Integrand size = 12, antiderivative size = 78 \[ \int x^2 \sinh \left (a+\frac {b}{x}\right ) \, dx=\frac {1}{6} b x^2 \cosh \left (a+\frac {b}{x}\right )-\frac {1}{6} b^3 \cosh (a) \text {Chi}\left (\frac {b}{x}\right )+\frac {1}{6} b^2 x \sinh \left (a+\frac {b}{x}\right )+\frac {1}{3} x^3 \sinh \left (a+\frac {b}{x}\right )-\frac {1}{6} b^3 \sinh (a) \text {Shi}\left (\frac {b}{x}\right ) \] Output:
1/6*b*x^2*cosh(a+b/x)-1/6*b^3*cosh(a)*Chi(b/x)+1/6*b^2*x*sinh(a+b/x)+1/3*x ^3*sinh(a+b/x)-1/6*b^3*sinh(a)*Shi(b/x)
Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90 \[ \int x^2 \sinh \left (a+\frac {b}{x}\right ) \, dx=\frac {1}{6} \left (-b^3 \cosh (a) \text {Chi}\left (\frac {b}{x}\right )+x \left (b x \cosh \left (a+\frac {b}{x}\right )+b^2 \sinh \left (a+\frac {b}{x}\right )+2 x^2 \sinh \left (a+\frac {b}{x}\right )\right )-b^3 \sinh (a) \text {Shi}\left (\frac {b}{x}\right )\right ) \] Input:
Integrate[x^2*Sinh[a + b/x],x]
Output:
(-(b^3*Cosh[a]*CoshIntegral[b/x]) + x*(b*x*Cosh[a + b/x] + b^2*Sinh[a + b/ x] + 2*x^2*Sinh[a + b/x]) - b^3*Sinh[a]*SinhIntegral[b/x])/6
Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.15, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.417, Rules used = {5843, 3042, 26, 3778, 3042, 3778, 26, 3042, 26, 3778, 3042, 3784, 26, 3042, 26, 3779, 3782}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 \sinh \left (a+\frac {b}{x}\right ) \, dx\) |
| \(\Big \downarrow \) 5843 |
| \(\displaystyle -\int x^4 \sinh \left (a+\frac {b}{x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int -i x^4 \sin \left (i a+\frac {i b}{x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int x^4 \sin \left (i a+\frac {i b}{x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle i \left (\frac {1}{3} i b \int x^3 \cosh \left (a+\frac {b}{x}\right )d\frac {1}{x}-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {1}{3} i b \int x^3 \sin \left (i a+\frac {i b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle i \left (\frac {1}{3} i b \left (-\frac {1}{2} x^2 \cosh \left (a+\frac {b}{x}\right )+\frac {1}{2} i b \int -i x^2 \sinh \left (a+\frac {b}{x}\right )d\frac {1}{x}\right )-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {1}{3} i b \left (\frac {1}{2} b \int x^2 \sinh \left (a+\frac {b}{x}\right )d\frac {1}{x}-\frac {1}{2} x^2 \cosh \left (a+\frac {b}{x}\right )\right )-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {1}{3} i b \left (-\frac {1}{2} x^2 \cosh \left (a+\frac {b}{x}\right )+\frac {1}{2} b \int -i x^2 \sin \left (i a+\frac {i b}{x}\right )d\frac {1}{x}\right )-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {1}{3} i b \left (-\frac {1}{2} x^2 \cosh \left (a+\frac {b}{x}\right )-\frac {1}{2} i b \int x^2 \sin \left (i a+\frac {i b}{x}\right )d\frac {1}{x}\right )-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle i \left (\frac {1}{3} i b \left (-\frac {1}{2} x^2 \cosh \left (a+\frac {b}{x}\right )-\frac {1}{2} i b \left (i b \int x \cosh \left (a+\frac {b}{x}\right )d\frac {1}{x}-i x \sinh \left (a+\frac {b}{x}\right )\right )\right )-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {1}{3} i b \left (-\frac {1}{2} x^2 \cosh \left (a+\frac {b}{x}\right )-\frac {1}{2} i b \left (i b \int x \sin \left (i a+\frac {i b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}-i x \sinh \left (a+\frac {b}{x}\right )\right )\right )-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle i \left (\frac {1}{3} i b \left (-\frac {1}{2} x^2 \cosh \left (a+\frac {b}{x}\right )-\frac {1}{2} i b \left (i b \left (\cosh (a) \int x \cosh \left (\frac {b}{x}\right )d\frac {1}{x}-i \sinh (a) \int i x \sinh \left (\frac {b}{x}\right )d\frac {1}{x}\right )-i x \sinh \left (a+\frac {b}{x}\right )\right )\right )-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {1}{3} i b \left (-\frac {1}{2} x^2 \cosh \left (a+\frac {b}{x}\right )-\frac {1}{2} i b \left (i b \left (\sinh (a) \int x \sinh \left (\frac {b}{x}\right )d\frac {1}{x}+\cosh (a) \int x \cosh \left (\frac {b}{x}\right )d\frac {1}{x}\right )-i x \sinh \left (a+\frac {b}{x}\right )\right )\right )-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {1}{3} i b \left (-\frac {1}{2} x^2 \cosh \left (a+\frac {b}{x}\right )-\frac {1}{2} i b \left (i b \left (\sinh (a) \int -i x \sin \left (\frac {i b}{x}\right )d\frac {1}{x}+\cosh (a) \int x \sin \left (\frac {i b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}\right )-i x \sinh \left (a+\frac {b}{x}\right )\right )\right )-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {1}{3} i b \left (-\frac {1}{2} x^2 \cosh \left (a+\frac {b}{x}\right )-\frac {1}{2} i b \left (i b \left (\cosh (a) \int x \sin \left (\frac {i b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}-i \sinh (a) \int x \sin \left (\frac {i b}{x}\right )d\frac {1}{x}\right )-i x \sinh \left (a+\frac {b}{x}\right )\right )\right )-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle i \left (\frac {1}{3} i b \left (-\frac {1}{2} x^2 \cosh \left (a+\frac {b}{x}\right )-\frac {1}{2} i b \left (i b \left (\sinh (a) \text {Shi}\left (\frac {b}{x}\right )+\cosh (a) \int x \sin \left (\frac {i b}{x}+\frac {\pi }{2}\right )d\frac {1}{x}\right )-i x \sinh \left (a+\frac {b}{x}\right )\right )\right )-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle i \left (\frac {1}{3} i b \left (-\frac {1}{2} x^2 \cosh \left (a+\frac {b}{x}\right )-\frac {1}{2} i b \left (i b \left (\cosh (a) \text {Chi}\left (\frac {b}{x}\right )+\sinh (a) \text {Shi}\left (\frac {b}{x}\right )\right )-i x \sinh \left (a+\frac {b}{x}\right )\right )\right )-\frac {1}{3} i x^3 \sinh \left (a+\frac {b}{x}\right )\right )\) |
Input:
Int[x^2*Sinh[a + b/x],x]
Output:
I*((-1/3*I)*x^3*Sinh[a + b/x] + (I/3)*b*(-1/2*(x^2*Cosh[a + b/x]) - (I/2)* b*((-I)*x*Sinh[a + b/x] + I*b*(Cosh[a]*CoshIntegral[b/x] + Sinh[a]*SinhInt egral[b/x]))))
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> 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]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> 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]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbo l] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sinh[c + d*x] )^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplif y[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplif y[(m + 1)/n], 0]))
Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.67
| method | result | size |
| risch | \(\frac {{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (\frac {b}{x}\right ) b^{3}}{12}-\frac {{\mathrm e}^{-\frac {a x +b}{x}} b^{2} x}{12}+\frac {{\mathrm e}^{-\frac {a x +b}{x}} b \,x^{2}}{12}-\frac {{\mathrm e}^{-\frac {a x +b}{x}} x^{3}}{6}+\frac {{\mathrm e}^{a} \operatorname {expIntegral}_{1}\left (-\frac {b}{x}\right ) b^{3}}{12}+\frac {{\mathrm e}^{\frac {a x +b}{x}} x \,b^{2}}{12}+\frac {{\mathrm e}^{\frac {a x +b}{x}} x^{2} b}{12}+\frac {{\mathrm e}^{\frac {a x +b}{x}} x^{3}}{6}\) | \(130\) |
| meijerg | \(\frac {b^{3} \sqrt {\pi }\, \cosh \left (a \right ) \left (\frac {8 x^{2}}{\sqrt {\pi }\, b^{2}}-\frac {4 \left (2 \gamma -\frac {11}{3}-2 \ln \left (x \right )+2 \ln \left (i b \right )\right )}{3 \sqrt {\pi }}-\frac {8 x^{2} \left (\frac {55 b^{2}}{2 x^{2}}+45\right )}{45 \sqrt {\pi }\, b^{2}}+\frac {8 x^{2} \cosh \left (\frac {b}{x}\right )}{3 \sqrt {\pi }\, b^{2}}+\frac {16 x^{3} \left (\frac {5 b^{2}}{2 x^{2}}+5\right ) \sinh \left (\frac {b}{x}\right )}{15 \sqrt {\pi }\, b^{3}}-\frac {8 \left (\operatorname {Chi}\left (\frac {b}{x}\right )-\ln \left (\frac {b}{x}\right )-\gamma \right )}{3 \sqrt {\pi }}\right )}{16}+\frac {i b^{3} \sqrt {\pi }\, \sinh \left (a \right ) \left (-\frac {8 i \left (\frac {b^{2}}{x^{2}}+2\right ) x^{3} \cosh \left (\frac {b}{x}\right )}{3 b^{3} \sqrt {\pi }}-\frac {8 i x^{2} \sinh \left (\frac {b}{x}\right )}{3 b^{2} \sqrt {\pi }}+\frac {8 i \operatorname {Shi}\left (\frac {b}{x}\right )}{3 \sqrt {\pi }}\right )}{16}\) | \(202\) |
Input:
int(x^2*sinh(a+b/x),x,method=_RETURNVERBOSE)
Output:
1/12*exp(-a)*Ei(1,b/x)*b^3-1/12*exp(-(a*x+b)/x)*b^2*x+1/12*exp(-(a*x+b)/x) *b*x^2-1/6*exp(-(a*x+b)/x)*x^3+1/12*exp(a)*Ei(1,-b/x)*b^3+1/12*exp((a*x+b) /x)*x*b^2+1/12*exp((a*x+b)/x)*x^2*b+1/6*exp((a*x+b)/x)*x^3
Time = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.19 \[ \int x^2 \sinh \left (a+\frac {b}{x}\right ) \, dx=\frac {1}{6} \, b x^{2} \cosh \left (\frac {a x + b}{x}\right ) - \frac {1}{12} \, {\left (b^{3} {\rm Ei}\left (\frac {b}{x}\right ) + b^{3} {\rm Ei}\left (-\frac {b}{x}\right )\right )} \cosh \left (a\right ) - \frac {1}{12} \, {\left (b^{3} {\rm Ei}\left (\frac {b}{x}\right ) - b^{3} {\rm Ei}\left (-\frac {b}{x}\right )\right )} \sinh \left (a\right ) + \frac {1}{6} \, {\left (b^{2} x + 2 \, x^{3}\right )} \sinh \left (\frac {a x + b}{x}\right ) \] Input:
integrate(x^2*sinh(a+b/x),x, algorithm="fricas")
Output:
1/6*b*x^2*cosh((a*x + b)/x) - 1/12*(b^3*Ei(b/x) + b^3*Ei(-b/x))*cosh(a) - 1/12*(b^3*Ei(b/x) - b^3*Ei(-b/x))*sinh(a) + 1/6*(b^2*x + 2*x^3)*sinh((a*x + b)/x)
\[ \int x^2 \sinh \left (a+\frac {b}{x}\right ) \, dx=\int x^{2} \sinh {\left (a + \frac {b}{x} \right )}\, dx \] Input:
integrate(x**2*sinh(a+b/x),x)
Output:
Integral(x**2*sinh(a + b/x), x)
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.60 \[ \int x^2 \sinh \left (a+\frac {b}{x}\right ) \, dx=\frac {1}{3} \, x^{3} \sinh \left (a + \frac {b}{x}\right ) + \frac {1}{6} \, {\left (b^{2} e^{\left (-a\right )} \Gamma \left (-2, \frac {b}{x}\right ) + b^{2} e^{a} \Gamma \left (-2, -\frac {b}{x}\right )\right )} b \] Input:
integrate(x^2*sinh(a+b/x),x, algorithm="maxima")
Output:
1/3*x^3*sinh(a + b/x) + 1/6*(b^2*e^(-a)*gamma(-2, b/x) + b^2*e^a*gamma(-2, -b/x))*b
Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (68) = 136\).
Time = 0.13 (sec) , antiderivative size = 534, normalized size of antiderivative = 6.85 \[ \int x^2 \sinh \left (a+\frac {b}{x}\right ) \, dx=-\frac {a^{3} b^{4} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )} + a^{3} b^{4} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a} - \frac {3 \, {\left (a x + b\right )} a^{2} b^{4} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )}}{x} - \frac {3 \, {\left (a x + b\right )} a^{2} b^{4} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a}}{x} + \frac {3 \, {\left (a x + b\right )}^{2} a b^{4} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )}}{x^{2}} + \frac {3 \, {\left (a x + b\right )}^{2} a b^{4} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a}}{x^{2}} + a^{2} b^{4} e^{\left (\frac {a x + b}{x}\right )} - a^{2} b^{4} e^{\left (-\frac {a x + b}{x}\right )} - \frac {{\left (a x + b\right )}^{3} b^{4} {\rm Ei}\left (a - \frac {a x + b}{x}\right ) e^{\left (-a\right )}}{x^{3}} - \frac {{\left (a x + b\right )}^{3} b^{4} {\rm Ei}\left (-a + \frac {a x + b}{x}\right ) e^{a}}{x^{3}} - a b^{4} e^{\left (\frac {a x + b}{x}\right )} - \frac {2 \, {\left (a x + b\right )} a b^{4} e^{\left (\frac {a x + b}{x}\right )}}{x} - a b^{4} e^{\left (-\frac {a x + b}{x}\right )} + \frac {2 \, {\left (a x + b\right )} a b^{4} e^{\left (-\frac {a x + b}{x}\right )}}{x} + 2 \, b^{4} e^{\left (\frac {a x + b}{x}\right )} + \frac {{\left (a x + b\right )}^{2} b^{4} e^{\left (\frac {a x + b}{x}\right )}}{x^{2}} + \frac {{\left (a x + b\right )} b^{4} e^{\left (\frac {a x + b}{x}\right )}}{x} - 2 \, b^{4} e^{\left (-\frac {a x + b}{x}\right )} - \frac {{\left (a x + b\right )}^{2} b^{4} e^{\left (-\frac {a x + b}{x}\right )}}{x^{2}} + \frac {{\left (a x + b\right )} b^{4} e^{\left (-\frac {a x + b}{x}\right )}}{x}}{12 \, {\left (a^{3} - \frac {3 \, {\left (a x + b\right )} a^{2}}{x} + \frac {3 \, {\left (a x + b\right )}^{2} a}{x^{2}} - \frac {{\left (a x + b\right )}^{3}}{x^{3}}\right )} b} \] Input:
integrate(x^2*sinh(a+b/x),x, algorithm="giac")
Output:
-1/12*(a^3*b^4*Ei(a - (a*x + b)/x)*e^(-a) + a^3*b^4*Ei(-a + (a*x + b)/x)*e ^a - 3*(a*x + b)*a^2*b^4*Ei(a - (a*x + b)/x)*e^(-a)/x - 3*(a*x + b)*a^2*b^ 4*Ei(-a + (a*x + b)/x)*e^a/x + 3*(a*x + b)^2*a*b^4*Ei(a - (a*x + b)/x)*e^( -a)/x^2 + 3*(a*x + b)^2*a*b^4*Ei(-a + (a*x + b)/x)*e^a/x^2 + a^2*b^4*e^((a *x + b)/x) - a^2*b^4*e^(-(a*x + b)/x) - (a*x + b)^3*b^4*Ei(a - (a*x + b)/x )*e^(-a)/x^3 - (a*x + b)^3*b^4*Ei(-a + (a*x + b)/x)*e^a/x^3 - a*b^4*e^((a* x + b)/x) - 2*(a*x + b)*a*b^4*e^((a*x + b)/x)/x - a*b^4*e^(-(a*x + b)/x) + 2*(a*x + b)*a*b^4*e^(-(a*x + b)/x)/x + 2*b^4*e^((a*x + b)/x) + (a*x + b)^ 2*b^4*e^((a*x + b)/x)/x^2 + (a*x + b)*b^4*e^((a*x + b)/x)/x - 2*b^4*e^(-(a *x + b)/x) - (a*x + b)^2*b^4*e^(-(a*x + b)/x)/x^2 + (a*x + b)*b^4*e^(-(a*x + b)/x)/x)/((a^3 - 3*(a*x + b)*a^2/x + 3*(a*x + b)^2*a/x^2 - (a*x + b)^3/ x^3)*b)
Timed out. \[ \int x^2 \sinh \left (a+\frac {b}{x}\right ) \, dx=\int x^2\,\mathrm {sinh}\left (a+\frac {b}{x}\right ) \,d x \] Input:
int(x^2*sinh(a + b/x),x)
Output:
int(x^2*sinh(a + b/x), x)
\[ \int x^2 \sinh \left (a+\frac {b}{x}\right ) \, dx=\frac {-e^{\frac {2 a x +b}{x}} \mathit {ei} \left (\frac {b}{x}\right ) b^{3}+e^{\frac {2 a x +2 b}{x}} b^{2} x +e^{\frac {2 a x +2 b}{x}} b \,x^{2}+2 e^{\frac {2 a x +2 b}{x}} x^{3}+e^{\frac {a x +b}{x}} \left (\int \frac {1}{e^{\frac {a x +b}{x}} x}d x \right ) b^{3}-b^{2} x +b \,x^{2}-2 x^{3}}{12 e^{\frac {a x +b}{x}}} \] Input:
int(x^2*sinh(a+b/x),x)
Output:
( - e**((2*a*x + b)/x)*ei(b/x)*b**3 + e**((2*a*x + 2*b)/x)*b**2*x + e**((2 *a*x + 2*b)/x)*b*x**2 + 2*e**((2*a*x + 2*b)/x)*x**3 + e**((a*x + b)/x)*int (1/(e**((a*x + b)/x)*x),x)*b**3 - b**2*x + b*x**2 - 2*x**3)/(12*e**((a*x + b)/x))