \left (f x + e\right )^{2} - {\left (4 \, \cos \left (f x + e\right )^{4} + 20 \, \cos \left (f x + e\right )^{3} - 30 \, \cos \left (f x + e\right )^{2} + 35 \, \cos \left (f x + e\right ) + 70\right )} \sin \left (f x + e\right ) + 35 \, \cos \left (f x + e\right ) + 70}{315 \, {\left (a^{6} f \cos \left (f x + e\right )^{5} + 5 \, a^{6} f \cos \left (f x + e\right )^{4} - 8 \, a^{6} f \cos \left (f x + e\right )^{3} - 20 \, a^{6} f \cos \left (f x + e\right )^{2} + 8 \, a^{6} f \cos \left (f x + e\right ) + 16 \, a^{6} f + {\left (a^{6} f \cos \left (f x + e\right )^{4} - 4 \, a^{6} f \cos \left (f x + e\right )^{3} - 12 \, a^{6} f \cos \left (f x + e\right )^{2} + 8 \, a^{6} f \cos \left (f x + e\right ) + 16 \, a^{6} f\right )} \sin \left (f x + e\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x, algorithm="fricas")

[Out]

1/315*(4*cos(f*x + e)^5 - 16*cos(f*x + e)^4 - 50*cos(f*x + e)^3 - 65*cos(f*x + e)^2 - (4*cos(f*x + e)^4 + 20*c
os(f*x + e)^3 - 30*cos(f*x + e)^2 + 35*cos(f*x + e) + 70)*sin(f*x + e) + 35*cos(f*x + e) + 70)/(a^6*f*cos(f*x
+ e)^5 + 5*a^6*f*cos(f*x + e)^4 - 8*a^6*f*cos(f*x + e)^3 - 20*a^6*f*cos(f*x + e)^2 + 8*a^6*f*cos(f*x + e) + 16
*a^6*f + (a^6*f*cos(f*x + e)^4 - 4*a^6*f*cos(f*x + e)^3 - 12*a^6*f*cos(f*x + e)^2 + 8*a^6*f*cos(f*x + e) + 16*
a^6*f)*sin(f*x + e))

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giac [A]  time = 0.25, size = 120, normalized size = 0.83 \[ -\frac {2 \, {\left (315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 315 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 945 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 441 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 609 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 81 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 99 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 11\right )}}{315 \, a^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x, algorithm="giac")

[Out]

-2/315*(315*tan(1/2*f*x + 1/2*e)^7 + 315*tan(1/2*f*x + 1/2*e)^6 + 945*tan(1/2*f*x + 1/2*e)^5 + 441*tan(1/2*f*x
+ 1/2*e)^4 + 609*tan(1/2*f*x + 1/2*e)^3 + 81*tan(1/2*f*x + 1/2*e)^2 + 99*tan(1/2*f*x + 1/2*e) + 11)/(a^6*f*(t
an(1/2*f*x + 1/2*e) + 1)^9)

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maple [A]  time = 0.48, size = 130, normalized size = 0.90 \[ \frac {-\frac {248}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{6}}+\frac {64}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{9}}+\frac {336}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{8}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {12}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {36}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}+\frac {464}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{7}}}{f \,a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x)

[Out]

4/f/a^6*(-62/3/(tan(1/2*f*x+1/2*e)+1)^6+16/9/(tan(1/2*f*x+1/2*e)+1)^9+84/5/(tan(1/2*f*x+1/2*e)+1)^5-8/(tan(1/2
*f*x+1/2*e)+1)^8-1/2/(tan(1/2*f*x+1/2*e)+1)^2+3/(tan(1/2*f*x+1/2*e)+1)^3-9/(tan(1/2*f*x+1/2*e)+1)^4+116/7/(tan
(1/2*f*x+1/2*e)+1)^7)

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maxima [B]  time = 0.36, size = 355, normalized size = 2.47 \[ -\frac {2 \, {\left (\frac {99 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {81 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {609 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {441 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {945 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {315 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {315 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + 11\right )}}{315 \, {\left (a^{6} + \frac {9 \, a^{6} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {36 \, a^{6} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {84 \, a^{6} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {126 \, a^{6} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {126 \, a^{6} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {84 \, a^{6} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {36 \, a^{6} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {9 \, a^{6} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {a^{6} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}}\right )} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*sin(f*x+e)/(a+a*sin(f*x+e))^6,x, algorithm="maxima")

[Out]

-2/315*(99*sin(f*x + e)/(cos(f*x + e) + 1) + 81*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 609*sin(f*x + e)^3/(cos(
f*x + e) + 1)^3 + 441*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 945*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 315*sin(
f*x + e)^6/(cos(f*x + e) + 1)^6 + 315*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 11)/((a^6 + 9*a^6*sin(f*x + e)/(co
s(f*x + e) + 1) + 36*a^6*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 84*a^6*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 12
6*a^6*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 126*a^6*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*a^6*sin(f*x + e)^
6/(cos(f*x + e) + 1)^6 + 36*a^6*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*a^6*sin(f*x + e)^8/(cos(f*x + e) + 1)^
8 + a^6*sin(f*x + e)^9/(cos(f*x + e) + 1)^9)*f)

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mupad [B]  time = 9.06, size = 207, normalized size = 1.44 \[ -\frac {2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (11\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+99\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )+81\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+609\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+441\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+945\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+315\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+315\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\right )}{315\,a^6\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(e + f*x)^2*sin(e + f*x))/(a + a*sin(e + f*x))^6,x)

[Out]

-(2*cos(e/2 + (f*x)/2)^2*(11*cos(e/2 + (f*x)/2)^7 + 315*sin(e/2 + (f*x)/2)^7 + 315*cos(e/2 + (f*x)/2)*sin(e/2
+ (f*x)/2)^6 + 99*cos(e/2 + (f*x)/2)^6*sin(e/2 + (f*x)/2) + 945*cos(e/2 + (f*x)/2)^2*sin(e/2 + (f*x)/2)^5 + 44
1*cos(e/2 + (f*x)/2)^3*sin(e/2 + (f*x)/2)^4 + 609*cos(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2)^3 + 81*cos(e/2 + (f*
x)/2)^5*sin(e/2 + (f*x)/2)^2))/(315*a^6*f*(cos(e/2 + (f*x)/2) + sin(e/2 + (f*x)/2))^9)

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sympy [A]  time = 109.19, size = 1501, normalized size = 10.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)**2*sin(f*x+e)/(a+a*sin(f*x+e))**6,x)

[Out]

Piecewise((-630*tan(e/2 + f*x/2)**7/(315*a**6*f*tan(e/2 + f*x/2)**9 + 2835*a**6*f*tan(e/2 + f*x/2)**8 + 11340*
a**6*f*tan(e/2 + f*x/2)**7 + 26460*a**6*f*tan(e/2 + f*x/2)**6 + 39690*a**6*f*tan(e/2 + f*x/2)**5 + 39690*a**6*
f*tan(e/2 + f*x/2)**4 + 26460*a**6*f*tan(e/2 + f*x/2)**3 + 11340*a**6*f*tan(e/2 + f*x/2)**2 + 2835*a**6*f*tan(
e/2 + f*x/2) + 315*a**6*f) - 630*tan(e/2 + f*x/2)**6/(315*a**6*f*tan(e/2 + f*x/2)**9 + 2835*a**6*f*tan(e/2 + f
*x/2)**8 + 11340*a**6*f*tan(e/2 + f*x/2)**7 + 26460*a**6*f*tan(e/2 + f*x/2)**6 + 39690*a**6*f*tan(e/2 + f*x/2)
**5 + 39690*a**6*f*tan(e/2 + f*x/2)**4 + 26460*a**6*f*tan(e/2 + f*x/2)**3 + 11340*a**6*f*tan(e/2 + f*x/2)**2 +
2835*a**6*f*tan(e/2 + f*x/2) + 315*a**6*f) - 1890*tan(e/2 + f*x/2)**5/(315*a**6*f*tan(e/2 + f*x/2)**9 + 2835*
a**6*f*tan(e/2 + f*x/2)**8 + 11340*a**6*f*tan(e/2 + f*x/2)**7 + 26460*a**6*f*tan(e/2 + f*x/2)**6 + 39690*a**6*
f*tan(e/2 + f*x/2)**5 + 39690*a**6*f*tan(e/2 + f*x/2)**4 + 26460*a**6*f*tan(e/2 + f*x/2)**3 + 11340*a**6*f*tan
(e/2 + f*x/2)**2 + 2835*a**6*f*tan(e/2 + f*x/2) + 315*a**6*f) - 882*tan(e/2 + f*x/2)**4/(315*a**6*f*tan(e/2 +
f*x/2)**9 + 2835*a**6*f*tan(e/2 + f*x/2)**8 + 11340*a**6*f*tan(e/2 + f*x/2)**7 + 26460*a**6*f*tan(e/2 + f*x/2)
**6 + 39690*a**6*f*tan(e/2 + f*x/2)**5 + 39690*a**6*f*tan(e/2 + f*x/2)**4 + 26460*a**6*f*tan(e/2 + f*x/2)**3 +
11340*a**6*f*tan(e/2 + f*x/2)**2 + 2835*a**6*f*tan(e/2 + f*x/2) + 315*a**6*f) - 1218*tan(e/2 + f*x/2)**3/(315
*a**6*f*tan(e/2 + f*x/2)**9 + 2835*a**6*f*tan(e/2 + f*x/2)**8 + 11340*a**6*f*tan(e/2 + f*x/2)**7 + 26460*a**6*
f*tan(e/2 + f*x/2)**6 + 39690*a**6*f*tan(e/2 + f*x/2)**5 + 39690*a**6*f*tan(e/2 + f*x/2)**4 + 26460*a**6*f*tan
(e/2 + f*x/2)**3 + 11340*a**6*f*tan(e/2 + f*x/2)**2 + 2835*a**6*f*tan(e/2 + f*x/2) + 315*a**6*f) - 162*tan(e/2
+ f*x/2)**2/(315*a**6*f*tan(e/2 + f*x/2)**9 + 2835*a**6*f*tan(e/2 + f*x/2)**8 + 11340*a**6*f*tan(e/2 + f*x/2)
**7 + 26460*a**6*f*tan(e/2 + f*x/2)**6 + 39690*a**6*f*tan(e/2 + f*x/2)**5 + 39690*a**6*f*tan(e/2 + f*x/2)**4 +
26460*a**6*f*tan(e/2 + f*x/2)**3 + 11340*a**6*f*tan(e/2 + f*x/2)**2 + 2835*a**6*f*tan(e/2 + f*x/2) + 315*a**6
*f) - 198*tan(e/2 + f*x/2)/(315*a**6*f*tan(e/2 + f*x/2)**9 + 2835*a**6*f*tan(e/2 + f*x/2)**8 + 11340*a**6*f*ta
n(e/2 + f*x/2)**7 + 26460*a**6*f*tan(e/2 + f*x/2)**6 + 39690*a**6*f*tan(e/2 + f*x/2)**5 + 39690*a**6*f*tan(e/2
+ f*x/2)**4 + 26460*a**6*f*tan(e/2 + f*x/2)**3 + 11340*a**6*f*tan(e/2 + f*x/2)**2 + 2835*a**6*f*tan(e/2 + f*x
/2) + 315*a**6*f) - 22/(315*a**6*f*tan(e/2 + f*x/2)**9 + 2835*a**6*f*tan(e/2 + f*x/2)**8 + 11340*a**6*f*tan(e/
2 + f*x/2)**7 + 26460*a**6*f*tan(e/2 + f*x/2)**6 + 39690*a**6*f*tan(e/2 + f*x/2)**5 + 39690*a**6*f*tan(e/2 + f
*x/2)**4 + 26460*a**6*f*tan(e/2 + f*x/2)**3 + 11340*a**6*f*tan(e/2 + f*x/2)**2 + 2835*a**6*f*tan(e/2 + f*x/2)
+ 315*a**6*f), Ne(f, 0)), (x*sin(e)*cos(e)**2/(a*sin(e) + a)**6, True))

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