3.106 \(\int (c+d x)^2 (a+a \cosh (e+f x))^2 \, dx\)

Optimal. Leaf size=168 \[ -\frac{a^2 d (c+d x) \cosh ^2(e+f x)}{2 f^2}-\frac{4 a^2 d (c+d x) \cosh (e+f x)}{f^2}+\frac{2 a^2 (c+d x)^2 \sinh (e+f x)}{f}+\frac{a^2 (c+d x)^2 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac{a^2 (c+d x)^3}{2 d}+\frac{4 a^2 d^2 \sinh (e+f x)}{f^3}+\frac{a^2 d^2 \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac{a^2 d^2 x}{4 f^2} \]

[Out]

(a^2*d^2*x)/(4*f^2) + (a^2*(c + d*x)^3)/(2*d) - (4*a^2*d*(c + d*x)*Cosh[e + f*x])/f^2 - (a^2*d*(c + d*x)*Cosh[
e + f*x]^2)/(2*f^2) + (4*a^2*d^2*Sinh[e + f*x])/f^3 + (2*a^2*(c + d*x)^2*Sinh[e + f*x])/f + (a^2*d^2*Cosh[e +
f*x]*Sinh[e + f*x])/(4*f^3) + (a^2*(c + d*x)^2*Cosh[e + f*x]*Sinh[e + f*x])/(2*f)

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Rubi [A]  time = 0.187681, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {3317, 3296, 2637, 3311, 32, 2635, 8} \[ -\frac{a^2 d (c+d x) \cosh ^2(e+f x)}{2 f^2}-\frac{4 a^2 d (c+d x) \cosh (e+f x)}{f^2}+\frac{2 a^2 (c+d x)^2 \sinh (e+f x)}{f}+\frac{a^2 (c+d x)^2 \sinh (e+f x) \cosh (e+f x)}{2 f}+\frac{a^2 (c+d x)^3}{2 d}+\frac{4 a^2 d^2 \sinh (e+f x)}{f^3}+\frac{a^2 d^2 \sinh (e+f x) \cosh (e+f x)}{4 f^3}+\frac{a^2 d^2 x}{4 f^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*(a + a*Cosh[e + f*x])^2,x]

[Out]

(a^2*d^2*x)/(4*f^2) + (a^2*(c + d*x)^3)/(2*d) - (4*a^2*d*(c + d*x)*Cosh[e + f*x])/f^2 - (a^2*d*(c + d*x)*Cosh[
e + f*x]^2)/(2*f^2) + (4*a^2*d^2*Sinh[e + f*x])/f^3 + (2*a^2*(c + d*x)^2*Sinh[e + f*x])/f + (a^2*d^2*Cosh[e +
f*x]*Sinh[e + f*x])/(4*f^3) + (a^2*(c + d*x)^2*Cosh[e + f*x]*Sinh[e + f*x])/(2*f)

Rule 3317

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

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 (c+d x)^2 (a+a \cosh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^2+2 a^2 (c+d x)^2 \cosh (e+f x)+a^2 (c+d x)^2 \cosh ^2(e+f x)\right ) \, dx\\ &=\frac{a^2 (c+d x)^3}{3 d}+a^2 \int (c+d x)^2 \cosh ^2(e+f x) \, dx+\left (2 a^2\right ) \int (c+d x)^2 \cosh (e+f x) \, dx\\ &=\frac{a^2 (c+d x)^3}{3 d}-\frac{a^2 d (c+d x) \cosh ^2(e+f x)}{2 f^2}+\frac{2 a^2 (c+d x)^2 \sinh (e+f x)}{f}+\frac{a^2 (c+d x)^2 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac{1}{2} a^2 \int (c+d x)^2 \, dx+\frac{\left (a^2 d^2\right ) \int \cosh ^2(e+f x) \, dx}{2 f^2}-\frac{\left (4 a^2 d\right ) \int (c+d x) \sinh (e+f x) \, dx}{f}\\ &=\frac{a^2 (c+d x)^3}{2 d}-\frac{4 a^2 d (c+d x) \cosh (e+f x)}{f^2}-\frac{a^2 d (c+d x) \cosh ^2(e+f x)}{2 f^2}+\frac{2 a^2 (c+d x)^2 \sinh (e+f x)}{f}+\frac{a^2 d^2 \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac{a^2 (c+d x)^2 \cosh (e+f x) \sinh (e+f x)}{2 f}+\frac{\left (a^2 d^2\right ) \int 1 \, dx}{4 f^2}+\frac{\left (4 a^2 d^2\right ) \int \cosh (e+f x) \, dx}{f^2}\\ &=\frac{a^2 d^2 x}{4 f^2}+\frac{a^2 (c+d x)^3}{2 d}-\frac{4 a^2 d (c+d x) \cosh (e+f x)}{f^2}-\frac{a^2 d (c+d x) \cosh ^2(e+f x)}{2 f^2}+\frac{4 a^2 d^2 \sinh (e+f x)}{f^3}+\frac{2 a^2 (c+d x)^2 \sinh (e+f x)}{f}+\frac{a^2 d^2 \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac{a^2 (c+d x)^2 \cosh (e+f x) \sinh (e+f x)}{2 f}\\ \end{align*}

Mathematica [A]  time = 0.521083, size = 192, normalized size = 1.14 \[ \frac{a^2 \left (16 c^2 f^2 \sinh (e+f x)+2 c^2 f^2 \sinh (2 (e+f x))+12 c^2 f^3 x+32 c d f^2 x \sinh (e+f x)+4 c d f^2 x \sinh (2 (e+f x))-32 d f (c+d x) \cosh (e+f x)-2 d f (c+d x) \cosh (2 (e+f x))+12 c d f^3 x^2+16 d^2 f^2 x^2 \sinh (e+f x)+2 d^2 f^2 x^2 \sinh (2 (e+f x))+32 d^2 \sinh (e+f x)+d^2 \sinh (2 (e+f x))+4 d^2 f^3 x^3\right )}{8 f^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2*(a + a*Cosh[e + f*x])^2,x]

[Out]

(a^2*(12*c^2*f^3*x + 12*c*d*f^3*x^2 + 4*d^2*f^3*x^3 - 32*d*f*(c + d*x)*Cosh[e + f*x] - 2*d*f*(c + d*x)*Cosh[2*
(e + f*x)] + 32*d^2*Sinh[e + f*x] + 16*c^2*f^2*Sinh[e + f*x] + 32*c*d*f^2*x*Sinh[e + f*x] + 16*d^2*f^2*x^2*Sin
h[e + f*x] + d^2*Sinh[2*(e + f*x)] + 2*c^2*f^2*Sinh[2*(e + f*x)] + 4*c*d*f^2*x*Sinh[2*(e + f*x)] + 2*d^2*f^2*x
^2*Sinh[2*(e + f*x)]))/(8*f^3)

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Maple [B]  time = 0.014, size = 541, 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((d*x+c)^2*(a+a*cosh(f*x+e))^2,x)

[Out]

1/f*(1/3/f^2*d^2*a^2*(f*x+e)^3+2/f^2*d^2*a^2*((f*x+e)^2*sinh(f*x+e)-2*(f*x+e)*cosh(f*x+e)+2*sinh(f*x+e))+1/f^2
*d^2*a^2*(1/2*(f*x+e)^2*cosh(f*x+e)*sinh(f*x+e)+1/6*(f*x+e)^3-1/2*(f*x+e)*cosh(f*x+e)^2+1/4*sinh(f*x+e)*cosh(f
*x+e)+1/4*f*x+1/4*e)-1/f^2*d^2*e*a^2*(f*x+e)^2-4/f^2*d^2*e*a^2*((f*x+e)*sinh(f*x+e)-cosh(f*x+e))-2/f^2*d^2*e*a
^2*(1/2*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+1/4*(f*x+e)^2-1/4*cosh(f*x+e)^2)+1/f^2*d^2*e^2*a^2*(f*x+e)+2/f^2*d^2*e
^2*a^2*sinh(f*x+e)+1/f^2*d^2*e^2*a^2*(1/2*sinh(f*x+e)*cosh(f*x+e)+1/2*f*x+1/2*e)+1/f*c*d*a^2*(f*x+e)^2+4/f*c*d
*a^2*((f*x+e)*sinh(f*x+e)-cosh(f*x+e))+2/f*c*d*a^2*(1/2*(f*x+e)*cosh(f*x+e)*sinh(f*x+e)+1/4*(f*x+e)^2-1/4*cosh
(f*x+e)^2)-2/f*c*d*e*a^2*(f*x+e)-4/f*c*d*e*a^2*sinh(f*x+e)-2/f*c*d*e*a^2*(1/2*sinh(f*x+e)*cosh(f*x+e)+1/2*f*x+
1/2*e)+a^2*c^2*(f*x+e)+2*a^2*c^2*sinh(f*x+e)+a^2*c^2*(1/2*sinh(f*x+e)*cosh(f*x+e)+1/2*f*x+1/2*e))

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Maxima [B]  time = 1.24887, size = 441, normalized size = 2.62 \begin{align*} \frac{1}{3} \, a^{2} d^{2} x^{3} + a^{2} c d x^{2} + \frac{1}{8} \,{\left (4 \, x^{2} + \frac{{\left (2 \, f x e^{\left (2 \, e\right )} - e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{2}} - \frac{{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{2}}\right )} a^{2} c d + \frac{1}{48} \,{\left (8 \, x^{3} + \frac{3 \,{\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} - 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{3}} - \frac{3 \,{\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{3}}\right )} a^{2} d^{2} + \frac{1}{8} \, a^{2} c^{2}{\left (4 \, x + \frac{e^{\left (2 \, f x + 2 \, e\right )}}{f} - \frac{e^{\left (-2 \, f x - 2 \, e\right )}}{f}\right )} + a^{2} c^{2} x + 2 \, a^{2} c d{\left (\frac{{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} - \frac{{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + a^{2} d^{2}{\left (\frac{{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} - \frac{{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac{2 \, a^{2} c^{2} \sinh \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2*(a+a*cosh(f*x+e))^2,x, algorithm="maxima")

[Out]

1/3*a^2*d^2*x^3 + a^2*c*d*x^2 + 1/8*(4*x^2 + (2*f*x*e^(2*e) - e^(2*e))*e^(2*f*x)/f^2 - (2*f*x + 1)*e^(-2*f*x -
2*e)/f^2)*a^2*c*d + 1/48*(8*x^3 + 3*(2*f^2*x^2*e^(2*e) - 2*f*x*e^(2*e) + e^(2*e))*e^(2*f*x)/f^3 - 3*(2*f^2*x^
2 + 2*f*x + 1)*e^(-2*f*x - 2*e)/f^3)*a^2*d^2 + 1/8*a^2*c^2*(4*x + e^(2*f*x + 2*e)/f - e^(-2*f*x - 2*e)/f) + a^
2*c^2*x + 2*a^2*c*d*((f*x*e^e - e^e)*e^(f*x)/f^2 - (f*x + 1)*e^(-f*x - e)/f^2) + a^2*d^2*((f^2*x^2*e^e - 2*f*x
*e^e + 2*e^e)*e^(f*x)/f^3 - (f^2*x^2 + 2*f*x + 2)*e^(-f*x - e)/f^3) + 2*a^2*c^2*sinh(f*x + e)/f

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Fricas [A]  time = 2.11826, size = 491, normalized size = 2.92 \begin{align*} \frac{2 \, a^{2} d^{2} f^{3} x^{3} + 6 \, a^{2} c d f^{3} x^{2} + 6 \, a^{2} c^{2} f^{3} x -{\left (a^{2} d^{2} f x + a^{2} c d f\right )} \cosh \left (f x + e\right )^{2} -{\left (a^{2} d^{2} f x + a^{2} c d f\right )} \sinh \left (f x + e\right )^{2} - 16 \,{\left (a^{2} d^{2} f x + a^{2} c d f\right )} \cosh \left (f x + e\right ) +{\left (8 \, a^{2} d^{2} f^{2} x^{2} + 16 \, a^{2} c d f^{2} x + 8 \, a^{2} c^{2} f^{2} + 16 \, a^{2} d^{2} +{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} + a^{2} d^{2}\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )}{4 \, f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2*(a+a*cosh(f*x+e))^2,x, algorithm="fricas")

[Out]

1/4*(2*a^2*d^2*f^3*x^3 + 6*a^2*c*d*f^3*x^2 + 6*a^2*c^2*f^3*x - (a^2*d^2*f*x + a^2*c*d*f)*cosh(f*x + e)^2 - (a^
2*d^2*f*x + a^2*c*d*f)*sinh(f*x + e)^2 - 16*(a^2*d^2*f*x + a^2*c*d*f)*cosh(f*x + e) + (8*a^2*d^2*f^2*x^2 + 16*
a^2*c*d*f^2*x + 8*a^2*c^2*f^2 + 16*a^2*d^2 + (2*a^2*d^2*f^2*x^2 + 4*a^2*c*d*f^2*x + 2*a^2*c^2*f^2 + a^2*d^2)*c
osh(f*x + e))*sinh(f*x + e))/f^3

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Sympy [A]  time = 3.24659, size = 456, normalized size = 2.71 \begin{align*} \begin{cases} - \frac{a^{2} c^{2} x \sinh ^{2}{\left (e + f x \right )}}{2} + \frac{a^{2} c^{2} x \cosh ^{2}{\left (e + f x \right )}}{2} + a^{2} c^{2} x + \frac{a^{2} c^{2} \sinh{\left (e + f x \right )} \cosh{\left (e + f x \right )}}{2 f} + \frac{2 a^{2} c^{2} \sinh{\left (e + f x \right )}}{f} - \frac{a^{2} c d x^{2} \sinh ^{2}{\left (e + f x \right )}}{2} + \frac{a^{2} c d x^{2} \cosh ^{2}{\left (e + f x \right )}}{2} + a^{2} c d x^{2} + \frac{a^{2} c d x \sinh{\left (e + f x \right )} \cosh{\left (e + f x \right )}}{f} + \frac{4 a^{2} c d x \sinh{\left (e + f x \right )}}{f} - \frac{a^{2} c d \sinh ^{2}{\left (e + f x \right )}}{2 f^{2}} - \frac{4 a^{2} c d \cosh{\left (e + f x \right )}}{f^{2}} - \frac{a^{2} d^{2} x^{3} \sinh ^{2}{\left (e + f x \right )}}{6} + \frac{a^{2} d^{2} x^{3} \cosh ^{2}{\left (e + f x \right )}}{6} + \frac{a^{2} d^{2} x^{3}}{3} + \frac{a^{2} d^{2} x^{2} \sinh{\left (e + f x \right )} \cosh{\left (e + f x \right )}}{2 f} + \frac{2 a^{2} d^{2} x^{2} \sinh{\left (e + f x \right )}}{f} - \frac{a^{2} d^{2} x \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac{a^{2} d^{2} x \cosh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac{4 a^{2} d^{2} x \cosh{\left (e + f x \right )}}{f^{2}} + \frac{a^{2} d^{2} \sinh{\left (e + f x \right )} \cosh{\left (e + f x \right )}}{4 f^{3}} + \frac{4 a^{2} d^{2} \sinh{\left (e + f x \right )}}{f^{3}} & \text{for}\: f \neq 0 \\\left (a \cosh{\left (e \right )} + a\right )^{2} \left (c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2*(a+a*cosh(f*x+e))**2,x)

[Out]

Piecewise((-a**2*c**2*x*sinh(e + f*x)**2/2 + a**2*c**2*x*cosh(e + f*x)**2/2 + a**2*c**2*x + a**2*c**2*sinh(e +
f*x)*cosh(e + f*x)/(2*f) + 2*a**2*c**2*sinh(e + f*x)/f - a**2*c*d*x**2*sinh(e + f*x)**2/2 + a**2*c*d*x**2*cos
h(e + f*x)**2/2 + a**2*c*d*x**2 + a**2*c*d*x*sinh(e + f*x)*cosh(e + f*x)/f + 4*a**2*c*d*x*sinh(e + f*x)/f - a*
*2*c*d*sinh(e + f*x)**2/(2*f**2) - 4*a**2*c*d*cosh(e + f*x)/f**2 - a**2*d**2*x**3*sinh(e + f*x)**2/6 + a**2*d*
*2*x**3*cosh(e + f*x)**2/6 + a**2*d**2*x**3/3 + a**2*d**2*x**2*sinh(e + f*x)*cosh(e + f*x)/(2*f) + 2*a**2*d**2
*x**2*sinh(e + f*x)/f - a**2*d**2*x*sinh(e + f*x)**2/(4*f**2) - a**2*d**2*x*cosh(e + f*x)**2/(4*f**2) - 4*a**2
*d**2*x*cosh(e + f*x)/f**2 + a**2*d**2*sinh(e + f*x)*cosh(e + f*x)/(4*f**3) + 4*a**2*d**2*sinh(e + f*x)/f**3,
Ne(f, 0)), ((a*cosh(e) + a)**2*(c**2*x + c*d*x**2 + d**2*x**3/3), True))

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Giac [B]  time = 1.2366, size = 450, normalized size = 2.68 \begin{align*} \frac{1}{2} \, a^{2} d^{2} x^{3} + \frac{3}{2} \, a^{2} c d x^{2} + \frac{3}{2} \, a^{2} c^{2} x + \frac{{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} - 2 \, a^{2} d^{2} f x - 2 \, a^{2} c d f + a^{2} d^{2}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{16 \, f^{3}} + \frac{{\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2} - 2 \, a^{2} d^{2} f x - 2 \, a^{2} c d f + 2 \, a^{2} d^{2}\right )} e^{\left (f x + e\right )}}{f^{3}} - \frac{{\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2} + 2 \, a^{2} d^{2} f x + 2 \, a^{2} c d f + 2 \, a^{2} d^{2}\right )} e^{\left (-f x - e\right )}}{f^{3}} - \frac{{\left (2 \, a^{2} d^{2} f^{2} x^{2} + 4 \, a^{2} c d f^{2} x + 2 \, a^{2} c^{2} f^{2} + 2 \, a^{2} d^{2} f x + 2 \, a^{2} c d f + a^{2} d^{2}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{16 \, f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2*(a+a*cosh(f*x+e))^2,x, algorithm="giac")

[Out]

1/2*a^2*d^2*x^3 + 3/2*a^2*c*d*x^2 + 3/2*a^2*c^2*x + 1/16*(2*a^2*d^2*f^2*x^2 + 4*a^2*c*d*f^2*x + 2*a^2*c^2*f^2
- 2*a^2*d^2*f*x - 2*a^2*c*d*f + a^2*d^2)*e^(2*f*x + 2*e)/f^3 + (a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^
2 - 2*a^2*d^2*f*x - 2*a^2*c*d*f + 2*a^2*d^2)*e^(f*x + e)/f^3 - (a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^
2 + 2*a^2*d^2*f*x + 2*a^2*c*d*f + 2*a^2*d^2)*e^(-f*x - e)/f^3 - 1/16*(2*a^2*d^2*f^2*x^2 + 4*a^2*c*d*f^2*x + 2*
a^2*c^2*f^2 + 2*a^2*d^2*f*x + 2*a^2*c*d*f + a^2*d^2)*e^(-2*f*x - 2*e)/f^3