3.269 \(\int \frac{(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx\)

Optimal. Leaf size=148 \[ \frac{35 c^5 \cos ^3(e+f x)}{a^2 f}-\frac{2 a^4 c^5 \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{f (a \sin (e+f x)+a)^4}+\frac{105 c^5 \sin (e+f x) \cos (e+f x)}{2 a^2 f}+\frac{105 c^5 x}{2 a^2}+\frac{42 c^5 \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^2} \]

[Out]

(105*c^5*x)/(2*a^2) + (35*c^5*Cos[e + f*x]^3)/(a^2*f) + (105*c^5*Cos[e + f*x]*Sin[e + f*x])/(2*a^2*f) - (2*a^4
*c^5*Cos[e + f*x]^9)/(3*f*(a + a*Sin[e + f*x])^6) + (6*a^2*c^5*Cos[e + f*x]^7)/(f*(a + a*Sin[e + f*x])^4) + (4
2*c^5*Cos[e + f*x]^5)/(f*(a + a*Sin[e + f*x])^2)

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Rubi [A]  time = 0.239855, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {2736, 2680, 2682, 2635, 8} \[ \frac{35 c^5 \cos ^3(e+f x)}{a^2 f}-\frac{2 a^4 c^5 \cos ^9(e+f x)}{3 f (a \sin (e+f x)+a)^6}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{f (a \sin (e+f x)+a)^4}+\frac{105 c^5 \sin (e+f x) \cos (e+f x)}{2 a^2 f}+\frac{105 c^5 x}{2 a^2}+\frac{42 c^5 \cos ^5(e+f x)}{f (a \sin (e+f x)+a)^2} \]

Antiderivative was successfully verified.

[In]

Int[(c - c*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^2,x]

[Out]

(105*c^5*x)/(2*a^2) + (35*c^5*Cos[e + f*x]^3)/(a^2*f) + (105*c^5*Cos[e + f*x]*Sin[e + f*x])/(2*a^2*f) - (2*a^4
*c^5*Cos[e + f*x]^9)/(3*f*(a + a*Sin[e + f*x])^6) + (6*a^2*c^5*Cos[e + f*x]^7)/(f*(a + a*Sin[e + f*x])^4) + (4
2*c^5*Cos[e + f*x]^5)/(f*(a + a*Sin[e + f*x])^2)

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 2680

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(2*g*(
g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(2*m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(2*m +
p + 1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 2682

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e
+ f*x])^(p - 1))/(b*f*(p - 1)), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

Rule 2635

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx &=\left (a^5 c^5\right ) \int \frac{\cos ^{10}(e+f x)}{(a+a \sin (e+f x))^7} \, dx\\ &=-\frac{2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}-\left (3 a^3 c^5\right ) \int \frac{\cos ^8(e+f x)}{(a+a \sin (e+f x))^5} \, dx\\ &=-\frac{2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\left (21 a c^5\right ) \int \frac{\cos ^6(e+f x)}{(a+a \sin (e+f x))^3} \, dx\\ &=-\frac{2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\frac{42 c^5 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^2}+\frac{\left (105 c^5\right ) \int \frac{\cos ^4(e+f x)}{a+a \sin (e+f x)} \, dx}{a}\\ &=\frac{35 c^5 \cos ^3(e+f x)}{a^2 f}-\frac{2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\frac{42 c^5 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^2}+\frac{\left (105 c^5\right ) \int \cos ^2(e+f x) \, dx}{a^2}\\ &=\frac{35 c^5 \cos ^3(e+f x)}{a^2 f}+\frac{105 c^5 \cos (e+f x) \sin (e+f x)}{2 a^2 f}-\frac{2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\frac{42 c^5 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^2}+\frac{\left (105 c^5\right ) \int 1 \, dx}{2 a^2}\\ &=\frac{105 c^5 x}{2 a^2}+\frac{35 c^5 \cos ^3(e+f x)}{a^2 f}+\frac{105 c^5 \cos (e+f x) \sin (e+f x)}{2 a^2 f}-\frac{2 a^4 c^5 \cos ^9(e+f x)}{3 f (a+a \sin (e+f x))^6}+\frac{6 a^2 c^5 \cos ^7(e+f x)}{f (a+a \sin (e+f x))^4}+\frac{42 c^5 \cos ^5(e+f x)}{f (a+a \sin (e+f x))^2}\\ \end{align*}

Mathematica [A]  time = 0.726095, size = 276, normalized size = 1.86 \[ \frac{(c-c \sin (e+f x))^5 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (256 \sin \left (\frac{1}{2} (e+f x)\right )+630 (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3+285 \cos (e+f x) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3-\cos (3 (e+f x)) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3-21 \sin (2 (e+f x)) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3-1664 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2-128 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )}{12 f (a \sin (e+f x)+a)^2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - c*Sin[e + f*x])^5/(a + a*Sin[e + f*x])^2,x]

[Out]

((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(c - c*Sin[e + f*x])^5*(256*Sin[(e + f*x)/2] - 128*(Cos[(e + f*x)/2] +
Sin[(e + f*x)/2]) - 1664*Sin[(e + f*x)/2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + 630*(e + f*x)*(Cos[(e + f*
x)/2] + Sin[(e + f*x)/2])^3 + 285*Cos[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - Cos[3*(e + f*x)]*(Cos
[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - 21*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3*Sin[2*(e + f*x)]))/(12*f*(Cos
[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(a + a*Sin[e + f*x])^2)

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Maple [A]  time = 0.108, size = 267, normalized size = 1.8 \begin{align*} 7\,{\frac{{c}^{5} \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{5}}{{a}^{2}f \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{3}}}+46\,{\frac{{c}^{5} \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{4}}{{a}^{2}f \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{3}}}+96\,{\frac{{c}^{5} \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}}{{a}^{2}f \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{3}}}-7\,{\frac{{c}^{5}\tan \left ( 1/2\,fx+e/2 \right ) }{{a}^{2}f \left ( 1+ \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2} \right ) ^{3}}}+{\frac{142\,{c}^{5}}{3\,{a}^{2}f} \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) ^{-3}}+105\,{\frac{{c}^{5}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{{a}^{2}f}}-{\frac{128\,{c}^{5}}{3\,{a}^{2}f} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-3}}+64\,{\frac{{c}^{5}}{{a}^{2}f \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) ^{2}}}+96\,{\frac{{c}^{5}}{{a}^{2}f \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x)

[Out]

7/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^5+46/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*
x+1/2*e)^4+96/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^3*tan(1/2*f*x+1/2*e)^2-7/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^3
*tan(1/2*f*x+1/2*e)+142/3/f*c^5/a^2/(1+tan(1/2*f*x+1/2*e)^2)^3+105/f*c^5/a^2*arctan(tan(1/2*f*x+1/2*e))-128/3/
f*c^5/a^2/(tan(1/2*f*x+1/2*e)+1)^3+64/f*c^5/a^2/(tan(1/2*f*x+1/2*e)+1)^2+96/f*c^5/a^2/(tan(1/2*f*x+1/2*e)+1)

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Maxima [B]  time = 3.30545, size = 1760, normalized size = 11.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

1/3*(5*c^5*((75*sin(f*x + e)/(cos(f*x + e) + 1) + 97*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 126*sin(f*x + e)^3/
(cos(f*x + e) + 1)^3 + 98*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 63*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 21*si
n(f*x + e)^6/(cos(f*x + e) + 1)^6 + 32)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 5*a^2*sin(f*x + e)^2/(c
os(f*x + e) + 1)^2 + 7*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 7*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5
*a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 3*a^2*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + a^2*sin(f*x + e)^7/(cos
(f*x + e) + 1)^7) + 21*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 2*c^5*((57*sin(f*x + e)/(cos(f*x + e) +
1) + 99*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 155*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 153*sin(f*x + e)^4/(co
s(f*x + e) + 1)^4 + 135*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 85*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 45*sin(
f*x + e)^7/(cos(f*x + e) + 1)^7 + 15*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + 24)/(a^2 + 3*a^2*sin(f*x + e)/(cos(
f*x + e) + 1) + 6*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 10*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 12*a^
2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 12*a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 10*a^2*sin(f*x + e)^6/(co
s(f*x + e) + 1)^6 + 6*a^2*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 3*a^2*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 + a^
2*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) + 15*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 40*c^5*((12*sin(f*x
+ e)/(cos(f*x + e) + 1) + 11*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 9*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*
sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 4*a^2*sin(f*x + e)^2/(
cos(f*x + e) + 1)^2 + 4*a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 3*a^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 +
a^2*sin(f*x + e)^5/(cos(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 20*c^5*((9*sin(f*x
+ e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 4)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e)
+ 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3) + 3*arctan(sin(f*x
+ e)/(cos(f*x + e) + 1))/a^2) - 2*c^5*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 3*sin(f*x + e)^2/(cos(f*x + e) + 1
)^2 + 2)/(a^2 + 3*a^2*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*
x + e)^3/(cos(f*x + e) + 1)^3) + 10*c^5*(3*sin(f*x + e)/(cos(f*x + e) + 1) + 1)/(a^2 + 3*a^2*sin(f*x + e)/(cos
(f*x + e) + 1) + 3*a^2*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3))/f

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Fricas [A]  time = 1.44187, size = 581, normalized size = 3.93 \begin{align*} -\frac{2 \, c^{5} \cos \left (f x + e\right )^{5} + 19 \, c^{5} \cos \left (f x + e\right )^{4} - 106 \, c^{5} \cos \left (f x + e\right )^{3} + 630 \, c^{5} f x - 64 \, c^{5} - 7 \,{\left (45 \, c^{5} f x - 77 \, c^{5}\right )} \cos \left (f x + e\right )^{2} +{\left (315 \, c^{5} f x + 598 \, c^{5}\right )} \cos \left (f x + e\right ) -{\left (2 \, c^{5} \cos \left (f x + e\right )^{4} - 17 \, c^{5} \cos \left (f x + e\right )^{3} - 630 \, c^{5} f x - 123 \, c^{5} \cos \left (f x + e\right )^{2} - 64 \, c^{5} -{\left (315 \, c^{5} f x + 662 \, c^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \,{\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f -{\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

-1/6*(2*c^5*cos(f*x + e)^5 + 19*c^5*cos(f*x + e)^4 - 106*c^5*cos(f*x + e)^3 + 630*c^5*f*x - 64*c^5 - 7*(45*c^5
*f*x - 77*c^5)*cos(f*x + e)^2 + (315*c^5*f*x + 598*c^5)*cos(f*x + e) - (2*c^5*cos(f*x + e)^4 - 17*c^5*cos(f*x
+ e)^3 - 630*c^5*f*x - 123*c^5*cos(f*x + e)^2 - 64*c^5 - (315*c^5*f*x + 662*c^5)*cos(f*x + e))*sin(f*x + e))/(
a^2*f*cos(f*x + e)^2 - a^2*f*cos(f*x + e) - 2*a^2*f - (a^2*f*cos(f*x + e) + 2*a^2*f)*sin(f*x + e))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sin(f*x+e))**5/(a+a*sin(f*x+e))**2,x)

[Out]

Timed out

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Giac [A]  time = 2.15119, size = 274, normalized size = 1.85 \begin{align*} \frac{\frac{315 \,{\left (f x + e\right )} c^{5}}{a^{2}} + \frac{2 \,{\left (309 \, c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} + 969 \, c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{7} + 1693 \, c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 3027 \, c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} + 2901 \, c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 3247 \, c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 1995 \, c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1173 \, c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 494 \, c^{5}\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}^{3} a^{2}}}{6 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c*sin(f*x+e))^5/(a+a*sin(f*x+e))^2,x, algorithm="giac")

[Out]

1/6*(315*(f*x + e)*c^5/a^2 + 2*(309*c^5*tan(1/2*f*x + 1/2*e)^8 + 969*c^5*tan(1/2*f*x + 1/2*e)^7 + 1693*c^5*tan
(1/2*f*x + 1/2*e)^6 + 3027*c^5*tan(1/2*f*x + 1/2*e)^5 + 2901*c^5*tan(1/2*f*x + 1/2*e)^4 + 3247*c^5*tan(1/2*f*x
+ 1/2*e)^3 + 1995*c^5*tan(1/2*f*x + 1/2*e)^2 + 1173*c^5*tan(1/2*f*x + 1/2*e) + 494*c^5)/((tan(1/2*f*x + 1/2*e
)^3 + tan(1/2*f*x + 1/2*e)^2 + tan(1/2*f*x + 1/2*e) + 1)^3*a^2))/f