3.462 \(\int \frac{(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx\)
Optimal. Leaf size=195 \[ \frac{2 d \left (8 c^2 d+c^3-20 c d^2+8 d^3\right ) \cos (e+f x)}{3 a^2 f}+\frac{d^2 \left (2 c^2+16 c d-21 d^2\right ) \sin (e+f x) \cos (e+f x)}{6 a^2 f}+\frac{d^2 x \left (12 c^2-16 c d+7 d^2\right )}{2 a^2}-\frac{(c-d) (c+8 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2} \]
[Out]
(d^2*(12*c^2 - 16*c*d + 7*d^2)*x)/(2*a^2) + (2*d*(c^3 + 8*c^2*d - 20*c*d^2 + 8*d^3)*Cos[e + f*x])/(3*a^2*f) +
(d^2*(2*c^2 + 16*c*d - 21*d^2)*Cos[e + f*x]*Sin[e + f*x])/(6*a^2*f) - ((c - d)*(c + 8*d)*Cos[e + f*x]*(c + d*S
in[e + f*x])^2)/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(3*f*(a + a*Sin[e
+ f*x])^2)
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Rubi [A] time = 0.362295, antiderivative size = 195, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used =
{2765, 2977, 2734} \[ \frac{2 d \left (8 c^2 d+c^3-20 c d^2+8 d^3\right ) \cos (e+f x)}{3 a^2 f}+\frac{d^2 \left (2 c^2+16 c d-21 d^2\right ) \sin (e+f x) \cos (e+f x)}{6 a^2 f}+\frac{d^2 x \left (12 c^2-16 c d+7 d^2\right )}{2 a^2}-\frac{(c-d) (c+8 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (\sin (e+f x)+1)}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a \sin (e+f x)+a)^2} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*Sin[e + f*x])^4/(a + a*Sin[e + f*x])^2,x]
[Out]
(d^2*(12*c^2 - 16*c*d + 7*d^2)*x)/(2*a^2) + (2*d*(c^3 + 8*c^2*d - 20*c*d^2 + 8*d^3)*Cos[e + f*x])/(3*a^2*f) +
(d^2*(2*c^2 + 16*c*d - 21*d^2)*Cos[e + f*x]*Sin[e + f*x])/(6*a^2*f) - ((c - d)*(c + 8*d)*Cos[e + f*x]*(c + d*S
in[e + f*x])^2)/(3*a^2*f*(1 + Sin[e + f*x])) - ((c - d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(3*f*(a + a*Sin[e
+ f*x])^2)
Rule 2765
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[((b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n - 1))/(a*f*(2*m + 1)), x] + Dist[1/
(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n - 2)*Simp[b*(c^2*(m + 1) + d^2*(n -
1)) + a*c*d*(m - n + 1) + d*(a*d*(m - n + 1) + b*c*(m + n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e,
f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ
[2*m, 2*n] || (IntegerQ[m] && EqQ[c, 0]))
Rule 2977
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x]
)^n)/(a*f*(2*m + 1)), x] - Dist[1/(a*b*(2*m + 1)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n -
1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ
[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Rule 2734
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Rubi steps
\begin{align*} \int \frac{(c+d \sin (e+f x))^4}{(a+a \sin (e+f x))^2} \, dx &=-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a+a \sin (e+f x))^2}-\frac{\int \frac{(c+d \sin (e+f x))^2 \left (-a \left (c^2+5 c d-3 d^2\right )+a (2 c-5 d) d \sin (e+f x)\right )}{a+a \sin (e+f x)} \, dx}{3 a^2}\\ &=-\frac{(c-d) (c+8 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (1+\sin (e+f x))}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a+a \sin (e+f x))^2}-\frac{\int (c+d \sin (e+f x)) \left (-a^2 (19 c-16 d) d^2+a^2 d \left (2 c^2+16 c d-21 d^2\right ) \sin (e+f x)\right ) \, dx}{3 a^4}\\ &=\frac{d^2 \left (12 c^2-16 c d+7 d^2\right ) x}{2 a^2}+\frac{2 d \left (c^3+8 c^2 d-20 c d^2+8 d^3\right ) \cos (e+f x)}{3 a^2 f}+\frac{d^2 \left (2 c^2+16 c d-21 d^2\right ) \cos (e+f x) \sin (e+f x)}{6 a^2 f}-\frac{(c-d) (c+8 d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f (1+\sin (e+f x))}-\frac{(c-d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 f (a+a \sin (e+f x))^2}\\ \end{align*}
Mathematica [A] time = 1.93449, size = 378, normalized size = 1.94 \[ \frac{\left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (3 d \cos \left (\frac{1}{2} (e+f x)\right ) \left (48 c^2 d (3 e+3 f x-4)+64 c^3-32 c d^2 (6 e+6 f x-5)+7 d^3 (12 e+12 f x-7)\right )-\cos \left (\frac{3}{2} (e+f x)\right ) \left (48 c^2 d^2 (3 e+3 f x-10)+128 c^3 d+16 c^4-16 c d^3 (12 e+12 f x-41)+d^4 (84 e+84 f x-239)\right )+3 \left (2 \sin \left (\frac{1}{2} (e+f x)\right ) \left (d^2 \cos (e+f x) \left (48 c^2 (e+f x)-64 c d (e+f x+1)+d^2 (28 e+28 f x+27)\right )+96 c^2 d^2 e+96 c^2 d^2 f x-144 c^2 d^2+32 c^3 d+8 c^4-2 d^3 (8 c-3 d) \cos (2 (e+f x))-128 c d^3 e-128 c d^3 f x+144 c d^3+d^4 \cos (3 (e+f x))+56 d^4 e+56 d^4 f x-50 d^4\right )+d^3 (16 c-5 d) \cos \left (\frac{5}{2} (e+f x)\right )+d^4 \cos \left (\frac{7}{2} (e+f x)\right )\right )\right )}{48 a^2 f (\sin (e+f x)+1)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*Sin[e + f*x])^4/(a + a*Sin[e + f*x])^2,x]
[Out]
((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(3*d*(64*c^3 + 48*c^2*d*(-4 + 3*e + 3*f*x) - 32*c*d^2*(-5 + 6*e + 6*f*x
) + 7*d^3*(-7 + 12*e + 12*f*x))*Cos[(e + f*x)/2] - (16*c^4 + 128*c^3*d + 48*c^2*d^2*(-10 + 3*e + 3*f*x) - 16*c
*d^3*(-41 + 12*e + 12*f*x) + d^4*(-239 + 84*e + 84*f*x))*Cos[(3*(e + f*x))/2] + 3*((16*c - 5*d)*d^3*Cos[(5*(e
+ f*x))/2] + d^4*Cos[(7*(e + f*x))/2] + 2*(8*c^4 + 32*c^3*d - 144*c^2*d^2 + 144*c*d^3 - 50*d^4 + 96*c^2*d^2*e
- 128*c*d^3*e + 56*d^4*e + 96*c^2*d^2*f*x - 128*c*d^3*f*x + 56*d^4*f*x + d^2*(48*c^2*(e + f*x) - 64*c*d*(1 + e
+ f*x) + d^2*(27 + 28*e + 28*f*x))*Cos[e + f*x] - 2*(8*c - 3*d)*d^3*Cos[2*(e + f*x)] + d^4*Cos[3*(e + f*x)])*
Sin[(e + f*x)/2])))/(48*a^2*f*(1 + Sin[e + f*x])^2)
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Maple [B] time = 0.087, size = 618, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e))^2,x)
[Out]
1/f/a^2*d^4/(1+tan(1/2*f*x+1/2*e)^2)^2*tan(1/2*f*x+1/2*e)^3-8/f/a^2*d^3/(1+tan(1/2*f*x+1/2*e)^2)^2*tan(1/2*f*x
+1/2*e)^2*c+4/f/a^2*d^4/(1+tan(1/2*f*x+1/2*e)^2)^2*tan(1/2*f*x+1/2*e)^2-1/f/a^2*d^4/(1+tan(1/2*f*x+1/2*e)^2)^2
*tan(1/2*f*x+1/2*e)-8/f/a^2*d^3/(1+tan(1/2*f*x+1/2*e)^2)^2*c+4/f/a^2*d^4/(1+tan(1/2*f*x+1/2*e)^2)^2+12/f/a^2*d
^2*arctan(tan(1/2*f*x+1/2*e))*c^2-16/f/a^2*d^3*arctan(tan(1/2*f*x+1/2*e))*c+7/f/a^2*d^4*arctan(tan(1/2*f*x+1/2
*e))-2/f/a^2/(tan(1/2*f*x+1/2*e)+1)*c^4+12/f/a^2/(tan(1/2*f*x+1/2*e)+1)*c^2*d^2-16/f/a^2/(tan(1/2*f*x+1/2*e)+1
)*c*d^3+6/f/a^2/(tan(1/2*f*x+1/2*e)+1)*d^4+2/f/a^2/(tan(1/2*f*x+1/2*e)+1)^2*c^4-8/f/a^2/(tan(1/2*f*x+1/2*e)+1)
^2*c^3*d+12/f/a^2/(tan(1/2*f*x+1/2*e)+1)^2*c^2*d^2-8/f/a^2/(tan(1/2*f*x+1/2*e)+1)^2*c*d^3+2/f/a^2/(tan(1/2*f*x
+1/2*e)+1)^2*d^4-4/3/f/a^2/(tan(1/2*f*x+1/2*e)+1)^3*c^4+16/3/f/a^2/(tan(1/2*f*x+1/2*e)+1)^3*c^3*d-8/f/a^2/(tan
(1/2*f*x+1/2*e)+1)^3*c^2*d^2+16/3/f/a^2/(tan(1/2*f*x+1/2*e)+1)^3*c*d^3-4/3/f/a^2/(tan(1/2*f*x+1/2*e)+1)^3*d^4
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Maxima [B] time = 1.7998, size = 1226, normalized size = 6.29 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e))^2,x, algorithm="maxima")
[Out]
1/3*(d^4*((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/(c
os(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*sin(
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/(cos
(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) - 16*c*d^3*((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/(c
os(f*x + e) + 1)^5) + 3*arctan(sin(f*x + e)/(cos(f*x + e) + 1))/a^2) + 12*c^2*d^2*((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^4*(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) - 8*c^3*d*(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 [B] time = 1.67779, size = 1000, normalized size = 5.13 \begin{align*} -\frac{3 \, d^{4} \cos \left (f x + e\right )^{4} - 2 \, c^{4} + 8 \, c^{3} d - 12 \, c^{2} d^{2} + 8 \, c d^{3} - 2 \, d^{4} + 6 \,{\left (4 \, c d^{3} - d^{4}\right )} \cos \left (f x + e\right )^{3} + 6 \,{\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} f x -{\left (2 \, c^{4} + 16 \, c^{3} d - 60 \, c^{2} d^{2} + 88 \, c d^{3} - 31 \, d^{4} + 3 \,{\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} f x\right )} \cos \left (f x + e\right )^{2} -{\left (4 \, c^{4} + 8 \, c^{3} d - 48 \, c^{2} d^{2} + 104 \, c d^{3} - 38 \, d^{4} - 3 \,{\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} f x\right )} \cos \left (f x + e\right ) +{\left (3 \, d^{4} \cos \left (f x + e\right )^{3} + 2 \, c^{4} - 8 \, c^{3} d + 12 \, c^{2} d^{2} - 8 \, c d^{3} + 2 \, d^{4} + 6 \,{\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} f x - 3 \,{\left (8 \, c d^{3} - 3 \, d^{4}\right )} \cos \left (f x + e\right )^{2} -{\left (2 \, c^{4} + 16 \, c^{3} d - 60 \, c^{2} d^{2} + 112 \, c d^{3} - 40 \, d^{4} - 3 \,{\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )} f x\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+d*sin(f*x+e))^4/(a+a*sin(f*x+e))^2,x, algorithm="fricas")
[Out]
-1/6*(3*d^4*cos(f*x + e)^4 - 2*c^4 + 8*c^3*d - 12*c^2*d^2 + 8*c*d^3 - 2*d^4 + 6*(4*c*d^3 - d^4)*cos(f*x + e)^3
+ 6*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*f*x - (2*c^4 + 16*c^3*d - 60*c^2*d^2 + 88*c*d^3 - 31*d^4 + 3*(12*c^2*d^2
- 16*c*d^3 + 7*d^4)*f*x)*cos(f*x + e)^2 - (4*c^4 + 8*c^3*d - 48*c^2*d^2 + 104*c*d^3 - 38*d^4 - 3*(12*c^2*d^2 -
16*c*d^3 + 7*d^4)*f*x)*cos(f*x + e) + (3*d^4*cos(f*x + e)^3 + 2*c^4 - 8*c^3*d + 12*c^2*d^2 - 8*c*d^3 + 2*d^4
+ 6*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*f*x - 3*(8*c*d^3 - 3*d^4)*cos(f*x + e)^2 - (2*c^4 + 16*c^3*d - 60*c^2*d^2
+ 112*c*d^3 - 40*d^4 - 3*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*f*x)*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))
________________________________________________________________________________________
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+d*sin(f*x+e))**4/(a+a*sin(f*x+e))**2,x)
[Out]
Timed out
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Giac [A] time = 1.32684, size = 456, normalized size = 2.34 \begin{align*} \frac{\frac{3 \,{\left (12 \, c^{2} d^{2} - 16 \, c d^{3} + 7 \, d^{4}\right )}{\left (f x + e\right )}}{a^{2}} + \frac{6 \,{\left (d^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 8 \, c d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 4 \, d^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - d^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 8 \, c d^{3} + 4 \, d^{4}\right )}}{{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 1\right )}^{2} a^{2}} - \frac{4 \,{\left (3 \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 18 \, c^{2} d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 24 \, c d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 9 \, d^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 3 \, c^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 12 \, c^{3} d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 54 \, c^{2} d^{2} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 60 \, c d^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 21 \, d^{4} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 2 \, c^{4} + 4 \, c^{3} d - 24 \, c^{2} d^{2} + 28 \, c d^{3} - 10 \, d^{4}\right )}}{a^{2}{\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}^{3}}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*sin(f*x+e))^4/(a+a*sin(f*x+e))^2,x, algorithm="giac")
[Out]
1/6*(3*(12*c^2*d^2 - 16*c*d^3 + 7*d^4)*(f*x + e)/a^2 + 6*(d^4*tan(1/2*f*x + 1/2*e)^3 - 8*c*d^3*tan(1/2*f*x + 1
/2*e)^2 + 4*d^4*tan(1/2*f*x + 1/2*e)^2 - d^4*tan(1/2*f*x + 1/2*e) - 8*c*d^3 + 4*d^4)/((tan(1/2*f*x + 1/2*e)^2
+ 1)^2*a^2) - 4*(3*c^4*tan(1/2*f*x + 1/2*e)^2 - 18*c^2*d^2*tan(1/2*f*x + 1/2*e)^2 + 24*c*d^3*tan(1/2*f*x + 1/2
*e)^2 - 9*d^4*tan(1/2*f*x + 1/2*e)^2 + 3*c^4*tan(1/2*f*x + 1/2*e) + 12*c^3*d*tan(1/2*f*x + 1/2*e) - 54*c^2*d^2
*tan(1/2*f*x + 1/2*e) + 60*c*d^3*tan(1/2*f*x + 1/2*e) - 21*d^4*tan(1/2*f*x + 1/2*e) + 2*c^4 + 4*c^3*d - 24*c^2
*d^2 + 28*c*d^3 - 10*d^4)/(a^2*(tan(1/2*f*x + 1/2*e) + 1)^3))/f