3.50.95 \(\int \frac {1+54 x^4+144 x^5+90 x^6-567 x^8-1296 x^9-729 x^{10}+e^{-4+2 x} (1-2 x+54 x^4+36 x^5-567 x^8-162 x^9)+e^{-2+x} (2-2 x-2 x^2+108 x^4+180 x^5+36 x^6-1134 x^8-1458 x^9-162 x^{10})}{x^2} \, dx\)

Optimal. Leaf size=26 \[ x-\frac {\left (1+e^{-2+x}+x\right )^2 \left (1-9 x^4\right )^2}{x} \]

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Rubi [B]  time = 1.06, antiderivative size = 129, normalized size of antiderivative = 4.96, number of steps used = 73, number of rules used = 6, integrand size = 115, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {14, 2199, 2194, 2177, 2178, 2176} \begin {gather*} -81 x^9-162 e^{x-2} x^8-162 x^8-162 e^{x-2} x^7-81 e^{2 x-4} x^7-81 x^7+18 x^5+36 e^{x-2} x^4+36 x^4+36 e^{x-2} x^3+18 e^{2 x-4} x^3+18 x^3-2 e^{x-2}-\frac {2 e^{x-2}}{x}-\frac {e^{2 x-4}}{x}-\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 54*x^4 + 144*x^5 + 90*x^6 - 567*x^8 - 1296*x^9 - 729*x^10 + E^(-4 + 2*x)*(1 - 2*x + 54*x^4 + 36*x^5 -
567*x^8 - 162*x^9) + E^(-2 + x)*(2 - 2*x - 2*x^2 + 108*x^4 + 180*x^5 + 36*x^6 - 1134*x^8 - 1458*x^9 - 162*x^1
0))/x^2,x]

[Out]

-2*E^(-2 + x) - x^(-1) - (2*E^(-2 + x))/x - E^(-4 + 2*x)/x + 18*x^3 + 36*E^(-2 + x)*x^3 + 18*E^(-4 + 2*x)*x^3
+ 36*x^4 + 36*E^(-2 + x)*x^4 + 18*x^5 - 81*x^7 - 162*E^(-2 + x)*x^7 - 81*E^(-4 + 2*x)*x^7 - 162*x^8 - 162*E^(-
2 + x)*x^8 - 81*x^9

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 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), 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] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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 {2 e^{-2+x} \left (-1+3 x^2\right ) \left (1+3 x^2\right ) \left (1-x-x^2+63 x^4+81 x^5+9 x^6\right )}{x^2}-\frac {e^{-4+2 x} \left (-1+2 x-54 x^4-36 x^5+567 x^8+162 x^9\right )}{x^2}+\frac {1+54 x^4+144 x^5+90 x^6-567 x^8-1296 x^9-729 x^{10}}{x^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{-2+x} \left (-1+3 x^2\right ) \left (1+3 x^2\right ) \left (1-x-x^2+63 x^4+81 x^5+9 x^6\right )}{x^2} \, dx\right )-\int \frac {e^{-4+2 x} \left (-1+2 x-54 x^4-36 x^5+567 x^8+162 x^9\right )}{x^2} \, dx+\int \frac {1+54 x^4+144 x^5+90 x^6-567 x^8-1296 x^9-729 x^{10}}{x^2} \, dx\\ &=-\left (2 \int \left (e^{-2+x}-\frac {e^{-2+x}}{x^2}+\frac {e^{-2+x}}{x}-54 e^{-2+x} x^2-90 e^{-2+x} x^3-18 e^{-2+x} x^4+567 e^{-2+x} x^6+729 e^{-2+x} x^7+81 e^{-2+x} x^8\right ) \, dx\right )-\int \left (-\frac {e^{-4+2 x}}{x^2}+\frac {2 e^{-4+2 x}}{x}-54 e^{-4+2 x} x^2-36 e^{-4+2 x} x^3+567 e^{-4+2 x} x^6+162 e^{-4+2 x} x^7\right ) \, dx+\int \left (\frac {1}{x^2}+54 x^2+144 x^3+90 x^4-567 x^6-1296 x^7-729 x^8\right ) \, dx\\ &=-\frac {1}{x}+18 x^3+36 x^4+18 x^5-81 x^7-162 x^8-81 x^9-2 \int e^{-2+x} \, dx+2 \int \frac {e^{-2+x}}{x^2} \, dx-2 \int \frac {e^{-2+x}}{x} \, dx-2 \int \frac {e^{-4+2 x}}{x} \, dx+36 \int e^{-4+2 x} x^3 \, dx+36 \int e^{-2+x} x^4 \, dx+54 \int e^{-4+2 x} x^2 \, dx+108 \int e^{-2+x} x^2 \, dx-162 \int e^{-4+2 x} x^7 \, dx-162 \int e^{-2+x} x^8 \, dx+180 \int e^{-2+x} x^3 \, dx-567 \int e^{-4+2 x} x^6 \, dx-1134 \int e^{-2+x} x^6 \, dx-1458 \int e^{-2+x} x^7 \, dx+\int \frac {e^{-4+2 x}}{x^2} \, dx\\ &=-2 e^{-2+x}-\frac {1}{x}-\frac {2 e^{-2+x}}{x}-\frac {e^{-4+2 x}}{x}+108 e^{-2+x} x^2+27 e^{-4+2 x} x^2+18 x^3+180 e^{-2+x} x^3+18 e^{-4+2 x} x^3+36 x^4+36 e^{-2+x} x^4+18 x^5-1134 e^{-2+x} x^6-\frac {567}{2} e^{-4+2 x} x^6-81 x^7-1458 e^{-2+x} x^7-81 e^{-4+2 x} x^7-162 x^8-162 e^{-2+x} x^8-81 x^9-\frac {2 \text {Ei}(x)}{e^2}-\frac {2 \text {Ei}(2 x)}{e^4}+2 \int \frac {e^{-2+x}}{x} \, dx+2 \int \frac {e^{-4+2 x}}{x} \, dx-54 \int e^{-4+2 x} x \, dx-54 \int e^{-4+2 x} x^2 \, dx-144 \int e^{-2+x} x^3 \, dx-216 \int e^{-2+x} x \, dx-540 \int e^{-2+x} x^2 \, dx+567 \int e^{-4+2 x} x^6 \, dx+1296 \int e^{-2+x} x^7 \, dx+1701 \int e^{-4+2 x} x^5 \, dx+6804 \int e^{-2+x} x^5 \, dx+10206 \int e^{-2+x} x^6 \, dx\\ &=-2 e^{-2+x}-\frac {1}{x}-\frac {2 e^{-2+x}}{x}-\frac {e^{-4+2 x}}{x}-216 e^{-2+x} x-27 e^{-4+2 x} x-432 e^{-2+x} x^2+18 x^3+36 e^{-2+x} x^3+18 e^{-4+2 x} x^3+36 x^4+36 e^{-2+x} x^4+18 x^5+6804 e^{-2+x} x^5+\frac {1701}{2} e^{-4+2 x} x^5+9072 e^{-2+x} x^6-81 x^7-162 e^{-2+x} x^7-81 e^{-4+2 x} x^7-162 x^8-162 e^{-2+x} x^8-81 x^9+27 \int e^{-4+2 x} \, dx+54 \int e^{-4+2 x} x \, dx+216 \int e^{-2+x} \, dx+432 \int e^{-2+x} x^2 \, dx+1080 \int e^{-2+x} x \, dx-1701 \int e^{-4+2 x} x^5 \, dx-\frac {8505}{2} \int e^{-4+2 x} x^4 \, dx-9072 \int e^{-2+x} x^6 \, dx-34020 \int e^{-2+x} x^4 \, dx-61236 \int e^{-2+x} x^5 \, dx\\ &=214 e^{-2+x}+\frac {27}{2} e^{-4+2 x}-\frac {1}{x}-\frac {2 e^{-2+x}}{x}-\frac {e^{-4+2 x}}{x}+864 e^{-2+x} x+18 x^3+36 e^{-2+x} x^3+18 e^{-4+2 x} x^3+36 x^4-33984 e^{-2+x} x^4-\frac {8505}{4} e^{-4+2 x} x^4+18 x^5-54432 e^{-2+x} x^5-81 x^7-162 e^{-2+x} x^7-81 e^{-4+2 x} x^7-162 x^8-162 e^{-2+x} x^8-81 x^9-27 \int e^{-4+2 x} \, dx-864 \int e^{-2+x} x \, dx-1080 \int e^{-2+x} \, dx+\frac {8505}{2} \int e^{-4+2 x} x^4 \, dx+8505 \int e^{-4+2 x} x^3 \, dx+54432 \int e^{-2+x} x^5 \, dx+136080 \int e^{-2+x} x^3 \, dx+306180 \int e^{-2+x} x^4 \, dx\\ &=-866 e^{-2+x}-\frac {1}{x}-\frac {2 e^{-2+x}}{x}-\frac {e^{-4+2 x}}{x}+18 x^3+136116 e^{-2+x} x^3+\frac {8541}{2} e^{-4+2 x} x^3+36 x^4+272196 e^{-2+x} x^4+18 x^5-81 x^7-162 e^{-2+x} x^7-81 e^{-4+2 x} x^7-162 x^8-162 e^{-2+x} x^8-81 x^9+864 \int e^{-2+x} \, dx-8505 \int e^{-4+2 x} x^3 \, dx-\frac {25515}{2} \int e^{-4+2 x} x^2 \, dx-272160 \int e^{-2+x} x^4 \, dx-408240 \int e^{-2+x} x^2 \, dx-1224720 \int e^{-2+x} x^3 \, dx\\ &=-2 e^{-2+x}-\frac {1}{x}-\frac {2 e^{-2+x}}{x}-\frac {e^{-4+2 x}}{x}-408240 e^{-2+x} x^2-\frac {25515}{4} e^{-4+2 x} x^2+18 x^3-1088604 e^{-2+x} x^3+18 e^{-4+2 x} x^3+36 x^4+36 e^{-2+x} x^4+18 x^5-81 x^7-162 e^{-2+x} x^7-81 e^{-4+2 x} x^7-162 x^8-162 e^{-2+x} x^8-81 x^9+\frac {25515}{2} \int e^{-4+2 x} x \, dx+\frac {25515}{2} \int e^{-4+2 x} x^2 \, dx+816480 \int e^{-2+x} x \, dx+1088640 \int e^{-2+x} x^3 \, dx+3674160 \int e^{-2+x} x^2 \, dx\\ &=-2 e^{-2+x}-\frac {1}{x}-\frac {2 e^{-2+x}}{x}-\frac {e^{-4+2 x}}{x}+816480 e^{-2+x} x+\frac {25515}{4} e^{-4+2 x} x+3265920 e^{-2+x} x^2+18 x^3+36 e^{-2+x} x^3+18 e^{-4+2 x} x^3+36 x^4+36 e^{-2+x} x^4+18 x^5-81 x^7-162 e^{-2+x} x^7-81 e^{-4+2 x} x^7-162 x^8-162 e^{-2+x} x^8-81 x^9-\frac {25515}{4} \int e^{-4+2 x} \, dx-\frac {25515}{2} \int e^{-4+2 x} x \, dx-816480 \int e^{-2+x} \, dx-3265920 \int e^{-2+x} x^2 \, dx-7348320 \int e^{-2+x} x \, dx\\ &=-816482 e^{-2+x}-\frac {25515}{8} e^{-4+2 x}-\frac {1}{x}-\frac {2 e^{-2+x}}{x}-\frac {e^{-4+2 x}}{x}-6531840 e^{-2+x} x+18 x^3+36 e^{-2+x} x^3+18 e^{-4+2 x} x^3+36 x^4+36 e^{-2+x} x^4+18 x^5-81 x^7-162 e^{-2+x} x^7-81 e^{-4+2 x} x^7-162 x^8-162 e^{-2+x} x^8-81 x^9+\frac {25515}{4} \int e^{-4+2 x} \, dx+6531840 \int e^{-2+x} x \, dx+7348320 \int e^{-2+x} \, dx\\ &=6531838 e^{-2+x}-\frac {1}{x}-\frac {2 e^{-2+x}}{x}-\frac {e^{-4+2 x}}{x}+18 x^3+36 e^{-2+x} x^3+18 e^{-4+2 x} x^3+36 x^4+36 e^{-2+x} x^4+18 x^5-81 x^7-162 e^{-2+x} x^7-81 e^{-4+2 x} x^7-162 x^8-162 e^{-2+x} x^8-81 x^9-6531840 \int e^{-2+x} \, dx\\ &=-2 e^{-2+x}-\frac {1}{x}-\frac {2 e^{-2+x}}{x}-\frac {e^{-4+2 x}}{x}+18 x^3+36 e^{-2+x} x^3+18 e^{-4+2 x} x^3+36 x^4+36 e^{-2+x} x^4+18 x^5-81 x^7-162 e^{-2+x} x^7-81 e^{-4+2 x} x^7-162 x^8-162 e^{-2+x} x^8-81 x^9\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.07, size = 79, normalized size = 3.04 \begin {gather*} \frac {-e^{2 x} \left (1-9 x^4\right )^2-2 e^{2+x} (1+x) \left (1-9 x^4\right )^2+e^4 \left (-1+18 x^4+36 x^5+18 x^6-81 x^8-162 x^9-81 x^{10}\right )}{e^4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 54*x^4 + 144*x^5 + 90*x^6 - 567*x^8 - 1296*x^9 - 729*x^10 + E^(-4 + 2*x)*(1 - 2*x + 54*x^4 + 36
*x^5 - 567*x^8 - 162*x^9) + E^(-2 + x)*(2 - 2*x - 2*x^2 + 108*x^4 + 180*x^5 + 36*x^6 - 1134*x^8 - 1458*x^9 - 1
62*x^10))/x^2,x]

[Out]

(-(E^(2*x)*(1 - 9*x^4)^2) - 2*E^(2 + x)*(1 + x)*(1 - 9*x^4)^2 + E^4*(-1 + 18*x^4 + 36*x^5 + 18*x^6 - 81*x^8 -
162*x^9 - 81*x^10))/(E^4*x)

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fricas [B]  time = 1.02, size = 85, normalized size = 3.27 \begin {gather*} -\frac {81 \, x^{10} + 162 \, x^{9} + 81 \, x^{8} - 18 \, x^{6} - 36 \, x^{5} - 18 \, x^{4} + {\left (81 \, x^{8} - 18 \, x^{4} + 1\right )} e^{\left (2 \, x - 4\right )} + 2 \, {\left (81 \, x^{9} + 81 \, x^{8} - 18 \, x^{5} - 18 \, x^{4} + x + 1\right )} e^{\left (x - 2\right )} + 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-162*x^9-567*x^8+36*x^5+54*x^4-2*x+1)*exp(x-2)^2+(-162*x^10-1458*x^9-1134*x^8+36*x^6+180*x^5+108*x
^4-2*x^2-2*x+2)*exp(x-2)-729*x^10-1296*x^9-567*x^8+90*x^6+144*x^5+54*x^4+1)/x^2,x, algorithm="fricas")

[Out]

-(81*x^10 + 162*x^9 + 81*x^8 - 18*x^6 - 36*x^5 - 18*x^4 + (81*x^8 - 18*x^4 + 1)*e^(2*x - 4) + 2*(81*x^9 + 81*x
^8 - 18*x^5 - 18*x^4 + x + 1)*e^(x - 2) + 1)/x

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giac [B]  time = 0.23, size = 129, normalized size = 4.96 \begin {gather*} -\frac {{\left (81 \, x^{10} e^{6} + 162 \, x^{9} e^{6} + 162 \, x^{9} e^{\left (x + 4\right )} + 81 \, x^{8} e^{6} + 81 \, x^{8} e^{\left (2 \, x + 2\right )} + 162 \, x^{8} e^{\left (x + 4\right )} - 18 \, x^{6} e^{6} - 36 \, x^{5} e^{6} - 36 \, x^{5} e^{\left (x + 4\right )} - 18 \, x^{4} e^{6} - 18 \, x^{4} e^{\left (2 \, x + 2\right )} - 36 \, x^{4} e^{\left (x + 4\right )} + 2 \, x e^{\left (x + 4\right )} + e^{6} + e^{\left (2 \, x + 2\right )} + 2 \, e^{\left (x + 4\right )}\right )} e^{\left (-6\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-162*x^9-567*x^8+36*x^5+54*x^4-2*x+1)*exp(x-2)^2+(-162*x^10-1458*x^9-1134*x^8+36*x^6+180*x^5+108*x
^4-2*x^2-2*x+2)*exp(x-2)-729*x^10-1296*x^9-567*x^8+90*x^6+144*x^5+54*x^4+1)/x^2,x, algorithm="giac")

[Out]

-(81*x^10*e^6 + 162*x^9*e^6 + 162*x^9*e^(x + 4) + 81*x^8*e^6 + 81*x^8*e^(2*x + 2) + 162*x^8*e^(x + 4) - 18*x^6
*e^6 - 36*x^5*e^6 - 36*x^5*e^(x + 4) - 18*x^4*e^6 - 18*x^4*e^(2*x + 2) - 36*x^4*e^(x + 4) + 2*x*e^(x + 4) + e^
6 + e^(2*x + 2) + 2*e^(x + 4))*e^(-6)/x

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maple [B]  time = 0.17, size = 92, normalized size = 3.54

method	result	size
risch	\(-81 x^{9}-162 x^{8}-81 x^{7}+18 x^{5}+36 x^{4}+18 x^{3}-\frac {1}{x}-\frac {\left (81 x^{8}-18 x^{4}+1\right ) {\mathrm e}^{2 x -4}}{x}-\frac {2 \left (81 x^{9}+81 x^{8}-18 x^{5}-18 x^{4}+x +1\right ) {\mathrm e}^{x -2}}{x}\)	\(92\)
norman	\(\frac {-1+18 x^{4}+36 x^{5}+18 x^{6}-81 x^{8}-162 x^{9}-81 x^{10}-{\mathrm e}^{2 x -4}-2 x \,{\mathrm e}^{x -2}+36 x^{4} {\mathrm e}^{x -2}+18 x^{4} {\mathrm e}^{2 x -4}+36 \,{\mathrm e}^{x -2} x^{5}-162 \,{\mathrm e}^{x -2} x^{8}-162 \,{\mathrm e}^{x -2} x^{9}-81 \,{\mathrm e}^{2 x -4} x^{8}-2 \,{\mathrm e}^{x -2}}{x}\)	\(116\)
derivativedivides	\(771984-385992 x -\frac {2 \,{\mathrm e}^{x -2}}{x}-715572 \left (x -2\right )^{2}-61346 \,{\mathrm e}^{x -2}-\frac {1}{x}-81 \,{\mathrm e}^{2 x -4} \left (x -2\right )^{7}-1134 \,{\mathrm e}^{2 x -4} \left (x -2\right )^{6}-6804 \,{\mathrm e}^{2 x -4} \left (x -2\right )^{5}-22680 \,{\mathrm e}^{2 x -4} \left (x -2\right )^{4}-45342 \,{\mathrm e}^{2 x -4} \left (x -2\right )^{3}-54324 \,{\mathrm e}^{2 x -4} \left (x -2\right )^{2}-36072 \,{\mathrm e}^{2 x -4} \left (x -2\right )-\frac {{\mathrm e}^{2 x -4}}{x}-10224 \,{\mathrm e}^{2 x -4}-162 \,{\mathrm e}^{x -2} \left (x -2\right )^{8}-2754 \,{\mathrm e}^{x -2} \left (x -2\right )^{7}-20412 \,{\mathrm e}^{x -2} \left (x -2\right )^{6}-86184 \,{\mathrm e}^{x -2} \left (x -2\right )^{5}-226764 \,{\mathrm e}^{x -2} \left (x -2\right )^{4}-380700 \,{\mathrm e}^{x -2} \left (x -2\right )^{3}-398088 \,{\mathrm e}^{x -2} \left (x -2\right )^{2}-236880 \,{\mathrm e}^{x -2} \left (x -2\right )-81 \left (x -2\right )^{9}-1620 \left (x -2\right )^{8}-14337 \left (x -2\right )^{7}-73710 \left (x -2\right )^{6}-242658 \left (x -2\right )^{5}-530496 \left (x -2\right )^{4}-770094 \left (x -2\right )^{3}\)	\(276\)
default	\(771984-385992 x -\frac {2 \,{\mathrm e}^{x -2}}{x}-715572 \left (x -2\right )^{2}-61346 \,{\mathrm e}^{x -2}-\frac {1}{x}-81 \,{\mathrm e}^{2 x -4} \left (x -2\right )^{7}-1134 \,{\mathrm e}^{2 x -4} \left (x -2\right )^{6}-6804 \,{\mathrm e}^{2 x -4} \left (x -2\right )^{5}-22680 \,{\mathrm e}^{2 x -4} \left (x -2\right )^{4}-45342 \,{\mathrm e}^{2 x -4} \left (x -2\right )^{3}-54324 \,{\mathrm e}^{2 x -4} \left (x -2\right )^{2}-36072 \,{\mathrm e}^{2 x -4} \left (x -2\right )-\frac {{\mathrm e}^{2 x -4}}{x}-10224 \,{\mathrm e}^{2 x -4}-162 \,{\mathrm e}^{x -2} \left (x -2\right )^{8}-2754 \,{\mathrm e}^{x -2} \left (x -2\right )^{7}-20412 \,{\mathrm e}^{x -2} \left (x -2\right )^{6}-86184 \,{\mathrm e}^{x -2} \left (x -2\right )^{5}-226764 \,{\mathrm e}^{x -2} \left (x -2\right )^{4}-380700 \,{\mathrm e}^{x -2} \left (x -2\right )^{3}-398088 \,{\mathrm e}^{x -2} \left (x -2\right )^{2}-236880 \,{\mathrm e}^{x -2} \left (x -2\right )-81 \left (x -2\right )^{9}-1620 \left (x -2\right )^{8}-14337 \left (x -2\right )^{7}-73710 \left (x -2\right )^{6}-242658 \left (x -2\right )^{5}-530496 \left (x -2\right )^{4}-770094 \left (x -2\right )^{3}\)	\(276\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-162*x^9-567*x^8+36*x^5+54*x^4-2*x+1)*exp(x-2)^2+(-162*x^10-1458*x^9-1134*x^8+36*x^6+180*x^5+108*x^4-2*x
^2-2*x+2)*exp(x-2)-729*x^10-1296*x^9-567*x^8+90*x^6+144*x^5+54*x^4+1)/x^2,x,method=_RETURNVERBOSE)

[Out]

-81*x^9-162*x^8-81*x^7+18*x^5+36*x^4+18*x^3-1/x-(81*x^8-18*x^4+1)/x*exp(2*x-4)-2*(81*x^9+81*x^8-18*x^5-18*x^4+
x+1)/x*exp(x-2)

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maxima [C]  time = 0.40, size = 370, normalized size = 14.23 \begin {gather*} -81 \, x^{9} - 162 \, x^{8} - 81 \, x^{7} + 18 \, x^{5} + 36 \, x^{4} + 18 \, x^{3} - 2 \, {\rm Ei}\relax (x) e^{\left (-2\right )} - 2 \, {\rm Ei}\left (2 \, x\right ) e^{\left (-4\right )} - \frac {81}{8} \, {\left (8 \, x^{7} - 28 \, x^{6} + 84 \, x^{5} - 210 \, x^{4} + 420 \, x^{3} - 630 \, x^{2} + 630 \, x - 315\right )} e^{\left (2 \, x - 4\right )} - \frac {567}{8} \, {\left (4 \, x^{6} - 12 \, x^{5} + 30 \, x^{4} - 60 \, x^{3} + 90 \, x^{2} - 90 \, x + 45\right )} e^{\left (2 \, x - 4\right )} + \frac {9}{2} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x - 4\right )} + \frac {27}{2} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x - 4\right )} - 162 \, {\left (x^{8} - 8 \, x^{7} + 56 \, x^{6} - 336 \, x^{5} + 1680 \, x^{4} - 6720 \, x^{3} + 20160 \, x^{2} - 40320 \, x + 40320\right )} e^{\left (x - 2\right )} - 1458 \, {\left (x^{7} - 7 \, x^{6} + 42 \, x^{5} - 210 \, x^{4} + 840 \, x^{3} - 2520 \, x^{2} + 5040 \, x - 5040\right )} e^{\left (x - 2\right )} - 1134 \, {\left (x^{6} - 6 \, x^{5} + 30 \, x^{4} - 120 \, x^{3} + 360 \, x^{2} - 720 \, x + 720\right )} e^{\left (x - 2\right )} + 36 \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{\left (x - 2\right )} + 180 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{\left (x - 2\right )} + 108 \, {\left (x^{2} - 2 \, x + 2\right )} e^{\left (x - 2\right )} + 2 \, e^{\left (-2\right )} \Gamma \left (-1, -x\right ) + 2 \, e^{\left (-4\right )} \Gamma \left (-1, -2 \, x\right ) - \frac {1}{x} - 2 \, e^{\left (x - 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-162*x^9-567*x^8+36*x^5+54*x^4-2*x+1)*exp(x-2)^2+(-162*x^10-1458*x^9-1134*x^8+36*x^6+180*x^5+108*x
^4-2*x^2-2*x+2)*exp(x-2)-729*x^10-1296*x^9-567*x^8+90*x^6+144*x^5+54*x^4+1)/x^2,x, algorithm="maxima")

[Out]

-81*x^9 - 162*x^8 - 81*x^7 + 18*x^5 + 36*x^4 + 18*x^3 - 2*Ei(x)*e^(-2) - 2*Ei(2*x)*e^(-4) - 81/8*(8*x^7 - 28*x
^6 + 84*x^5 - 210*x^4 + 420*x^3 - 630*x^2 + 630*x - 315)*e^(2*x - 4) - 567/8*(4*x^6 - 12*x^5 + 30*x^4 - 60*x^3
+ 90*x^2 - 90*x + 45)*e^(2*x - 4) + 9/2*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x - 4) + 27/2*(2*x^2 - 2*x + 1)*e^(2*x
- 4) - 162*(x^8 - 8*x^7 + 56*x^6 - 336*x^5 + 1680*x^4 - 6720*x^3 + 20160*x^2 - 40320*x + 40320)*e^(x - 2) - 1
458*(x^7 - 7*x^6 + 42*x^5 - 210*x^4 + 840*x^3 - 2520*x^2 + 5040*x - 5040)*e^(x - 2) - 1134*(x^6 - 6*x^5 + 30*x
^4 - 120*x^3 + 360*x^2 - 720*x + 720)*e^(x - 2) + 36*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^(x - 2) + 180*(x^3 -
3*x^2 + 6*x - 6)*e^(x - 2) + 108*(x^2 - 2*x + 2)*e^(x - 2) + 2*e^(-2)*gamma(-1, -x) + 2*e^(-4)*gamma(-1, -2*x
) - 1/x - 2*e^(x - 2)

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mupad [B]  time = 0.13, size = 102, normalized size = 3.92 \begin {gather*} x^3\,\left (36\,{\mathrm {e}}^{x-2}+18\,{\mathrm {e}}^{2\,x-4}+18\right )-2\,{\mathrm {e}}^{x-2}-x^7\,\left (162\,{\mathrm {e}}^{x-2}+81\,{\mathrm {e}}^{2\,x-4}+81\right )+x^4\,\left (36\,{\mathrm {e}}^{x-2}+36\right )-x^8\,\left (162\,{\mathrm {e}}^{x-2}+162\right )-\frac {2\,{\mathrm {e}}^{x-2}+{\mathrm {e}}^{2\,x-4}+1}{x}+18\,x^5-81\,x^9 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x - 4)*(2*x - 54*x^4 - 36*x^5 + 567*x^8 + 162*x^9 - 1) + exp(x - 2)*(2*x + 2*x^2 - 108*x^4 - 180*x
^5 - 36*x^6 + 1134*x^8 + 1458*x^9 + 162*x^10 - 2) - 54*x^4 - 144*x^5 - 90*x^6 + 567*x^8 + 1296*x^9 + 729*x^10
- 1)/x^2,x)

[Out]

x^3*(36*exp(x - 2) + 18*exp(2*x - 4) + 18) - 2*exp(x - 2) - x^7*(162*exp(x - 2) + 81*exp(2*x - 4) + 81) + x^4*
(36*exp(x - 2) + 36) - x^8*(162*exp(x - 2) + 162) - (2*exp(x - 2) + exp(2*x - 4) + 1)/x + 18*x^5 - 81*x^9

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sympy [B]  time = 0.19, size = 88, normalized size = 3.38 \begin {gather*} - 81 x^{9} - 162 x^{8} - 81 x^{7} + 18 x^{5} + 36 x^{4} + 18 x^{3} - \frac {1}{x} + \frac {\left (- 81 x^{9} + 18 x^{5} - x\right ) e^{2 x - 4} + \left (- 162 x^{10} - 162 x^{9} + 36 x^{6} + 36 x^{5} - 2 x^{2} - 2 x\right ) e^{x - 2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-162*x**9-567*x**8+36*x**5+54*x**4-2*x+1)*exp(x-2)**2+(-162*x**10-1458*x**9-1134*x**8+36*x**6+180*
x**5+108*x**4-2*x**2-2*x+2)*exp(x-2)-729*x**10-1296*x**9-567*x**8+90*x**6+144*x**5+54*x**4+1)/x**2,x)

[Out]

-81*x**9 - 162*x**8 - 81*x**7 + 18*x**5 + 36*x**4 + 18*x**3 - 1/x + ((-81*x**9 + 18*x**5 - x)*exp(2*x - 4) + (
-162*x**10 - 162*x**9 + 36*x**6 + 36*x**5 - 2*x**2 - 2*x)*exp(x - 2))/x**2

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