3.64.96 \(\int \frac {-336 x^2-2528 x^3-7284 x^4-9984 x^5-6400 x^6-1536 x^7+e^{8/x} (288+384 x+80 x^2-32 x^3)+e^{6/x} (-864-2880 x-2208 x^2+128 x^3+384 x^4)+e^{4/x} (864+4608 x+7296 x^2+1920 x^3-3072 x^4-1536 x^5)+e^{2/x} (-288-2112 x-4832 x^2-1792 x^3+6912 x^4+8192 x^5+2560 x^6)}{x^2} \, dx\)

Optimal. Leaf size=29 \[ 4 \left (x^3-\left (-1+e^{2/x}-2 x\right )^4 (3+2 x)^2\right ) \]

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Rubi [C]  time = 1.61, antiderivative size = 217, normalized size of antiderivative = 7.48, number of steps used = 46, number of rules used = 8, integrand size = 157, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {14, 2288, 6742, 2206, 2210, 2209, 2214, 2218} \begin {gather*} 6144 \text {Ei}\left (\frac {2}{x}\right )-16384 \text {Ei}\left (\frac {4}{x}\right )-256 x^6-1280 x^5-2496 x^4+2304 e^{2/x} x^3-1024 e^{4/x} x^3+128 e^{6/x} x^3-2428 x^3+1408 e^{2/x} x^2-1088 e^{4/x} x^2+448 e^{6/x} x^2-1264 x^2-2016 e^{2/x} x+2944 e^{4/x} x+480 e^{6/x} x-336 x+144 e^{2/x}-216 e^{4/x}+144 e^{6/x}-4 e^{8/x} (2 x+3)^2-81920 \Gamma \left (-5,-\frac {2}{x}\right )-393216 \Gamma \left (-4,-\frac {4}{x}\right )+131072 \Gamma \left (-4,-\frac {2}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-336*x^2 - 2528*x^3 - 7284*x^4 - 9984*x^5 - 6400*x^6 - 1536*x^7 + E^(8/x)*(288 + 384*x + 80*x^2 - 32*x^3)
+ E^(6/x)*(-864 - 2880*x - 2208*x^2 + 128*x^3 + 384*x^4) + E^(4/x)*(864 + 4608*x + 7296*x^2 + 1920*x^3 - 3072
*x^4 - 1536*x^5) + E^(2/x)*(-288 - 2112*x - 4832*x^2 - 1792*x^3 + 6912*x^4 + 8192*x^5 + 2560*x^6))/x^2,x]

[Out]

144*E^(2/x) - 216*E^(4/x) + 144*E^(6/x) - 336*x - 2016*E^(2/x)*x + 2944*E^(4/x)*x + 480*E^(6/x)*x - 1264*x^2 +
1408*E^(2/x)*x^2 - 1088*E^(4/x)*x^2 + 448*E^(6/x)*x^2 - 2428*x^3 + 2304*E^(2/x)*x^3 - 1024*E^(4/x)*x^3 + 128*
E^(6/x)*x^3 - 2496*x^4 - 1280*x^5 - 256*x^6 - 4*E^(8/x)*(3 + 2*x)^2 + 6144*ExpIntegralEi[2/x] - 16384*ExpInteg
ralEi[4/x] - 81920*Gamma[-5, -2/x] - 393216*Gamma[-4, -4/x] + 131072*Gamma[-4, -2/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 2206

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[((c + d*x)*F^(a + b*(c + d*x)^n))/d, x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {16 e^{8/x} (3+2 x) \left (-6-4 x+x^2\right )}{x^2}-\frac {96 e^{4/x} (1+2 x) (3+2 x) \left (-3-8 x+4 x^3\right )}{x^2}+\frac {32 e^{6/x} (3+2 x) \left (-9-24 x-7 x^2+6 x^3\right )}{x^2}+\frac {32 e^{2/x} (1+2 x)^2 (3+2 x) \left (-3-8 x+7 x^2+10 x^3\right )}{x^2}-4 \left (84+632 x+1821 x^2+2496 x^3+1600 x^4+384 x^5\right )\right ) \, dx\\ &=-\left (4 \int \left (84+632 x+1821 x^2+2496 x^3+1600 x^4+384 x^5\right ) \, dx\right )-16 \int \frac {e^{8/x} (3+2 x) \left (-6-4 x+x^2\right )}{x^2} \, dx+32 \int \frac {e^{6/x} (3+2 x) \left (-9-24 x-7 x^2+6 x^3\right )}{x^2} \, dx+32 \int \frac {e^{2/x} (1+2 x)^2 (3+2 x) \left (-3-8 x+7 x^2+10 x^3\right )}{x^2} \, dx-96 \int \frac {e^{4/x} (1+2 x) (3+2 x) \left (-3-8 x+4 x^3\right )}{x^2} \, dx\\ &=-336 x-1264 x^2-2428 x^3-2496 x^4-1280 x^5-256 x^6-4 e^{8/x} (3+2 x)^2+32 \int \left (-69 e^{6/x}-\frac {27 e^{6/x}}{x^2}-\frac {90 e^{6/x}}{x}+4 e^{6/x} x+12 e^{6/x} x^2\right ) \, dx+32 \int \left (-151 e^{2/x}-\frac {9 e^{2/x}}{x^2}-\frac {66 e^{2/x}}{x}-56 e^{2/x} x+216 e^{2/x} x^2+256 e^{2/x} x^3+80 e^{2/x} x^4\right ) \, dx-96 \int \left (-76 e^{4/x}-\frac {9 e^{4/x}}{x^2}-\frac {48 e^{4/x}}{x}-20 e^{4/x} x+32 e^{4/x} x^2+16 e^{4/x} x^3\right ) \, dx\\ &=-336 x-1264 x^2-2428 x^3-2496 x^4-1280 x^5-256 x^6-4 e^{8/x} (3+2 x)^2+128 \int e^{6/x} x \, dx-288 \int \frac {e^{2/x}}{x^2} \, dx+384 \int e^{6/x} x^2 \, dx+864 \int \frac {e^{4/x}}{x^2} \, dx-864 \int \frac {e^{6/x}}{x^2} \, dx-1536 \int e^{4/x} x^3 \, dx-1792 \int e^{2/x} x \, dx+1920 \int e^{4/x} x \, dx-2112 \int \frac {e^{2/x}}{x} \, dx-2208 \int e^{6/x} \, dx+2560 \int e^{2/x} x^4 \, dx-2880 \int \frac {e^{6/x}}{x} \, dx-3072 \int e^{4/x} x^2 \, dx+4608 \int \frac {e^{4/x}}{x} \, dx-4832 \int e^{2/x} \, dx+6912 \int e^{2/x} x^2 \, dx+7296 \int e^{4/x} \, dx+8192 \int e^{2/x} x^3 \, dx\\ &=144 e^{2/x}-216 e^{4/x}+144 e^{6/x}-336 x-4832 e^{2/x} x+7296 e^{4/x} x-2208 e^{6/x} x-1264 x^2-896 e^{2/x} x^2+960 e^{4/x} x^2+64 e^{6/x} x^2-2428 x^3+2304 e^{2/x} x^3-1024 e^{4/x} x^3+128 e^{6/x} x^3-2496 x^4-1280 x^5-256 x^6-4 e^{8/x} (3+2 x)^2+2112 \text {Ei}\left (\frac {2}{x}\right )-4608 \text {Ei}\left (\frac {4}{x}\right )+2880 \text {Ei}\left (\frac {6}{x}\right )-81920 \Gamma \left (-5,-\frac {2}{x}\right )-393216 \Gamma \left (-4,-\frac {4}{x}\right )+131072 \Gamma \left (-4,-\frac {2}{x}\right )+384 \int e^{6/x} \, dx+768 \int e^{6/x} x \, dx-1792 \int e^{2/x} \, dx+3840 \int e^{4/x} \, dx-4096 \int e^{4/x} x \, dx+4608 \int e^{2/x} x \, dx-9664 \int \frac {e^{2/x}}{x} \, dx-13248 \int \frac {e^{6/x}}{x} \, dx+29184 \int \frac {e^{4/x}}{x} \, dx\\ &=144 e^{2/x}-216 e^{4/x}+144 e^{6/x}-336 x-6624 e^{2/x} x+11136 e^{4/x} x-1824 e^{6/x} x-1264 x^2+1408 e^{2/x} x^2-1088 e^{4/x} x^2+448 e^{6/x} x^2-2428 x^3+2304 e^{2/x} x^3-1024 e^{4/x} x^3+128 e^{6/x} x^3-2496 x^4-1280 x^5-256 x^6-4 e^{8/x} (3+2 x)^2+11776 \text {Ei}\left (\frac {2}{x}\right )-33792 \text {Ei}\left (\frac {4}{x}\right )+16128 \text {Ei}\left (\frac {6}{x}\right )-81920 \Gamma \left (-5,-\frac {2}{x}\right )-393216 \Gamma \left (-4,-\frac {4}{x}\right )+131072 \Gamma \left (-4,-\frac {2}{x}\right )+2304 \int e^{6/x} \, dx+2304 \int \frac {e^{6/x}}{x} \, dx-3584 \int \frac {e^{2/x}}{x} \, dx+4608 \int e^{2/x} \, dx-8192 \int e^{4/x} \, dx+15360 \int \frac {e^{4/x}}{x} \, dx\\ &=144 e^{2/x}-216 e^{4/x}+144 e^{6/x}-336 x-2016 e^{2/x} x+2944 e^{4/x} x+480 e^{6/x} x-1264 x^2+1408 e^{2/x} x^2-1088 e^{4/x} x^2+448 e^{6/x} x^2-2428 x^3+2304 e^{2/x} x^3-1024 e^{4/x} x^3+128 e^{6/x} x^3-2496 x^4-1280 x^5-256 x^6-4 e^{8/x} (3+2 x)^2+15360 \text {Ei}\left (\frac {2}{x}\right )-49152 \text {Ei}\left (\frac {4}{x}\right )+13824 \text {Ei}\left (\frac {6}{x}\right )-81920 \Gamma \left (-5,-\frac {2}{x}\right )-393216 \Gamma \left (-4,-\frac {4}{x}\right )+131072 \Gamma \left (-4,-\frac {2}{x}\right )+9216 \int \frac {e^{2/x}}{x} \, dx+13824 \int \frac {e^{6/x}}{x} \, dx-32768 \int \frac {e^{4/x}}{x} \, dx\\ &=144 e^{2/x}-216 e^{4/x}+144 e^{6/x}-336 x-2016 e^{2/x} x+2944 e^{4/x} x+480 e^{6/x} x-1264 x^2+1408 e^{2/x} x^2-1088 e^{4/x} x^2+448 e^{6/x} x^2-2428 x^3+2304 e^{2/x} x^3-1024 e^{4/x} x^3+128 e^{6/x} x^3-2496 x^4-1280 x^5-256 x^6-4 e^{8/x} (3+2 x)^2+6144 \text {Ei}\left (\frac {2}{x}\right )-16384 \text {Ei}\left (\frac {4}{x}\right )-81920 \Gamma \left (-5,-\frac {2}{x}\right )-393216 \Gamma \left (-4,-\frac {4}{x}\right )+131072 \Gamma \left (-4,-\frac {2}{x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.09, size = 111, normalized size = 3.83 \begin {gather*} -4 \left (84 x+316 x^2+607 x^3+624 x^4+320 x^5+64 x^6+e^{8/x} (3+2 x)^2-4 e^{6/x} (1+2 x) (3+2 x)^2-4 e^{2/x} (1+2 x)^3 (3+2 x)^2+6 e^{4/x} \left (3+8 x+4 x^2\right )^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-336*x^2 - 2528*x^3 - 7284*x^4 - 9984*x^5 - 6400*x^6 - 1536*x^7 + E^(8/x)*(288 + 384*x + 80*x^2 - 3
2*x^3) + E^(6/x)*(-864 - 2880*x - 2208*x^2 + 128*x^3 + 384*x^4) + E^(4/x)*(864 + 4608*x + 7296*x^2 + 1920*x^3
- 3072*x^4 - 1536*x^5) + E^(2/x)*(-288 - 2112*x - 4832*x^2 - 1792*x^3 + 6912*x^4 + 8192*x^5 + 2560*x^6))/x^2,x
]

[Out]

-4*(84*x + 316*x^2 + 607*x^3 + 624*x^4 + 320*x^5 + 64*x^6 + E^(8/x)*(3 + 2*x)^2 - 4*E^(6/x)*(1 + 2*x)*(3 + 2*x
)^2 - 4*E^(2/x)*(1 + 2*x)^3*(3 + 2*x)^2 + 6*E^(4/x)*(3 + 8*x + 4*x^2)^2)

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fricas [B]  time = 0.69, size = 131, normalized size = 4.52 \begin {gather*} -256 \, x^{6} - 1280 \, x^{5} - 2496 \, x^{4} - 2428 \, x^{3} - 1264 \, x^{2} - 4 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} e^{\frac {8}{x}} + 16 \, {\left (8 \, x^{3} + 28 \, x^{2} + 30 \, x + 9\right )} e^{\frac {6}{x}} - 24 \, {\left (16 \, x^{4} + 64 \, x^{3} + 88 \, x^{2} + 48 \, x + 9\right )} e^{\frac {4}{x}} + 16 \, {\left (32 \, x^{5} + 144 \, x^{4} + 240 \, x^{3} + 184 \, x^{2} + 66 \, x + 9\right )} e^{\frac {2}{x}} - 336 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x^3+80*x^2+384*x+288)*exp(2/x)^4+(384*x^4+128*x^3-2208*x^2-2880*x-864)*exp(2/x)^3+(-1536*x^5-3
072*x^4+1920*x^3+7296*x^2+4608*x+864)*exp(2/x)^2+(2560*x^6+8192*x^5+6912*x^4-1792*x^3-4832*x^2-2112*x-288)*exp
(2/x)-1536*x^7-6400*x^6-9984*x^5-7284*x^4-2528*x^3-336*x^2)/x^2,x, algorithm="fricas")

[Out]

-256*x^6 - 1280*x^5 - 2496*x^4 - 2428*x^3 - 1264*x^2 - 4*(4*x^2 + 12*x + 9)*e^(8/x) + 16*(8*x^3 + 28*x^2 + 30*
x + 9)*e^(6/x) - 24*(16*x^4 + 64*x^3 + 88*x^2 + 48*x + 9)*e^(4/x) + 16*(32*x^5 + 144*x^4 + 240*x^3 + 184*x^2 +
66*x + 9)*e^(2/x) - 336*x

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giac [B]  time = 0.17, size = 230, normalized size = 7.93 \begin {gather*} 4 \, x^{6} {\left (\frac {128 \, e^{\frac {2}{x}}}{x} - \frac {320}{x} - \frac {96 \, e^{\frac {4}{x}}}{x^{2}} + \frac {576 \, e^{\frac {2}{x}}}{x^{2}} - \frac {624}{x^{2}} + \frac {32 \, e^{\frac {6}{x}}}{x^{3}} - \frac {384 \, e^{\frac {4}{x}}}{x^{3}} + \frac {960 \, e^{\frac {2}{x}}}{x^{3}} - \frac {607}{x^{3}} - \frac {4 \, e^{\frac {8}{x}}}{x^{4}} + \frac {112 \, e^{\frac {6}{x}}}{x^{4}} - \frac {528 \, e^{\frac {4}{x}}}{x^{4}} + \frac {736 \, e^{\frac {2}{x}}}{x^{4}} - \frac {316}{x^{4}} - \frac {12 \, e^{\frac {8}{x}}}{x^{5}} + \frac {120 \, e^{\frac {6}{x}}}{x^{5}} - \frac {288 \, e^{\frac {4}{x}}}{x^{5}} + \frac {264 \, e^{\frac {2}{x}}}{x^{5}} - \frac {84}{x^{5}} - \frac {9 \, e^{\frac {8}{x}}}{x^{6}} + \frac {36 \, e^{\frac {6}{x}}}{x^{6}} - \frac {54 \, e^{\frac {4}{x}}}{x^{6}} + \frac {36 \, e^{\frac {2}{x}}}{x^{6}} - 64\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x^3+80*x^2+384*x+288)*exp(2/x)^4+(384*x^4+128*x^3-2208*x^2-2880*x-864)*exp(2/x)^3+(-1536*x^5-3
072*x^4+1920*x^3+7296*x^2+4608*x+864)*exp(2/x)^2+(2560*x^6+8192*x^5+6912*x^4-1792*x^3-4832*x^2-2112*x-288)*exp
(2/x)-1536*x^7-6400*x^6-9984*x^5-7284*x^4-2528*x^3-336*x^2)/x^2,x, algorithm="giac")

[Out]

4*x^6*(128*e^(2/x)/x - 320/x - 96*e^(4/x)/x^2 + 576*e^(2/x)/x^2 - 624/x^2 + 32*e^(6/x)/x^3 - 384*e^(4/x)/x^3 +
960*e^(2/x)/x^3 - 607/x^3 - 4*e^(8/x)/x^4 + 112*e^(6/x)/x^4 - 528*e^(4/x)/x^4 + 736*e^(2/x)/x^4 - 316/x^4 - 1
2*e^(8/x)/x^5 + 120*e^(6/x)/x^5 - 288*e^(4/x)/x^5 + 264*e^(2/x)/x^5 - 84/x^5 - 9*e^(8/x)/x^6 + 36*e^(6/x)/x^6
- 54*e^(4/x)/x^6 + 36*e^(2/x)/x^6 - 64)

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maple [B]  time = 0.24, size = 128, normalized size = 4.41

method	result	size
risch	\(-256 x^{6}-1280 x^{5}-2496 x^{4}-2428 x^{3}-1264 x^{2}-336 x +\left (-16 x^{2}-48 x -36\right ) {\mathrm e}^{\frac {8}{x}}+\left (128 x^{3}+448 x^{2}+480 x +144\right ) {\mathrm e}^{\frac {6}{x}}+\left (-384 x^{4}-1536 x^{3}-2112 x^{2}-1152 x -216\right ) {\mathrm e}^{\frac {4}{x}}+\left (512 x^{5}+2304 x^{4}+3840 x^{3}+2944 x^{2}+1056 x +144\right ) {\mathrm e}^{\frac {2}{x}}\)	\(128\)
derivativedivides	\(-256 x^{6}-1280 x^{5}-2496 x^{4}-2428 x^{3}-1264 x^{2}-336 x -36 \,{\mathrm e}^{\frac {8}{x}}-16 \,{\mathrm e}^{\frac {8}{x}} x^{2}-48 \,{\mathrm e}^{\frac {8}{x}} x +144 \,{\mathrm e}^{\frac {6}{x}}+128 \,{\mathrm e}^{\frac {6}{x}} x^{3}+448 \,{\mathrm e}^{\frac {6}{x}} x^{2}+480 \,{\mathrm e}^{\frac {6}{x}} x -216 \,{\mathrm e}^{\frac {4}{x}}-384 x^{4} {\mathrm e}^{\frac {4}{x}}-1536 \,{\mathrm e}^{\frac {4}{x}} x^{3}-2112 x^{2} {\mathrm e}^{\frac {4}{x}}-1152 x \,{\mathrm e}^{\frac {4}{x}}+512 \,{\mathrm e}^{\frac {2}{x}} x^{5}+2304 \,{\mathrm e}^{\frac {2}{x}} x^{4}+3840 x^{3} {\mathrm e}^{\frac {2}{x}}+2944 x^{2} {\mathrm e}^{\frac {2}{x}}+1056 x \,{\mathrm e}^{\frac {2}{x}}+144 \,{\mathrm e}^{\frac {2}{x}}\)	\(232\)
default	\(-256 x^{6}-1280 x^{5}-2496 x^{4}-2428 x^{3}-1264 x^{2}-336 x -36 \,{\mathrm e}^{\frac {8}{x}}-16 \,{\mathrm e}^{\frac {8}{x}} x^{2}-48 \,{\mathrm e}^{\frac {8}{x}} x +144 \,{\mathrm e}^{\frac {6}{x}}+128 \,{\mathrm e}^{\frac {6}{x}} x^{3}+448 \,{\mathrm e}^{\frac {6}{x}} x^{2}+480 \,{\mathrm e}^{\frac {6}{x}} x -216 \,{\mathrm e}^{\frac {4}{x}}-384 x^{4} {\mathrm e}^{\frac {4}{x}}-1536 \,{\mathrm e}^{\frac {4}{x}} x^{3}-2112 x^{2} {\mathrm e}^{\frac {4}{x}}-1152 x \,{\mathrm e}^{\frac {4}{x}}+512 \,{\mathrm e}^{\frac {2}{x}} x^{5}+2304 \,{\mathrm e}^{\frac {2}{x}} x^{4}+3840 x^{3} {\mathrm e}^{\frac {2}{x}}+2944 x^{2} {\mathrm e}^{\frac {2}{x}}+1056 x \,{\mathrm e}^{\frac {2}{x}}+144 \,{\mathrm e}^{\frac {2}{x}}\)	\(232\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-32*x^3+80*x^2+384*x+288)*exp(2/x)^4+(384*x^4+128*x^3-2208*x^2-2880*x-864)*exp(2/x)^3+(-1536*x^5-3072*x^
4+1920*x^3+7296*x^2+4608*x+864)*exp(2/x)^2+(2560*x^6+8192*x^5+6912*x^4-1792*x^3-4832*x^2-2112*x-288)*exp(2/x)-
1536*x^7-6400*x^6-9984*x^5-7284*x^4-2528*x^3-336*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-256*x^6-1280*x^5-2496*x^4-2428*x^3-1264*x^2-336*x+(-16*x^2-48*x-36)*exp(8/x)+(128*x^3+448*x^2+480*x+144)*exp(
6/x)+(-384*x^4-1536*x^3-2112*x^2-1152*x-216)*exp(4/x)+(512*x^5+2304*x^4+3840*x^3+2944*x^2+1056*x+144)*exp(2/x)

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maxima [C]  time = 0.42, size = 219, normalized size = 7.55 \begin {gather*} -256 \, x^{6} - 1280 \, x^{5} - 2496 \, x^{4} - 2428 \, x^{3} - 1264 \, x^{2} - 336 \, x - 384 \, {\rm Ei}\left (\frac {8}{x}\right ) + 2880 \, {\rm Ei}\left (\frac {6}{x}\right ) - 4608 \, {\rm Ei}\left (\frac {4}{x}\right ) + 2112 \, {\rm Ei}\left (\frac {2}{x}\right ) - 36 \, e^{\frac {8}{x}} + 144 \, e^{\frac {6}{x}} - 216 \, e^{\frac {4}{x}} + 144 \, e^{\frac {2}{x}} + 9664 \, \Gamma \left (-1, -\frac {2}{x}\right ) - 29184 \, \Gamma \left (-1, -\frac {4}{x}\right ) + 13248 \, \Gamma \left (-1, -\frac {6}{x}\right ) - 640 \, \Gamma \left (-1, -\frac {8}{x}\right ) - 7168 \, \Gamma \left (-2, -\frac {2}{x}\right ) + 30720 \, \Gamma \left (-2, -\frac {4}{x}\right ) + 4608 \, \Gamma \left (-2, -\frac {6}{x}\right ) - 2048 \, \Gamma \left (-2, -\frac {8}{x}\right ) - 55296 \, \Gamma \left (-3, -\frac {2}{x}\right ) + 196608 \, \Gamma \left (-3, -\frac {4}{x}\right ) - 82944 \, \Gamma \left (-3, -\frac {6}{x}\right ) + 131072 \, \Gamma \left (-4, -\frac {2}{x}\right ) - 393216 \, \Gamma \left (-4, -\frac {4}{x}\right ) - 81920 \, \Gamma \left (-5, -\frac {2}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x^3+80*x^2+384*x+288)*exp(2/x)^4+(384*x^4+128*x^3-2208*x^2-2880*x-864)*exp(2/x)^3+(-1536*x^5-3
072*x^4+1920*x^3+7296*x^2+4608*x+864)*exp(2/x)^2+(2560*x^6+8192*x^5+6912*x^4-1792*x^3-4832*x^2-2112*x-288)*exp
(2/x)-1536*x^7-6400*x^6-9984*x^5-7284*x^4-2528*x^3-336*x^2)/x^2,x, algorithm="maxima")

[Out]

-256*x^6 - 1280*x^5 - 2496*x^4 - 2428*x^3 - 1264*x^2 - 336*x - 384*Ei(8/x) + 2880*Ei(6/x) - 4608*Ei(4/x) + 211
2*Ei(2/x) - 36*e^(8/x) + 144*e^(6/x) - 216*e^(4/x) + 144*e^(2/x) + 9664*gamma(-1, -2/x) - 29184*gamma(-1, -4/x
) + 13248*gamma(-1, -6/x) - 640*gamma(-1, -8/x) - 7168*gamma(-2, -2/x) + 30720*gamma(-2, -4/x) + 4608*gamma(-2
, -6/x) - 2048*gamma(-2, -8/x) - 55296*gamma(-3, -2/x) + 196608*gamma(-3, -4/x) - 82944*gamma(-3, -6/x) + 1310
72*gamma(-4, -2/x) - 393216*gamma(-4, -4/x) - 81920*gamma(-5, -2/x)

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mupad [B]  time = 4.85, size = 207, normalized size = 7.14 \begin {gather*} 144\,{\mathrm {e}}^{2/x}-336\,x-216\,{\mathrm {e}}^{4/x}+144\,{\mathrm {e}}^{6/x}-36\,{\mathrm {e}}^{8/x}+1056\,x\,{\mathrm {e}}^{2/x}-1152\,x\,{\mathrm {e}}^{4/x}+480\,x\,{\mathrm {e}}^{6/x}-48\,x\,{\mathrm {e}}^{8/x}+2944\,x^2\,{\mathrm {e}}^{2/x}+3840\,x^3\,{\mathrm {e}}^{2/x}-2112\,x^2\,{\mathrm {e}}^{4/x}+2304\,x^4\,{\mathrm {e}}^{2/x}-1536\,x^3\,{\mathrm {e}}^{4/x}+512\,x^5\,{\mathrm {e}}^{2/x}+448\,x^2\,{\mathrm {e}}^{6/x}-384\,x^4\,{\mathrm {e}}^{4/x}+128\,x^3\,{\mathrm {e}}^{6/x}-16\,x^2\,{\mathrm {e}}^{8/x}-1264\,x^2-2428\,x^3-2496\,x^4-1280\,x^5-256\,x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2/x)*(2112*x + 4832*x^2 + 1792*x^3 - 6912*x^4 - 8192*x^5 - 2560*x^6 + 288) - exp(4/x)*(4608*x + 7296
*x^2 + 1920*x^3 - 3072*x^4 - 1536*x^5 + 864) - exp(8/x)*(384*x + 80*x^2 - 32*x^3 + 288) + exp(6/x)*(2880*x + 2
208*x^2 - 128*x^3 - 384*x^4 + 864) + 336*x^2 + 2528*x^3 + 7284*x^4 + 9984*x^5 + 6400*x^6 + 1536*x^7)/x^2,x)

[Out]

144*exp(2/x) - 336*x - 216*exp(4/x) + 144*exp(6/x) - 36*exp(8/x) + 1056*x*exp(2/x) - 1152*x*exp(4/x) + 480*x*e
xp(6/x) - 48*x*exp(8/x) + 2944*x^2*exp(2/x) + 3840*x^3*exp(2/x) - 2112*x^2*exp(4/x) + 2304*x^4*exp(2/x) - 1536
*x^3*exp(4/x) + 512*x^5*exp(2/x) + 448*x^2*exp(6/x) - 384*x^4*exp(4/x) + 128*x^3*exp(6/x) - 16*x^2*exp(8/x) -
1264*x^2 - 2428*x^3 - 2496*x^4 - 1280*x^5 - 256*x^6

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sympy [B]  time = 0.26, size = 122, normalized size = 4.21 \begin {gather*} - 256 x^{6} - 1280 x^{5} - 2496 x^{4} - 2428 x^{3} - 1264 x^{2} - 336 x + \left (- 16 x^{2} - 48 x - 36\right ) e^{\frac {8}{x}} + \left (128 x^{3} + 448 x^{2} + 480 x + 144\right ) e^{\frac {6}{x}} + \left (- 384 x^{4} - 1536 x^{3} - 2112 x^{2} - 1152 x - 216\right ) e^{\frac {4}{x}} + \left (512 x^{5} + 2304 x^{4} + 3840 x^{3} + 2944 x^{2} + 1056 x + 144\right ) e^{\frac {2}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-32*x**3+80*x**2+384*x+288)*exp(2/x)**4+(384*x**4+128*x**3-2208*x**2-2880*x-864)*exp(2/x)**3+(-153
6*x**5-3072*x**4+1920*x**3+7296*x**2+4608*x+864)*exp(2/x)**2+(2560*x**6+8192*x**5+6912*x**4-1792*x**3-4832*x**
2-2112*x-288)*exp(2/x)-1536*x**7-6400*x**6-9984*x**5-7284*x**4-2528*x**3-336*x**2)/x**2,x)

[Out]

-256*x**6 - 1280*x**5 - 2496*x**4 - 2428*x**3 - 1264*x**2 - 336*x + (-16*x**2 - 48*x - 36)*exp(8/x) + (128*x**
3 + 448*x**2 + 480*x + 144)*exp(6/x) + (-384*x**4 - 1536*x**3 - 2112*x**2 - 1152*x - 216)*exp(4/x) + (512*x**5
+ 2304*x**4 + 3840*x**3 + 2944*x**2 + 1056*x + 144)*exp(2/x)

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