3.73.80 \(\int \frac {135-72 x+90 x^2-12 x^3-x^4+4 x^6+e^{2 e^2} (15-3 x^2)+e^{2 x} (15-6 x-3 x^2+2 x^3)+e^x (-90+42 x-24 x^2+2 x^3+6 x^4-4 x^5)+e^{e^2} (-90+24 x-18 x^2-4 x^3+4 x^4+e^x (30-6 x-6 x^2+2 x^3))}{x^6} \, dx\)

Optimal. Leaf size=34 \[ \left (-2-\frac {3-e^{e^2}-e^x-x}{x^2}\right )^2 \left (-\frac {3}{x}+x\right ) \]

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Rubi [C]  time = 0.93, antiderivative size = 406, normalized size of antiderivative = 11.94, number of steps used = 47, number of rules used = 4, integrand size = 139, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {14, 2199, 2177, 2178} \begin {gather*} -\left (4+e^{e^2}\right ) \text {Ei}(x)+\left (1+e^{e^2}\right ) \text {Ei}(x)+\frac {1}{4} \left (7-e^{e^2}\right ) \text {Ei}(x)-\frac {1}{4} \left (3-e^{e^2}\right ) \text {Ei}(x)+2 \text {Ei}(x)-\frac {3 e^{2 x}}{x^5}-\frac {3 \left (3-e^{e^2}\right )^2}{x^5}+\frac {6 \left (3-e^{e^2}\right ) e^x}{x^5}-\frac {3 \left (7-e^{e^2}\right ) e^x}{2 x^4}+\frac {3 \left (3-e^{e^2}\right ) e^x}{2 x^4}+\frac {6 \left (3-e^{e^2}\right )}{x^4}+\frac {e^{2 x}}{x^3}-\frac {30-6 e^{e^2}-e^{2 e^2}}{x^3}+\frac {2 \left (4+e^{e^2}\right ) e^x}{x^3}-\frac {\left (7-e^{e^2}\right ) e^x}{2 x^3}+\frac {\left (3-e^{e^2}\right ) e^x}{2 x^3}+\frac {\left (4+e^{e^2}\right ) e^x}{x^2}+\frac {2 \left (3+e^{e^2}\right )}{x^2}-\frac {\left (1+e^{e^2}\right ) e^x}{x^2}-\frac {\left (7-e^{e^2}\right ) e^x}{4 x^2}+\frac {\left (3-e^{e^2}\right ) e^x}{4 x^2}+4 x-\frac {6 e^x}{x}+\frac {\left (4+e^{e^2}\right ) e^x}{x}-\frac {\left (1+e^{e^2}\right ) e^x}{x}-\frac {\left (7-e^{e^2}\right ) e^x}{4 x}+\frac {\left (3-e^{e^2}\right ) e^x}{4 x}+\frac {1-4 e^{e^2}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(135 - 72*x + 90*x^2 - 12*x^3 - x^4 + 4*x^6 + E^(2*E^2)*(15 - 3*x^2) + E^(2*x)*(15 - 6*x - 3*x^2 + 2*x^3)
+ E^x*(-90 + 42*x - 24*x^2 + 2*x^3 + 6*x^4 - 4*x^5) + E^E^2*(-90 + 24*x - 18*x^2 - 4*x^3 + 4*x^4 + E^x*(30 - 6
*x - 6*x^2 + 2*x^3)))/x^6,x]

[Out]

(-3*E^(2*x))/x^5 + (6*E^x*(3 - E^E^2))/x^5 - (3*(3 - E^E^2)^2)/x^5 + (6*(3 - E^E^2))/x^4 + (3*E^x*(3 - E^E^2))
/(2*x^4) - (3*E^x*(7 - E^E^2))/(2*x^4) + E^(2*x)/x^3 + (E^x*(3 - E^E^2))/(2*x^3) - (E^x*(7 - E^E^2))/(2*x^3) +
(2*E^x*(4 + E^E^2))/x^3 - (30 - 6*E^E^2 - E^(2*E^2))/x^3 + (E^x*(3 - E^E^2))/(4*x^2) - (E^x*(7 - E^E^2))/(4*x
^2) - (E^x*(1 + E^E^2))/x^2 + (2*(3 + E^E^2))/x^2 + (E^x*(4 + E^E^2))/x^2 - (6*E^x)/x + (1 - 4*E^E^2)/x + (E^x
*(3 - E^E^2))/(4*x) - (E^x*(7 - E^E^2))/(4*x) - (E^x*(1 + E^E^2))/x + (E^x*(4 + E^E^2))/x + 4*x + 2*ExpIntegra
lEi[x] - ((3 - E^E^2)*ExpIntegralEi[x])/4 + ((7 - E^E^2)*ExpIntegralEi[x])/4 + (1 + E^E^2)*ExpIntegralEi[x] -
(4 + E^E^2)*ExpIntegralEi[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^{2 x} \left (15-6 x-3 x^2+2 x^3\right )}{x^6}+\frac {2 e^x \left (-45 \left (1-\frac {e^{e^2}}{3}\right )+21 \left (1-\frac {e^{e^2}}{7}\right ) x-12 \left (1+\frac {e^{e^2}}{4}\right ) x^2+\left (1+e^{e^2}\right ) x^3+3 x^4-2 x^5\right )}{x^6}+\frac {135 \left (1+\frac {1}{9} e^{e^2} \left (-6+e^{e^2}\right )\right )-72 \left (1-\frac {e^{e^2}}{3}\right ) x+90 \left (1-\frac {1}{30} e^{e^2} \left (6+e^{e^2}\right )\right ) x^2-12 \left (1+\frac {e^{e^2}}{3}\right ) x^3-\left (1-4 e^{e^2}\right ) x^4+4 x^6}{x^6}\right ) \, dx\\ &=2 \int \frac {e^x \left (-45 \left (1-\frac {e^{e^2}}{3}\right )+21 \left (1-\frac {e^{e^2}}{7}\right ) x-12 \left (1+\frac {e^{e^2}}{4}\right ) x^2+\left (1+e^{e^2}\right ) x^3+3 x^4-2 x^5\right )}{x^6} \, dx+\int \frac {e^{2 x} \left (15-6 x-3 x^2+2 x^3\right )}{x^6} \, dx+\int \frac {135 \left (1+\frac {1}{9} e^{e^2} \left (-6+e^{e^2}\right )\right )-72 \left (1-\frac {e^{e^2}}{3}\right ) x+90 \left (1-\frac {1}{30} e^{e^2} \left (6+e^{e^2}\right )\right ) x^2-12 \left (1+\frac {e^{e^2}}{3}\right ) x^3-\left (1-4 e^{e^2}\right ) x^4+4 x^6}{x^6} \, dx\\ &=2 \int \left (\frac {15 e^x \left (-3+e^{e^2}\right )}{x^6}-\frac {3 e^x \left (-7+e^{e^2}\right )}{x^5}-\frac {3 e^x \left (4+e^{e^2}\right )}{x^4}+\frac {e^x \left (1+e^{e^2}\right )}{x^3}+\frac {3 e^x}{x^2}-\frac {2 e^x}{x}\right ) \, dx+\int \left (\frac {15 e^{2 x}}{x^6}-\frac {6 e^{2 x}}{x^5}-\frac {3 e^{2 x}}{x^4}+\frac {2 e^{2 x}}{x^3}\right ) \, dx+\int \left (4+\frac {15 \left (-3+e^{e^2}\right )^2}{x^6}+\frac {24 \left (-3+e^{e^2}\right )}{x^5}-\frac {3 \left (-30+6 e^{e^2}+e^{2 e^2}\right )}{x^4}-\frac {4 \left (3+e^{e^2}\right )}{x^3}+\frac {-1+4 e^{e^2}}{x^2}\right ) \, dx\\ &=-\frac {3 \left (3-e^{e^2}\right )^2}{x^5}+\frac {6 \left (3-e^{e^2}\right )}{x^4}-\frac {30-6 e^{e^2}-e^{2 e^2}}{x^3}+\frac {2 \left (3+e^{e^2}\right )}{x^2}+\frac {1-4 e^{e^2}}{x}+4 x+2 \int \frac {e^{2 x}}{x^3} \, dx-3 \int \frac {e^{2 x}}{x^4} \, dx-4 \int \frac {e^x}{x} \, dx-6 \int \frac {e^{2 x}}{x^5} \, dx+6 \int \frac {e^x}{x^2} \, dx+15 \int \frac {e^{2 x}}{x^6} \, dx-\left (30 \left (3-e^{e^2}\right )\right ) \int \frac {e^x}{x^6} \, dx+\left (6 \left (7-e^{e^2}\right )\right ) \int \frac {e^x}{x^5} \, dx+\left (2 \left (1+e^{e^2}\right )\right ) \int \frac {e^x}{x^3} \, dx-\left (6 \left (4+e^{e^2}\right )\right ) \int \frac {e^x}{x^4} \, dx\\ &=-\frac {3 e^{2 x}}{x^5}+\frac {6 e^x \left (3-e^{e^2}\right )}{x^5}-\frac {3 \left (3-e^{e^2}\right )^2}{x^5}+\frac {3 e^{2 x}}{2 x^4}+\frac {6 \left (3-e^{e^2}\right )}{x^4}-\frac {3 e^x \left (7-e^{e^2}\right )}{2 x^4}+\frac {e^{2 x}}{x^3}+\frac {2 e^x \left (4+e^{e^2}\right )}{x^3}-\frac {30-6 e^{e^2}-e^{2 e^2}}{x^3}-\frac {e^{2 x}}{x^2}-\frac {e^x \left (1+e^{e^2}\right )}{x^2}+\frac {2 \left (3+e^{e^2}\right )}{x^2}-\frac {6 e^x}{x}+\frac {1-4 e^{e^2}}{x}+4 x-4 \text {Ei}(x)-2 \int \frac {e^{2 x}}{x^3} \, dx+2 \int \frac {e^{2 x}}{x^2} \, dx-3 \int \frac {e^{2 x}}{x^4} \, dx+6 \int \frac {e^{2 x}}{x^5} \, dx+6 \int \frac {e^x}{x} \, dx-\left (6 \left (3-e^{e^2}\right )\right ) \int \frac {e^x}{x^5} \, dx+\frac {1}{2} \left (3 \left (7-e^{e^2}\right )\right ) \int \frac {e^x}{x^4} \, dx+\left (1+e^{e^2}\right ) \int \frac {e^x}{x^2} \, dx-\left (2 \left (4+e^{e^2}\right )\right ) \int \frac {e^x}{x^3} \, dx\\ &=-\frac {3 e^{2 x}}{x^5}+\frac {6 e^x \left (3-e^{e^2}\right )}{x^5}-\frac {3 \left (3-e^{e^2}\right )^2}{x^5}+\frac {6 \left (3-e^{e^2}\right )}{x^4}+\frac {3 e^x \left (3-e^{e^2}\right )}{2 x^4}-\frac {3 e^x \left (7-e^{e^2}\right )}{2 x^4}+\frac {2 e^{2 x}}{x^3}-\frac {e^x \left (7-e^{e^2}\right )}{2 x^3}+\frac {2 e^x \left (4+e^{e^2}\right )}{x^3}-\frac {30-6 e^{e^2}-e^{2 e^2}}{x^3}-\frac {e^x \left (1+e^{e^2}\right )}{x^2}+\frac {2 \left (3+e^{e^2}\right )}{x^2}+\frac {e^x \left (4+e^{e^2}\right )}{x^2}-\frac {6 e^x}{x}-\frac {2 e^{2 x}}{x}+\frac {1-4 e^{e^2}}{x}-\frac {e^x \left (1+e^{e^2}\right )}{x}+4 x+2 \text {Ei}(x)-2 \int \frac {e^{2 x}}{x^3} \, dx-2 \int \frac {e^{2 x}}{x^2} \, dx+3 \int \frac {e^{2 x}}{x^4} \, dx+4 \int \frac {e^{2 x}}{x} \, dx-\frac {1}{2} \left (3 \left (3-e^{e^2}\right )\right ) \int \frac {e^x}{x^4} \, dx+\frac {1}{2} \left (7-e^{e^2}\right ) \int \frac {e^x}{x^3} \, dx+\left (1+e^{e^2}\right ) \int \frac {e^x}{x} \, dx-\left (4+e^{e^2}\right ) \int \frac {e^x}{x^2} \, dx\\ &=-\frac {3 e^{2 x}}{x^5}+\frac {6 e^x \left (3-e^{e^2}\right )}{x^5}-\frac {3 \left (3-e^{e^2}\right )^2}{x^5}+\frac {6 \left (3-e^{e^2}\right )}{x^4}+\frac {3 e^x \left (3-e^{e^2}\right )}{2 x^4}-\frac {3 e^x \left (7-e^{e^2}\right )}{2 x^4}+\frac {e^{2 x}}{x^3}+\frac {e^x \left (3-e^{e^2}\right )}{2 x^3}-\frac {e^x \left (7-e^{e^2}\right )}{2 x^3}+\frac {2 e^x \left (4+e^{e^2}\right )}{x^3}-\frac {30-6 e^{e^2}-e^{2 e^2}}{x^3}+\frac {e^{2 x}}{x^2}-\frac {e^x \left (7-e^{e^2}\right )}{4 x^2}-\frac {e^x \left (1+e^{e^2}\right )}{x^2}+\frac {2 \left (3+e^{e^2}\right )}{x^2}+\frac {e^x \left (4+e^{e^2}\right )}{x^2}-\frac {6 e^x}{x}+\frac {1-4 e^{e^2}}{x}-\frac {e^x \left (1+e^{e^2}\right )}{x}+\frac {e^x \left (4+e^{e^2}\right )}{x}+4 x+2 \text {Ei}(x)+\left (1+e^{e^2}\right ) \text {Ei}(x)+4 \text {Ei}(2 x)+2 \int \frac {e^{2 x}}{x^3} \, dx-2 \int \frac {e^{2 x}}{x^2} \, dx-4 \int \frac {e^{2 x}}{x} \, dx-\frac {1}{2} \left (3-e^{e^2}\right ) \int \frac {e^x}{x^3} \, dx+\frac {1}{4} \left (7-e^{e^2}\right ) \int \frac {e^x}{x^2} \, dx-\left (4+e^{e^2}\right ) \int \frac {e^x}{x} \, dx\\ &=-\frac {3 e^{2 x}}{x^5}+\frac {6 e^x \left (3-e^{e^2}\right )}{x^5}-\frac {3 \left (3-e^{e^2}\right )^2}{x^5}+\frac {6 \left (3-e^{e^2}\right )}{x^4}+\frac {3 e^x \left (3-e^{e^2}\right )}{2 x^4}-\frac {3 e^x \left (7-e^{e^2}\right )}{2 x^4}+\frac {e^{2 x}}{x^3}+\frac {e^x \left (3-e^{e^2}\right )}{2 x^3}-\frac {e^x \left (7-e^{e^2}\right )}{2 x^3}+\frac {2 e^x \left (4+e^{e^2}\right )}{x^3}-\frac {30-6 e^{e^2}-e^{2 e^2}}{x^3}+\frac {e^x \left (3-e^{e^2}\right )}{4 x^2}-\frac {e^x \left (7-e^{e^2}\right )}{4 x^2}-\frac {e^x \left (1+e^{e^2}\right )}{x^2}+\frac {2 \left (3+e^{e^2}\right )}{x^2}+\frac {e^x \left (4+e^{e^2}\right )}{x^2}-\frac {6 e^x}{x}+\frac {2 e^{2 x}}{x}+\frac {1-4 e^{e^2}}{x}-\frac {e^x \left (7-e^{e^2}\right )}{4 x}-\frac {e^x \left (1+e^{e^2}\right )}{x}+\frac {e^x \left (4+e^{e^2}\right )}{x}+4 x+2 \text {Ei}(x)+\left (1+e^{e^2}\right ) \text {Ei}(x)-\left (4+e^{e^2}\right ) \text {Ei}(x)+2 \int \frac {e^{2 x}}{x^2} \, dx-4 \int \frac {e^{2 x}}{x} \, dx-\frac {1}{4} \left (3-e^{e^2}\right ) \int \frac {e^x}{x^2} \, dx+\frac {1}{4} \left (7-e^{e^2}\right ) \int \frac {e^x}{x} \, dx\\ &=-\frac {3 e^{2 x}}{x^5}+\frac {6 e^x \left (3-e^{e^2}\right )}{x^5}-\frac {3 \left (3-e^{e^2}\right )^2}{x^5}+\frac {6 \left (3-e^{e^2}\right )}{x^4}+\frac {3 e^x \left (3-e^{e^2}\right )}{2 x^4}-\frac {3 e^x \left (7-e^{e^2}\right )}{2 x^4}+\frac {e^{2 x}}{x^3}+\frac {e^x \left (3-e^{e^2}\right )}{2 x^3}-\frac {e^x \left (7-e^{e^2}\right )}{2 x^3}+\frac {2 e^x \left (4+e^{e^2}\right )}{x^3}-\frac {30-6 e^{e^2}-e^{2 e^2}}{x^3}+\frac {e^x \left (3-e^{e^2}\right )}{4 x^2}-\frac {e^x \left (7-e^{e^2}\right )}{4 x^2}-\frac {e^x \left (1+e^{e^2}\right )}{x^2}+\frac {2 \left (3+e^{e^2}\right )}{x^2}+\frac {e^x \left (4+e^{e^2}\right )}{x^2}-\frac {6 e^x}{x}+\frac {1-4 e^{e^2}}{x}+\frac {e^x \left (3-e^{e^2}\right )}{4 x}-\frac {e^x \left (7-e^{e^2}\right )}{4 x}-\frac {e^x \left (1+e^{e^2}\right )}{x}+\frac {e^x \left (4+e^{e^2}\right )}{x}+4 x+2 \text {Ei}(x)+\frac {1}{4} \left (7-e^{e^2}\right ) \text {Ei}(x)+\left (1+e^{e^2}\right ) \text {Ei}(x)-\left (4+e^{e^2}\right ) \text {Ei}(x)-4 \text {Ei}(2 x)+4 \int \frac {e^{2 x}}{x} \, dx-\frac {1}{4} \left (3-e^{e^2}\right ) \int \frac {e^x}{x} \, dx\\ &=-\frac {3 e^{2 x}}{x^5}+\frac {6 e^x \left (3-e^{e^2}\right )}{x^5}-\frac {3 \left (3-e^{e^2}\right )^2}{x^5}+\frac {6 \left (3-e^{e^2}\right )}{x^4}+\frac {3 e^x \left (3-e^{e^2}\right )}{2 x^4}-\frac {3 e^x \left (7-e^{e^2}\right )}{2 x^4}+\frac {e^{2 x}}{x^3}+\frac {e^x \left (3-e^{e^2}\right )}{2 x^3}-\frac {e^x \left (7-e^{e^2}\right )}{2 x^3}+\frac {2 e^x \left (4+e^{e^2}\right )}{x^3}-\frac {30-6 e^{e^2}-e^{2 e^2}}{x^3}+\frac {e^x \left (3-e^{e^2}\right )}{4 x^2}-\frac {e^x \left (7-e^{e^2}\right )}{4 x^2}-\frac {e^x \left (1+e^{e^2}\right )}{x^2}+\frac {2 \left (3+e^{e^2}\right )}{x^2}+\frac {e^x \left (4+e^{e^2}\right )}{x^2}-\frac {6 e^x}{x}+\frac {1-4 e^{e^2}}{x}+\frac {e^x \left (3-e^{e^2}\right )}{4 x}-\frac {e^x \left (7-e^{e^2}\right )}{4 x}-\frac {e^x \left (1+e^{e^2}\right )}{x}+\frac {e^x \left (4+e^{e^2}\right )}{x}+4 x+2 \text {Ei}(x)-\frac {1}{4} \left (3-e^{e^2}\right ) \text {Ei}(x)+\frac {1}{4} \left (7-e^{e^2}\right ) \text {Ei}(x)+\left (1+e^{e^2}\right ) \text {Ei}(x)-\left (4+e^{e^2}\right ) \text {Ei}(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.09, size = 113, normalized size = 3.32 \begin {gather*} \frac {-27+18 x-30 x^2+6 x^3+x^4+4 x^6+e^{2 e^2} \left (-3+x^2\right )+e^{2 x} \left (-3+x^2\right )+2 e^{e^2+x} \left (-3+x^2\right )+2 e^{e^2} \left (9-3 x+3 x^2+x^3-2 x^4\right )+2 e^x \left (9-3 x+3 x^2+x^3-2 x^4\right )}{x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(135 - 72*x + 90*x^2 - 12*x^3 - x^4 + 4*x^6 + E^(2*E^2)*(15 - 3*x^2) + E^(2*x)*(15 - 6*x - 3*x^2 + 2
*x^3) + E^x*(-90 + 42*x - 24*x^2 + 2*x^3 + 6*x^4 - 4*x^5) + E^E^2*(-90 + 24*x - 18*x^2 - 4*x^3 + 4*x^4 + E^x*(
30 - 6*x - 6*x^2 + 2*x^3)))/x^6,x]

[Out]

(-27 + 18*x - 30*x^2 + 6*x^3 + x^4 + 4*x^6 + E^(2*E^2)*(-3 + x^2) + E^(2*x)*(-3 + x^2) + 2*E^(E^2 + x)*(-3 + x
^2) + 2*E^E^2*(9 - 3*x + 3*x^2 + x^3 - 2*x^4) + 2*E^x*(9 - 3*x + 3*x^2 + x^3 - 2*x^4))/x^5

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fricas [B]  time = 0.74, size = 106, normalized size = 3.12 \begin {gather*} \frac {4 \, x^{6} + x^{4} + 6 \, x^{3} - 30 \, x^{2} + {\left (x^{2} - 3\right )} e^{\left (2 \, x\right )} - 2 \, {\left (2 \, x^{4} - x^{3} - 3 \, x^{2} + 3 \, x - 9\right )} e^{x} + {\left (x^{2} - 3\right )} e^{\left (2 \, e^{2}\right )} - 2 \, {\left (2 \, x^{4} - x^{3} - 3 \, x^{2} - {\left (x^{2} - 3\right )} e^{x} + 3 \, x - 9\right )} e^{\left (e^{2}\right )} + 18 \, x - 27}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+15)*exp(exp(2))^2+((2*x^3-6*x^2-6*x+30)*exp(x)+4*x^4-4*x^3-18*x^2+24*x-90)*exp(exp(2))+(2*x
^3-3*x^2-6*x+15)*exp(x)^2+(-4*x^5+6*x^4+2*x^3-24*x^2+42*x-90)*exp(x)+4*x^6-x^4-12*x^3+90*x^2-72*x+135)/x^6,x,
algorithm="fricas")

[Out]

(4*x^6 + x^4 + 6*x^3 - 30*x^2 + (x^2 - 3)*e^(2*x) - 2*(2*x^4 - x^3 - 3*x^2 + 3*x - 9)*e^x + (x^2 - 3)*e^(2*e^2
) - 2*(2*x^4 - x^3 - 3*x^2 - (x^2 - 3)*e^x + 3*x - 9)*e^(e^2) + 18*x - 27)/x^5

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giac [B]  time = 0.22, size = 139, normalized size = 4.09 \begin {gather*} \frac {4 \, x^{6} - 4 \, x^{4} e^{x} - 4 \, x^{4} e^{\left (e^{2}\right )} + x^{4} + 2 \, x^{3} e^{x} + 2 \, x^{3} e^{\left (e^{2}\right )} + 6 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 2 \, x^{2} e^{\left (x + e^{2}\right )} + 6 \, x^{2} e^{x} + x^{2} e^{\left (2 \, e^{2}\right )} + 6 \, x^{2} e^{\left (e^{2}\right )} - 30 \, x^{2} - 6 \, x e^{x} - 6 \, x e^{\left (e^{2}\right )} + 18 \, x - 3 \, e^{\left (2 \, x\right )} - 6 \, e^{\left (x + e^{2}\right )} + 18 \, e^{x} - 3 \, e^{\left (2 \, e^{2}\right )} + 18 \, e^{\left (e^{2}\right )} - 27}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+15)*exp(exp(2))^2+((2*x^3-6*x^2-6*x+30)*exp(x)+4*x^4-4*x^3-18*x^2+24*x-90)*exp(exp(2))+(2*x
^3-3*x^2-6*x+15)*exp(x)^2+(-4*x^5+6*x^4+2*x^3-24*x^2+42*x-90)*exp(x)+4*x^6-x^4-12*x^3+90*x^2-72*x+135)/x^6,x,
algorithm="giac")

[Out]

(4*x^6 - 4*x^4*e^x - 4*x^4*e^(e^2) + x^4 + 2*x^3*e^x + 2*x^3*e^(e^2) + 6*x^3 + x^2*e^(2*x) + 2*x^2*e^(x + e^2)
+ 6*x^2*e^x + x^2*e^(2*e^2) + 6*x^2*e^(e^2) - 30*x^2 - 6*x*e^x - 6*x*e^(e^2) + 18*x - 3*e^(2*x) - 6*e^(x + e^
2) + 18*e^x - 3*e^(2*e^2) + 18*e^(e^2) - 27)/x^5

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maple [B]  time = 0.07, size = 120, normalized size = 3.53

method	result	size
risch	\(4 x +\frac {\left (-4 \,{\mathrm e}^{{\mathrm e}^{2}}+1\right ) x^{4}+\left (6+2 \,{\mathrm e}^{{\mathrm e}^{2}}\right ) x^{3}+\left ({\mathrm e}^{2 \,{\mathrm e}^{2}}+6 \,{\mathrm e}^{{\mathrm e}^{2}}-30\right ) x^{2}+\left (18-6 \,{\mathrm e}^{{\mathrm e}^{2}}\right ) x -3 \,{\mathrm e}^{2 \,{\mathrm e}^{2}}+18 \,{\mathrm e}^{{\mathrm e}^{2}}-27}{x^{5}}+\frac {\left (x^{2}-3\right ) {\mathrm e}^{2 x}}{x^{5}}+\frac {2 \left (-2 x^{4}+x^{2} {\mathrm e}^{{\mathrm e}^{2}}+x^{3}+3 x^{2}-3 \,{\mathrm e}^{{\mathrm e}^{2}}-3 x +9\right ) {\mathrm e}^{x}}{x^{5}}\)	\(120\)
norman	\(\frac {\left (6+2 \,{\mathrm e}^{{\mathrm e}^{2}}\right ) x^{3}+\left (18-6 \,{\mathrm e}^{{\mathrm e}^{2}}\right ) x +\left (18-6 \,{\mathrm e}^{{\mathrm e}^{2}}\right ) {\mathrm e}^{x}+\left (-4 \,{\mathrm e}^{{\mathrm e}^{2}}+1\right ) x^{4}+\left ({\mathrm e}^{2 \,{\mathrm e}^{2}}+6 \,{\mathrm e}^{{\mathrm e}^{2}}-30\right ) x^{2}+{\mathrm e}^{2 x} x^{2}+\left (6+2 \,{\mathrm e}^{{\mathrm e}^{2}}\right ) x^{2} {\mathrm e}^{x}+4 x^{6}-3 \,{\mathrm e}^{2 x}-6 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x} x^{3}-4 \,{\mathrm e}^{x} x^{4}-3 \,{\mathrm e}^{2 \,{\mathrm e}^{2}}+18 \,{\mathrm e}^{{\mathrm e}^{2}}-27}{x^{5}}\)	\(127\)
default	\(4 x -\frac {27}{x^{5}}+\frac {18}{x^{4}}-\frac {30}{x^{3}}+\frac {6}{x^{2}}+\frac {1}{x}+\frac {18 \,{\mathrm e}^{x}}{x^{5}}-\frac {6 \,{\mathrm e}^{x}}{x^{4}}+\frac {6 \,{\mathrm e}^{x}}{x^{3}}+\frac {2 \,{\mathrm e}^{x}}{x^{2}}-\frac {4 \,{\mathrm e}^{x}}{x}+\frac {18 \,{\mathrm e}^{{\mathrm e}^{2}}}{x^{5}}-\frac {3 \,{\mathrm e}^{2 \,{\mathrm e}^{2}}}{x^{5}}-\frac {6 \,{\mathrm e}^{{\mathrm e}^{2}}}{x^{4}}+\frac {6 \,{\mathrm e}^{{\mathrm e}^{2}}}{x^{3}}+\frac {2 \,{\mathrm e}^{{\mathrm e}^{2}}}{x^{2}}-\frac {3 \,{\mathrm e}^{2 x}}{x^{5}}+\frac {{\mathrm e}^{2 x}}{x^{3}}-\frac {4 \,{\mathrm e}^{{\mathrm e}^{2}}}{x}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{2}}}{x^{3}}+30 \,{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x}}{5 x^{5}}-\frac {{\mathrm e}^{x}}{20 x^{4}}-\frac {{\mathrm e}^{x}}{60 x^{3}}-\frac {{\mathrm e}^{x}}{120 x^{2}}-\frac {{\mathrm e}^{x}}{120 x}-\frac {\expIntegralEi \left (1, -x \right )}{120}\right )-6 \,{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x}}{4 x^{4}}-\frac {{\mathrm e}^{x}}{12 x^{3}}-\frac {{\mathrm e}^{x}}{24 x^{2}}-\frac {{\mathrm e}^{x}}{24 x}-\frac {\expIntegralEi \left (1, -x \right )}{24}\right )-6 \,{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x}}{3 x^{3}}-\frac {{\mathrm e}^{x}}{6 x^{2}}-\frac {{\mathrm e}^{x}}{6 x}-\frac {\expIntegralEi \left (1, -x \right )}{6}\right )+2 \,{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x}}{2 x^{2}}-\frac {{\mathrm e}^{x}}{2 x}-\frac {\expIntegralEi \left (1, -x \right )}{2}\right )\)	\(289\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x^2+15)*exp(exp(2))^2+((2*x^3-6*x^2-6*x+30)*exp(x)+4*x^4-4*x^3-18*x^2+24*x-90)*exp(exp(2))+(2*x^3-3*x
^2-6*x+15)*exp(x)^2+(-4*x^5+6*x^4+2*x^3-24*x^2+42*x-90)*exp(x)+4*x^6-x^4-12*x^3+90*x^2-72*x+135)/x^6,x,method=
_RETURNVERBOSE)

[Out]

4*x+((-4*exp(exp(2))+1)*x^4+(6+2*exp(exp(2)))*x^3+(exp(2*exp(2))+6*exp(exp(2))-30)*x^2+(18-6*exp(exp(2)))*x-3*
exp(2*exp(2))+18*exp(exp(2))-27)/x^5+(x^2-3)/x^5*exp(2*x)+2*(-2*x^4+x^2*exp(exp(2))+x^3+3*x^2-3*exp(exp(2))-3*
x+9)/x^5*exp(x)

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maxima [C]  time = 0.43, size = 193, normalized size = 5.68 \begin {gather*} -2 \, e^{\left (e^{2}\right )} \Gamma \left (-2, -x\right ) - 6 \, e^{\left (e^{2}\right )} \Gamma \left (-3, -x\right ) + 6 \, e^{\left (e^{2}\right )} \Gamma \left (-4, -x\right ) + 30 \, e^{\left (e^{2}\right )} \Gamma \left (-5, -x\right ) + 4 \, x - \frac {4 \, e^{\left (e^{2}\right )}}{x} + \frac {1}{x} + \frac {2 \, e^{\left (e^{2}\right )}}{x^{2}} + \frac {6}{x^{2}} + \frac {e^{\left (2 \, e^{2}\right )}}{x^{3}} + \frac {6 \, e^{\left (e^{2}\right )}}{x^{3}} - \frac {30}{x^{3}} - \frac {6 \, e^{\left (e^{2}\right )}}{x^{4}} + \frac {18}{x^{4}} - \frac {3 \, e^{\left (2 \, e^{2}\right )}}{x^{5}} + \frac {18 \, e^{\left (e^{2}\right )}}{x^{5}} - \frac {27}{x^{5}} - 4 \, {\rm Ei}\relax (x) + 6 \, \Gamma \left (-1, -x\right ) - 2 \, \Gamma \left (-2, -x\right ) - 8 \, \Gamma \left (-2, -2 \, x\right ) - 24 \, \Gamma \left (-3, -x\right ) - 24 \, \Gamma \left (-3, -2 \, x\right ) - 42 \, \Gamma \left (-4, -x\right ) + 96 \, \Gamma \left (-4, -2 \, x\right ) - 90 \, \Gamma \left (-5, -x\right ) + 480 \, \Gamma \left (-5, -2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x^2+15)*exp(exp(2))^2+((2*x^3-6*x^2-6*x+30)*exp(x)+4*x^4-4*x^3-18*x^2+24*x-90)*exp(exp(2))+(2*x
^3-3*x^2-6*x+15)*exp(x)^2+(-4*x^5+6*x^4+2*x^3-24*x^2+42*x-90)*exp(x)+4*x^6-x^4-12*x^3+90*x^2-72*x+135)/x^6,x,
algorithm="maxima")

[Out]

-2*e^(e^2)*gamma(-2, -x) - 6*e^(e^2)*gamma(-3, -x) + 6*e^(e^2)*gamma(-4, -x) + 30*e^(e^2)*gamma(-5, -x) + 4*x
- 4*e^(e^2)/x + 1/x + 2*e^(e^2)/x^2 + 6/x^2 + e^(2*e^2)/x^3 + 6*e^(e^2)/x^3 - 30/x^3 - 6*e^(e^2)/x^4 + 18/x^4
- 3*e^(2*e^2)/x^5 + 18*e^(e^2)/x^5 - 27/x^5 - 4*Ei(x) + 6*gamma(-1, -x) - 2*gamma(-2, -x) - 8*gamma(-2, -2*x)
- 24*gamma(-3, -x) - 24*gamma(-3, -2*x) - 42*gamma(-4, -x) + 96*gamma(-4, -2*x) - 90*gamma(-5, -x) + 480*gamma
(-5, -2*x)

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mupad [B]  time = 4.37, size = 96, normalized size = 2.82 \begin {gather*} 4\,x+\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^2}+{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{x+{\mathrm {e}}^2}+6\,{\mathrm {e}}^{{\mathrm {e}}^2}+6\,{\mathrm {e}}^x-30}{x^3}-\frac {3\,{\left ({\mathrm {e}}^{{\mathrm {e}}^2}+{\mathrm {e}}^x-3\right )}^2}{x^5}-\frac {4\,{\mathrm {e}}^{{\mathrm {e}}^2}+4\,{\mathrm {e}}^x-1}{x}+\frac {2\,{\mathrm {e}}^{{\mathrm {e}}^2}+2\,{\mathrm {e}}^x+6}{x^2}-\frac {6\,{\mathrm {e}}^{{\mathrm {e}}^2}+6\,{\mathrm {e}}^x-18}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(72*x + exp(2*exp(2))*(3*x^2 - 15) + exp(2*x)*(6*x + 3*x^2 - 2*x^3 - 15) - exp(x)*(42*x - 24*x^2 + 2*x^3
+ 6*x^4 - 4*x^5 - 90) - 90*x^2 + 12*x^3 + x^4 - 4*x^6 + exp(exp(2))*(18*x^2 - 24*x + 4*x^3 - 4*x^4 + exp(x)*(6
*x + 6*x^2 - 2*x^3 - 30) + 90) - 135)/x^6,x)

[Out]

4*x + (exp(2*exp(2)) + exp(2*x) + 2*exp(x + exp(2)) + 6*exp(exp(2)) + 6*exp(x) - 30)/x^3 - (3*(exp(exp(2)) + e
xp(x) - 3)^2)/x^5 - (4*exp(exp(2)) + 4*exp(x) - 1)/x + (2*exp(exp(2)) + 2*exp(x) + 6)/x^2 - (6*exp(exp(2)) + 6
*exp(x) - 18)/x^4

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sympy [B]  time = 2.48, size = 141, normalized size = 4.15 \begin {gather*} 4 x + \frac {x^{4} \left (1 - 4 e^{e^{2}}\right ) + x^{3} \left (6 + 2 e^{e^{2}}\right ) + x^{2} \left (-30 + 6 e^{e^{2}} + e^{2 e^{2}}\right ) + x \left (18 - 6 e^{e^{2}}\right ) - 3 e^{2 e^{2}} - 27 + 18 e^{e^{2}}}{x^{5}} + \frac {\left (x^{7} - 3 x^{5}\right ) e^{2 x} + \left (- 4 x^{9} + 2 x^{8} + 6 x^{7} + 2 x^{7} e^{e^{2}} - 6 x^{6} - 6 x^{5} e^{e^{2}} + 18 x^{5}\right ) e^{x}}{x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x**2+15)*exp(exp(2))**2+((2*x**3-6*x**2-6*x+30)*exp(x)+4*x**4-4*x**3-18*x**2+24*x-90)*exp(exp(2
))+(2*x**3-3*x**2-6*x+15)*exp(x)**2+(-4*x**5+6*x**4+2*x**3-24*x**2+42*x-90)*exp(x)+4*x**6-x**4-12*x**3+90*x**2
-72*x+135)/x**6,x)

[Out]

4*x + (x**4*(1 - 4*exp(exp(2))) + x**3*(6 + 2*exp(exp(2))) + x**2*(-30 + 6*exp(exp(2)) + exp(2*exp(2))) + x*(1
8 - 6*exp(exp(2))) - 3*exp(2*exp(2)) - 27 + 18*exp(exp(2)))/x**5 + ((x**7 - 3*x**5)*exp(2*x) + (-4*x**9 + 2*x*
*8 + 6*x**7 + 2*x**7*exp(exp(2)) - 6*x**6 - 6*x**5*exp(exp(2)) + 18*x**5)*exp(x))/x**10

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