3.89 \(\int \frac{(a+b x^3)^2 \cosh (c+d x)}{x} \, dx\)

Optimal. Leaf size=160 \[ a^2 \cosh (c) \text{Chi}(d x)+a^2 \sinh (c) \text{Shi}(d x)+\frac{4 a b \sinh (c+d x)}{d^3}-\frac{4 a b x \cosh (c+d x)}{d^2}+\frac{2 a b x^2 \sinh (c+d x)}{d}+\frac{20 b^2 x^3 \sinh (c+d x)}{d^3}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}-\frac{60 b^2 x^2 \cosh (c+d x)}{d^4}+\frac{120 b^2 x \sinh (c+d x)}{d^5}-\frac{120 b^2 \cosh (c+d x)}{d^6}+\frac{b^2 x^5 \sinh (c+d x)}{d} \]

[Out]

(-120*b^2*Cosh[c + d*x])/d^6 - (4*a*b*x*Cosh[c + d*x])/d^2 - (60*b^2*x^2*Cosh[c + d*x])/d^4 - (5*b^2*x^4*Cosh[
c + d*x])/d^2 + a^2*Cosh[c]*CoshIntegral[d*x] + (4*a*b*Sinh[c + d*x])/d^3 + (120*b^2*x*Sinh[c + d*x])/d^5 + (2
*a*b*x^2*Sinh[c + d*x])/d + (20*b^2*x^3*Sinh[c + d*x])/d^3 + (b^2*x^5*Sinh[c + d*x])/d + a^2*Sinh[c]*SinhInteg
ral[d*x]

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Rubi [A]  time = 0.282343, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {5287, 3303, 3298, 3301, 3296, 2637, 2638} \[ a^2 \cosh (c) \text{Chi}(d x)+a^2 \sinh (c) \text{Shi}(d x)+\frac{4 a b \sinh (c+d x)}{d^3}-\frac{4 a b x \cosh (c+d x)}{d^2}+\frac{2 a b x^2 \sinh (c+d x)}{d}+\frac{20 b^2 x^3 \sinh (c+d x)}{d^3}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}-\frac{60 b^2 x^2 \cosh (c+d x)}{d^4}+\frac{120 b^2 x \sinh (c+d x)}{d^5}-\frac{120 b^2 \cosh (c+d x)}{d^6}+\frac{b^2 x^5 \sinh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^2*Cosh[c + d*x])/x,x]

[Out]

(-120*b^2*Cosh[c + d*x])/d^6 - (4*a*b*x*Cosh[c + d*x])/d^2 - (60*b^2*x^2*Cosh[c + d*x])/d^4 - (5*b^2*x^4*Cosh[
c + d*x])/d^2 + a^2*Cosh[c]*CoshIntegral[d*x] + (4*a*b*Sinh[c + d*x])/d^3 + (120*b^2*x*Sinh[c + d*x])/d^5 + (2
*a*b*x^2*Sinh[c + d*x])/d + (20*b^2*x^3*Sinh[c + d*x])/d^3 + (b^2*x^5*Sinh[c + d*x])/d + a^2*Sinh[c]*SinhInteg
ral[d*x]

Rule 5287

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[Cosh[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rule 3303

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

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 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 2638

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^2 \cosh (c+d x)}{x} \, dx &=\int \left (\frac{a^2 \cosh (c+d x)}{x}+2 a b x^2 \cosh (c+d x)+b^2 x^5 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int \frac{\cosh (c+d x)}{x} \, dx+(2 a b) \int x^2 \cosh (c+d x) \, dx+b^2 \int x^5 \cosh (c+d x) \, dx\\ &=\frac{2 a b x^2 \sinh (c+d x)}{d}+\frac{b^2 x^5 \sinh (c+d x)}{d}-\frac{(4 a b) \int x \sinh (c+d x) \, dx}{d}-\frac{\left (5 b^2\right ) \int x^4 \sinh (c+d x) \, dx}{d}+\left (a^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx+\left (a^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx\\ &=-\frac{4 a b x \cosh (c+d x)}{d^2}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text{Chi}(d x)+\frac{2 a b x^2 \sinh (c+d x)}{d}+\frac{b^2 x^5 \sinh (c+d x)}{d}+a^2 \sinh (c) \text{Shi}(d x)+\frac{(4 a b) \int \cosh (c+d x) \, dx}{d^2}+\frac{\left (20 b^2\right ) \int x^3 \cosh (c+d x) \, dx}{d^2}\\ &=-\frac{4 a b x \cosh (c+d x)}{d^2}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text{Chi}(d x)+\frac{4 a b \sinh (c+d x)}{d^3}+\frac{2 a b x^2 \sinh (c+d x)}{d}+\frac{20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac{b^2 x^5 \sinh (c+d x)}{d}+a^2 \sinh (c) \text{Shi}(d x)-\frac{\left (60 b^2\right ) \int x^2 \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{4 a b x \cosh (c+d x)}{d^2}-\frac{60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text{Chi}(d x)+\frac{4 a b \sinh (c+d x)}{d^3}+\frac{2 a b x^2 \sinh (c+d x)}{d}+\frac{20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac{b^2 x^5 \sinh (c+d x)}{d}+a^2 \sinh (c) \text{Shi}(d x)+\frac{\left (120 b^2\right ) \int x \cosh (c+d x) \, dx}{d^4}\\ &=-\frac{4 a b x \cosh (c+d x)}{d^2}-\frac{60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text{Chi}(d x)+\frac{4 a b \sinh (c+d x)}{d^3}+\frac{120 b^2 x \sinh (c+d x)}{d^5}+\frac{2 a b x^2 \sinh (c+d x)}{d}+\frac{20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac{b^2 x^5 \sinh (c+d x)}{d}+a^2 \sinh (c) \text{Shi}(d x)-\frac{\left (120 b^2\right ) \int \sinh (c+d x) \, dx}{d^5}\\ &=-\frac{120 b^2 \cosh (c+d x)}{d^6}-\frac{4 a b x \cosh (c+d x)}{d^2}-\frac{60 b^2 x^2 \cosh (c+d x)}{d^4}-\frac{5 b^2 x^4 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text{Chi}(d x)+\frac{4 a b \sinh (c+d x)}{d^3}+\frac{120 b^2 x \sinh (c+d x)}{d^5}+\frac{2 a b x^2 \sinh (c+d x)}{d}+\frac{20 b^2 x^3 \sinh (c+d x)}{d^3}+\frac{b^2 x^5 \sinh (c+d x)}{d}+a^2 \sinh (c) \text{Shi}(d x)\\ \end{align*}

Mathematica [A]  time = 0.468257, size = 108, normalized size = 0.68 \[ a^2 \cosh (c) \text{Chi}(d x)+a^2 \sinh (c) \text{Shi}(d x)+\frac{b \left (2 a d^2 \left (d^2 x^2+2\right )+b x \left (d^4 x^4+20 d^2 x^2+120\right )\right ) \sinh (c+d x)}{d^5}-\frac{b \left (4 a d^4 x+5 b \left (d^4 x^4+12 d^2 x^2+24\right )\right ) \cosh (c+d x)}{d^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^2*Cosh[c + d*x])/x,x]

[Out]

-((b*(4*a*d^4*x + 5*b*(24 + 12*d^2*x^2 + d^4*x^4))*Cosh[c + d*x])/d^6) + a^2*Cosh[c]*CoshIntegral[d*x] + (b*(2
*a*d^2*(2 + d^2*x^2) + b*x*(120 + 20*d^2*x^2 + d^4*x^4))*Sinh[c + d*x])/d^5 + a^2*Sinh[c]*SinhIntegral[d*x]

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Maple [B]  time = 0.086, size = 335, normalized size = 2.1 \begin{align*} -{\frac{{b}^{2}{{\rm e}^{-dx-c}}{x}^{5}}{2\,d}}-{\frac{5\,{b}^{2}{{\rm e}^{-dx-c}}{x}^{4}}{2\,{d}^{2}}}-10\,{\frac{{b}^{2}{{\rm e}^{-dx-c}}{x}^{3}}{{d}^{3}}}-30\,{\frac{{b}^{2}{{\rm e}^{-dx-c}}{x}^{2}}{{d}^{4}}}-60\,{\frac{{b}^{2}{{\rm e}^{-dx-c}}x}{{d}^{5}}}-{\frac{{a}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{{a}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}-60\,{\frac{{{\rm e}^{dx+c}}{b}^{2}}{{d}^{6}}}-2\,{\frac{ab{{\rm e}^{-dx-c}}}{{d}^{3}}}+{\frac{ab{{\rm e}^{dx+c}}{x}^{2}}{d}}-2\,{\frac{ab{{\rm e}^{dx+c}}x}{{d}^{2}}}+2\,{\frac{ab{{\rm e}^{dx+c}}}{{d}^{3}}}-60\,{\frac{{b}^{2}{{\rm e}^{-dx-c}}}{{d}^{6}}}-30\,{\frac{{{\rm e}^{dx+c}}{b}^{2}{x}^{2}}{{d}^{4}}}+60\,{\frac{{{\rm e}^{dx+c}}{b}^{2}x}{{d}^{5}}}+{\frac{{{\rm e}^{dx+c}}{b}^{2}{x}^{5}}{2\,d}}-{\frac{5\,{{\rm e}^{dx+c}}{b}^{2}{x}^{4}}{2\,{d}^{2}}}+10\,{\frac{{{\rm e}^{dx+c}}{b}^{2}{x}^{3}}{{d}^{3}}}-{\frac{ab{{\rm e}^{-dx-c}}{x}^{2}}{d}}-2\,{\frac{ab{{\rm e}^{-dx-c}}x}{{d}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2*cosh(d*x+c)/x,x)

[Out]

-1/2/d*b^2*exp(-d*x-c)*x^5-5/2/d^2*b^2*exp(-d*x-c)*x^4-10/d^3*b^2*exp(-d*x-c)*x^3-30/d^4*b^2*exp(-d*x-c)*x^2-6
0/d^5*b^2*exp(-d*x-c)*x-1/2*a^2*exp(-c)*Ei(1,d*x)-1/2*a^2*exp(c)*Ei(1,-d*x)-60/d^6*b^2*exp(d*x+c)-2/d^3*a*b*ex
p(-d*x-c)+1/d*a*b*exp(d*x+c)*x^2-2/d^2*a*b*exp(d*x+c)*x+2/d^3*a*b*exp(d*x+c)-60/d^6*b^2*exp(-d*x-c)-30/d^4*b^2
*exp(d*x+c)*x^2+60/d^5*b^2*exp(d*x+c)*x+1/2/d*b^2*exp(d*x+c)*x^5-5/2/d^2*b^2*exp(d*x+c)*x^4+10/d^3*b^2*exp(d*x
+c)*x^3-1/d*a*b*exp(-d*x-c)*x^2-2/d^2*a*b*exp(-d*x-c)*x

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Maxima [A]  time = 1.23268, size = 390, normalized size = 2.44 \begin{align*} -\frac{1}{12} \,{\left (4 \, a b{\left (\frac{{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} e^{\left (d x\right )}}{d^{4}} + \frac{{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} e^{\left (-d x - c\right )}}{d^{4}}\right )} + b^{2}{\left (\frac{{\left (d^{6} x^{6} e^{c} - 6 \, d^{5} x^{5} e^{c} + 30 \, d^{4} x^{4} e^{c} - 120 \, d^{3} x^{3} e^{c} + 360 \, d^{2} x^{2} e^{c} - 720 \, d x e^{c} + 720 \, e^{c}\right )} e^{\left (d x\right )}}{d^{7}} + \frac{{\left (d^{6} x^{6} + 6 \, d^{5} x^{5} + 30 \, d^{4} x^{4} + 120 \, d^{3} x^{3} + 360 \, d^{2} x^{2} + 720 \, d x + 720\right )} e^{\left (-d x - c\right )}}{d^{7}}\right )} + \frac{4 \, a^{2} \cosh \left (d x + c\right ) \log \left (x^{3}\right )}{d} - \frac{6 \,{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} +{\rm Ei}\left (d x\right ) e^{c}\right )} a^{2}}{d}\right )} d + \frac{1}{6} \,{\left (b^{2} x^{6} + 4 \, a b x^{3} + 2 \, a^{2} \log \left (x^{3}\right )\right )} \cosh \left (d x + c\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*cosh(d*x+c)/x,x, algorithm="maxima")

[Out]

-1/12*(4*a*b*((d^3*x^3*e^c - 3*d^2*x^2*e^c + 6*d*x*e^c - 6*e^c)*e^(d*x)/d^4 + (d^3*x^3 + 3*d^2*x^2 + 6*d*x + 6
)*e^(-d*x - c)/d^4) + b^2*((d^6*x^6*e^c - 6*d^5*x^5*e^c + 30*d^4*x^4*e^c - 120*d^3*x^3*e^c + 360*d^2*x^2*e^c -
720*d*x*e^c + 720*e^c)*e^(d*x)/d^7 + (d^6*x^6 + 6*d^5*x^5 + 30*d^4*x^4 + 120*d^3*x^3 + 360*d^2*x^2 + 720*d*x
+ 720)*e^(-d*x - c)/d^7) + 4*a^2*cosh(d*x + c)*log(x^3)/d - 6*(Ei(-d*x)*e^(-c) + Ei(d*x)*e^c)*a^2/d)*d + 1/6*(
b^2*x^6 + 4*a*b*x^3 + 2*a^2*log(x^3))*cosh(d*x + c)

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Fricas [A]  time = 1.78628, size = 365, normalized size = 2.28 \begin{align*} -\frac{2 \,{\left (5 \, b^{2} d^{4} x^{4} + 4 \, a b d^{4} x + 60 \, b^{2} d^{2} x^{2} + 120 \, b^{2}\right )} \cosh \left (d x + c\right ) -{\left (a^{2} d^{6}{\rm Ei}\left (d x\right ) + a^{2} d^{6}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - 2 \,{\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{2} + 20 \, b^{2} d^{3} x^{3} + 4 \, a b d^{3} + 120 \, b^{2} d x\right )} \sinh \left (d x + c\right ) -{\left (a^{2} d^{6}{\rm Ei}\left (d x\right ) - a^{2} d^{6}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*cosh(d*x+c)/x,x, algorithm="fricas")

[Out]

-1/2*(2*(5*b^2*d^4*x^4 + 4*a*b*d^4*x + 60*b^2*d^2*x^2 + 120*b^2)*cosh(d*x + c) - (a^2*d^6*Ei(d*x) + a^2*d^6*Ei
(-d*x))*cosh(c) - 2*(b^2*d^5*x^5 + 2*a*b*d^5*x^2 + 20*b^2*d^3*x^3 + 4*a*b*d^3 + 120*b^2*d*x)*sinh(d*x + c) - (
a^2*d^6*Ei(d*x) - a^2*d^6*Ei(-d*x))*sinh(c))/d^6

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Sympy [A]  time = 11.71, size = 168, normalized size = 1.05 \begin{align*} a^{2} \sinh{\left (c \right )} \operatorname{Shi}{\left (d x \right )} + a^{2} \cosh{\left (c \right )} \operatorname{Chi}\left (d x\right ) + 2 a b \left (\begin{cases} \frac{x^{2} \sinh{\left (c + d x \right )}}{d} - \frac{2 x \cosh{\left (c + d x \right )}}{d^{2}} + \frac{2 \sinh{\left (c + d x \right )}}{d^{3}} & \text{for}\: d \neq 0 \\\frac{x^{3} \cosh{\left (c \right )}}{3} & \text{otherwise} \end{cases}\right ) + b^{2} \left (\begin{cases} \frac{x^{5} \sinh{\left (c + d x \right )}}{d} - \frac{5 x^{4} \cosh{\left (c + d x \right )}}{d^{2}} + \frac{20 x^{3} \sinh{\left (c + d x \right )}}{d^{3}} - \frac{60 x^{2} \cosh{\left (c + d x \right )}}{d^{4}} + \frac{120 x \sinh{\left (c + d x \right )}}{d^{5}} - \frac{120 \cosh{\left (c + d x \right )}}{d^{6}} & \text{for}\: d \neq 0 \\\frac{x^{6} \cosh{\left (c \right )}}{6} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2*cosh(d*x+c)/x,x)

[Out]

a**2*sinh(c)*Shi(d*x) + a**2*cosh(c)*Chi(d*x) + 2*a*b*Piecewise((x**2*sinh(c + d*x)/d - 2*x*cosh(c + d*x)/d**2
+ 2*sinh(c + d*x)/d**3, Ne(d, 0)), (x**3*cosh(c)/3, True)) + b**2*Piecewise((x**5*sinh(c + d*x)/d - 5*x**4*co
sh(c + d*x)/d**2 + 20*x**3*sinh(c + d*x)/d**3 - 60*x**2*cosh(c + d*x)/d**4 + 120*x*sinh(c + d*x)/d**5 - 120*co
sh(c + d*x)/d**6, Ne(d, 0)), (x**6*cosh(c)/6, True))

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Giac [B]  time = 1.21348, size = 447, normalized size = 2.79 \begin{align*} \frac{b^{2} d^{5} x^{5} e^{\left (d x + c\right )} - b^{2} d^{5} x^{5} e^{\left (-d x - c\right )} - 5 \, b^{2} d^{4} x^{4} e^{\left (d x + c\right )} - 5 \, b^{2} d^{4} x^{4} e^{\left (-d x - c\right )} + 2 \, a b d^{5} x^{2} e^{\left (d x + c\right )} - 2 \, a b d^{5} x^{2} e^{\left (-d x - c\right )} + a^{2} d^{6}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{6}{\rm Ei}\left (d x\right ) e^{c} + 20 \, b^{2} d^{3} x^{3} e^{\left (d x + c\right )} - 20 \, b^{2} d^{3} x^{3} e^{\left (-d x - c\right )} - 4 \, a b d^{4} x e^{\left (d x + c\right )} - 4 \, a b d^{4} x e^{\left (-d x - c\right )} - 60 \, b^{2} d^{2} x^{2} e^{\left (d x + c\right )} - 60 \, b^{2} d^{2} x^{2} e^{\left (-d x - c\right )} + 4 \, a b d^{3} e^{\left (d x + c\right )} - 4 \, a b d^{3} e^{\left (-d x - c\right )} + 120 \, b^{2} d x e^{\left (d x + c\right )} - 120 \, b^{2} d x e^{\left (-d x - c\right )} - 120 \, b^{2} e^{\left (d x + c\right )} - 120 \, b^{2} e^{\left (-d x - c\right )}}{2 \, d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*cosh(d*x+c)/x,x, algorithm="giac")

[Out]

1/2*(b^2*d^5*x^5*e^(d*x + c) - b^2*d^5*x^5*e^(-d*x - c) - 5*b^2*d^4*x^4*e^(d*x + c) - 5*b^2*d^4*x^4*e^(-d*x -
c) + 2*a*b*d^5*x^2*e^(d*x + c) - 2*a*b*d^5*x^2*e^(-d*x - c) + a^2*d^6*Ei(-d*x)*e^(-c) + a^2*d^6*Ei(d*x)*e^c +
20*b^2*d^3*x^3*e^(d*x + c) - 20*b^2*d^3*x^3*e^(-d*x - c) - 4*a*b*d^4*x*e^(d*x + c) - 4*a*b*d^4*x*e^(-d*x - c)
- 60*b^2*d^2*x^2*e^(d*x + c) - 60*b^2*d^2*x^2*e^(-d*x - c) + 4*a*b*d^3*e^(d*x + c) - 4*a*b*d^3*e^(-d*x - c) +
120*b^2*d*x*e^(d*x + c) - 120*b^2*d*x*e^(-d*x - c) - 120*b^2*e^(d*x + c) - 120*b^2*e^(-d*x - c))/d^6