3.38.100 \(\int \frac {-8 e^{2/x}+x^2+512 x^3-192 x^4+16 x^5+e^{\frac {1}{x}} (-64 x+72 x^2-16 x^3)+e^{2 x} (448 x^2-120 x^3+8 x^4)+e^x (-512 x^2-256 x^3+104 x^4-8 x^5+e^{\frac {1}{x}} (64-8 x-56 x^2+8 x^3))}{x^2} \, dx\)

Optimal. Leaf size=27 \[ 4 \left (-e^{\frac {1}{x}}+(8-x) \left (e^x-x\right )\right )^2+x \]

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Rubi [F]  time = 1.77, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-8 e^{2/x}+x^2+512 x^3-192 x^4+16 x^5+e^{\frac {1}{x}} \left (-64 x+72 x^2-16 x^3\right )+e^{2 x} \left (448 x^2-120 x^3+8 x^4\right )+e^x \left (-512 x^2-256 x^3+104 x^4-8 x^5+e^{\frac {1}{x}} \left (64-8 x-56 x^2+8 x^3\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*E^(2/x) + x^2 + 512*x^3 - 192*x^4 + 16*x^5 + E^x^(-1)*(-64*x + 72*x^2 - 16*x^3) + E^(2*x)*(448*x^2 - 1
20*x^3 + 8*x^4) + E^x*(-512*x^2 - 256*x^3 + 104*x^4 - 8*x^5 + E^x^(-1)*(64 - 8*x - 56*x^2 + 8*x^3)))/x^2,x]

[Out]

4*E^(2/x) + 256*E^(2*x) + x + 64*E^x^(-1)*x - 512*E^x*x - 64*E^(2*x)*x + 256*x^2 - 8*E^x^(-1)*x^2 + 128*E^x*x^
2 + 4*E^(2*x)*x^2 - 64*x^3 - 8*E^x*x^3 + 4*x^4 - 56*Defer[Int][E^(x^(-1) + x), x] + 64*Defer[Int][E^(x^(-1) +
x)/x^2, x] - 8*Defer[Int][E^(x^(-1) + x)/x, x] + 8*Defer[Int][E^(x^(-1) + x)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (8 e^{2 x} (-8+x) (-7+x)-\frac {8 e^x \left (-8 e^{\frac {1}{x}}+e^{\frac {1}{x}} x+64 x^2+7 e^{\frac {1}{x}} x^2+32 x^3-e^{\frac {1}{x}} x^3-13 x^4+x^5\right )}{x^2}+\frac {-8 e^{2/x}-64 e^{\frac {1}{x}} x+x^2+72 e^{\frac {1}{x}} x^2+512 x^3-16 e^{\frac {1}{x}} x^3-192 x^4+16 x^5}{x^2}\right ) \, dx\\ &=8 \int e^{2 x} (-8+x) (-7+x) \, dx-8 \int \frac {e^x \left (-8 e^{\frac {1}{x}}+e^{\frac {1}{x}} x+64 x^2+7 e^{\frac {1}{x}} x^2+32 x^3-e^{\frac {1}{x}} x^3-13 x^4+x^5\right )}{x^2} \, dx+\int \frac {-8 e^{2/x}-64 e^{\frac {1}{x}} x+x^2+72 e^{\frac {1}{x}} x^2+512 x^3-16 e^{\frac {1}{x}} x^3-192 x^4+16 x^5}{x^2} \, dx\\ &=8 \int \left (56 e^{2 x}-15 e^{2 x} x+e^{2 x} x^2\right ) \, dx-8 \int e^x \left (64+e^{\frac {1}{x}} \left (7-\frac {8}{x^2}+\frac {1}{x}-x\right )+32 x-13 x^2+x^3\right ) \, dx+\int \left (1-\frac {8 e^{2/x}}{x^2}+512 x-192 x^2+16 x^3-\frac {8 e^{\frac {1}{x}} \left (8-9 x+2 x^2\right )}{x}\right ) \, dx\\ &=x+256 x^2-64 x^3+4 x^4-8 \int \frac {e^{2/x}}{x^2} \, dx+8 \int e^{2 x} x^2 \, dx-8 \int \frac {e^{\frac {1}{x}} \left (8-9 x+2 x^2\right )}{x} \, dx-8 \int \left (64 e^x+32 e^x x-13 e^x x^2+e^x x^3-\frac {e^{\frac {1}{x}+x} \left (8-x-7 x^2+x^3\right )}{x^2}\right ) \, dx-120 \int e^{2 x} x \, dx+448 \int e^{2 x} \, dx\\ &=4 e^{2/x}+224 e^{2 x}+x-60 e^{2 x} x+256 x^2+4 e^{2 x} x^2-64 x^3+4 x^4-8 \int e^{2 x} x \, dx-8 \int e^x x^3 \, dx-8 \int \left (-9 e^{\frac {1}{x}}+\frac {8 e^{\frac {1}{x}}}{x}+2 e^{\frac {1}{x}} x\right ) \, dx+8 \int \frac {e^{\frac {1}{x}+x} \left (8-x-7 x^2+x^3\right )}{x^2} \, dx+60 \int e^{2 x} \, dx+104 \int e^x x^2 \, dx-256 \int e^x x \, dx-512 \int e^x \, dx\\ &=4 e^{2/x}-512 e^x+254 e^{2 x}+x-256 e^x x-64 e^{2 x} x+256 x^2+104 e^x x^2+4 e^{2 x} x^2-64 x^3-8 e^x x^3+4 x^4+4 \int e^{2 x} \, dx+8 \int \left (-7 e^{\frac {1}{x}+x}+\frac {8 e^{\frac {1}{x}+x}}{x^2}-\frac {e^{\frac {1}{x}+x}}{x}+e^{\frac {1}{x}+x} x\right ) \, dx-16 \int e^{\frac {1}{x}} x \, dx+24 \int e^x x^2 \, dx-64 \int \frac {e^{\frac {1}{x}}}{x} \, dx+72 \int e^{\frac {1}{x}} \, dx-208 \int e^x x \, dx+256 \int e^x \, dx\\ &=4 e^{2/x}-256 e^x+256 e^{2 x}+x+72 e^{\frac {1}{x}} x-464 e^x x-64 e^{2 x} x+256 x^2-8 e^{\frac {1}{x}} x^2+128 e^x x^2+4 e^{2 x} x^2-64 x^3-8 e^x x^3+4 x^4+64 \text {Ei}\left (\frac {1}{x}\right )-8 \int e^{\frac {1}{x}} \, dx-8 \int \frac {e^{\frac {1}{x}+x}}{x} \, dx+8 \int e^{\frac {1}{x}+x} x \, dx-48 \int e^x x \, dx-56 \int e^{\frac {1}{x}+x} \, dx+64 \int \frac {e^{\frac {1}{x}+x}}{x^2} \, dx+72 \int \frac {e^{\frac {1}{x}}}{x} \, dx+208 \int e^x \, dx\\ &=4 e^{2/x}-48 e^x+256 e^{2 x}+x+64 e^{\frac {1}{x}} x-512 e^x x-64 e^{2 x} x+256 x^2-8 e^{\frac {1}{x}} x^2+128 e^x x^2+4 e^{2 x} x^2-64 x^3-8 e^x x^3+4 x^4-8 \text {Ei}\left (\frac {1}{x}\right )-8 \int \frac {e^{\frac {1}{x}}}{x} \, dx-8 \int \frac {e^{\frac {1}{x}+x}}{x} \, dx+8 \int e^{\frac {1}{x}+x} x \, dx+48 \int e^x \, dx-56 \int e^{\frac {1}{x}+x} \, dx+64 \int \frac {e^{\frac {1}{x}+x}}{x^2} \, dx\\ &=4 e^{2/x}+256 e^{2 x}+x+64 e^{\frac {1}{x}} x-512 e^x x-64 e^{2 x} x+256 x^2-8 e^{\frac {1}{x}} x^2+128 e^x x^2+4 e^{2 x} x^2-64 x^3-8 e^x x^3+4 x^4-8 \int \frac {e^{\frac {1}{x}+x}}{x} \, dx+8 \int e^{\frac {1}{x}+x} x \, dx-56 \int e^{\frac {1}{x}+x} \, dx+64 \int \frac {e^{\frac {1}{x}+x}}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.18, size = 88, normalized size = 3.26 \begin {gather*} 4 e^{2/x}+8 e^{\frac {1}{x}+x} (-8+x)+x+256 x^2-64 x^3+4 x^4+8 e^{2 x} \left (32-8 x+\frac {x^2}{2}\right )-8 e^{\frac {1}{x}} \left (-8 x+x^2\right )-8 e^x \left (64 x-16 x^2+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*E^(2/x) + x^2 + 512*x^3 - 192*x^4 + 16*x^5 + E^x^(-1)*(-64*x + 72*x^2 - 16*x^3) + E^(2*x)*(448*x
^2 - 120*x^3 + 8*x^4) + E^x*(-512*x^2 - 256*x^3 + 104*x^4 - 8*x^5 + E^x^(-1)*(64 - 8*x - 56*x^2 + 8*x^3)))/x^2
,x]

[Out]

4*E^(2/x) + 8*E^(x^(-1) + x)*(-8 + x) + x + 256*x^2 - 64*x^3 + 4*x^4 + 8*E^(2*x)*(32 - 8*x + x^2/2) - 8*E^x^(-
1)*(-8*x + x^2) - 8*E^x*(64*x - 16*x^2 + x^3)

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fricas [B]  time = 1.18, size = 77, normalized size = 2.85 \begin {gather*} 4 \, x^{4} - 64 \, x^{3} + 256 \, x^{2} + 4 \, {\left (x^{2} - 16 \, x + 64\right )} e^{\left (2 \, x\right )} - 8 \, {\left (x^{3} - 16 \, x^{2} - {\left (x - 8\right )} e^{\frac {1}{x}} + 64 \, x\right )} e^{x} - 8 \, {\left (x^{2} - 8 \, x\right )} e^{\frac {1}{x}} + x + 4 \, e^{\frac {2}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^4-120*x^3+448*x^2)*exp(x)^2+((8*x^3-56*x^2-8*x+64)*exp(1/x)-8*x^5+104*x^4-256*x^3-512*x^2)*exp
(x)-8*exp(1/x)^2+(-16*x^3+72*x^2-64*x)*exp(1/x)+16*x^5-192*x^4+512*x^3+x^2)/x^2,x, algorithm="fricas")

[Out]

4*x^4 - 64*x^3 + 256*x^2 + 4*(x^2 - 16*x + 64)*e^(2*x) - 8*(x^3 - 16*x^2 - (x - 8)*e^(1/x) + 64*x)*e^x - 8*(x^
2 - 8*x)*e^(1/x) + x + 4*e^(2/x)

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giac [B]  time = 0.20, size = 107, normalized size = 3.96 \begin {gather*} 4 \, x^{4} - 8 \, x^{3} e^{x} - 64 \, x^{3} + 4 \, x^{2} e^{\left (2 \, x\right )} + 128 \, x^{2} e^{x} - 8 \, x^{2} e^{\frac {1}{x}} + 256 \, x^{2} - 64 \, x e^{\left (2 \, x\right )} - 512 \, x e^{x} + 8 \, x e^{\left (\frac {x^{2} + 1}{x}\right )} + 64 \, x e^{\frac {1}{x}} + x + 256 \, e^{\left (2 \, x\right )} - 64 \, e^{\left (\frac {x^{2} + 1}{x}\right )} + 4 \, e^{\frac {2}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^4-120*x^3+448*x^2)*exp(x)^2+((8*x^3-56*x^2-8*x+64)*exp(1/x)-8*x^5+104*x^4-256*x^3-512*x^2)*exp
(x)-8*exp(1/x)^2+(-16*x^3+72*x^2-64*x)*exp(1/x)+16*x^5-192*x^4+512*x^3+x^2)/x^2,x, algorithm="giac")

[Out]

4*x^4 - 8*x^3*e^x - 64*x^3 + 4*x^2*e^(2*x) + 128*x^2*e^x - 8*x^2*e^(1/x) + 256*x^2 - 64*x*e^(2*x) - 512*x*e^x
+ 8*x*e^((x^2 + 1)/x) + 64*x*e^(1/x) + x + 256*e^(2*x) - 64*e^((x^2 + 1)/x) + 4*e^(2/x)

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maple [B]  time = 0.14, size = 81, normalized size = 3.00

method	result	size
risch	\(4 x^{4}-64 x^{3}+256 x^{2}+x +\left (4 x^{2}-64 x +256\right ) {\mathrm e}^{2 x}+\left (-8 x^{3}+128 x^{2}-512 x \right ) {\mathrm e}^{x}+4 \,{\mathrm e}^{\frac {2}{x}}+\left (-8 x^{2}+8 \,{\mathrm e}^{x} x +64 x -64 \,{\mathrm e}^{x}\right ) {\mathrm e}^{\frac {1}{x}}\)	\(81\)
derivativedivides	\(4 x^{4}-64 x^{3}+256 x^{2}+x +4 \,{\mathrm e}^{\frac {2}{x}}+4 \,{\mathrm e}^{2 x} x^{2}-64 x \,{\mathrm e}^{2 x}+256 \,{\mathrm e}^{2 x}-8 x^{2} {\mathrm e}^{\frac {1}{x}}+64 x \,{\mathrm e}^{\frac {1}{x}}+\left (\frac {128 \,{\mathrm e}^{x}}{x^{2}}-\frac {512 \,{\mathrm e}^{x}}{x^{3}}-\frac {8 \,{\mathrm e}^{x}}{x}+\frac {8 \,{\mathrm e}^{\frac {1}{x}} {\mathrm e}^{x}}{x^{3}}-\frac {64 \,{\mathrm e}^{\frac {1}{x}} {\mathrm e}^{x}}{x^{4}}\right ) x^{4}\)	\(112\)
default	\(4 x^{4}-64 x^{3}+256 x^{2}+x +4 \,{\mathrm e}^{\frac {2}{x}}+4 \,{\mathrm e}^{2 x} x^{2}-64 x \,{\mathrm e}^{2 x}+256 \,{\mathrm e}^{2 x}-8 x^{2} {\mathrm e}^{\frac {1}{x}}+64 x \,{\mathrm e}^{\frac {1}{x}}+\left (\frac {128 \,{\mathrm e}^{x}}{x^{2}}-\frac {512 \,{\mathrm e}^{x}}{x^{3}}-\frac {8 \,{\mathrm e}^{x}}{x}+\frac {8 \,{\mathrm e}^{\frac {1}{x}} {\mathrm e}^{x}}{x^{3}}-\frac {64 \,{\mathrm e}^{\frac {1}{x}} {\mathrm e}^{x}}{x^{4}}\right ) x^{4}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^4-120*x^3+448*x^2)*exp(x)^2+((8*x^3-56*x^2-8*x+64)*exp(1/x)-8*x^5+104*x^4-256*x^3-512*x^2)*exp(x)-8*
exp(1/x)^2+(-16*x^3+72*x^2-64*x)*exp(1/x)+16*x^5-192*x^4+512*x^3+x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

4*x^4-64*x^3+256*x^2+x+(4*x^2-64*x+256)*exp(2*x)+(-8*x^3+128*x^2-512*x)*exp(x)+4*exp(2/x)+(-8*x^2+8*exp(x)*x+6
4*x-64*exp(x))*exp(1/x)

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maxima [C]  time = 0.45, size = 133, normalized size = 4.93 \begin {gather*} 4 \, x^{4} - 64 \, x^{3} + 256 \, x^{2} + 2 \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} - 30 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x - 8\right )} e^{\left (x + \frac {1}{x}\right )} - 8 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + 104 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} - 256 \, {\left (x - 1\right )} e^{x} + x + 64 \, {\rm Ei}\left (\frac {1}{x}\right ) + 224 \, e^{\left (2 \, x\right )} - 512 \, e^{x} + 4 \, e^{\frac {2}{x}} - 72 \, \Gamma \left (-1, -\frac {1}{x}\right ) - 16 \, \Gamma \left (-2, -\frac {1}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^4-120*x^3+448*x^2)*exp(x)^2+((8*x^3-56*x^2-8*x+64)*exp(1/x)-8*x^5+104*x^4-256*x^3-512*x^2)*exp
(x)-8*exp(1/x)^2+(-16*x^3+72*x^2-64*x)*exp(1/x)+16*x^5-192*x^4+512*x^3+x^2)/x^2,x, algorithm="maxima")

[Out]

4*x^4 - 64*x^3 + 256*x^2 + 2*(2*x^2 - 2*x + 1)*e^(2*x) - 30*(2*x - 1)*e^(2*x) + 8*(x - 8)*e^(x + 1/x) - 8*(x^3
- 3*x^2 + 6*x - 6)*e^x + 104*(x^2 - 2*x + 2)*e^x - 256*(x - 1)*e^x + x + 64*Ei(1/x) + 224*e^(2*x) - 512*e^x +
4*e^(2/x) - 72*gamma(-1, -1/x) - 16*gamma(-2, -1/x)

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mupad [B]  time = 2.75, size = 66, normalized size = 2.44 \begin {gather*} x+4\,{\mathrm {e}}^{2/x}+4\,{\mathrm {e}}^{2\,x}\,{\left (x-8\right )}^2+256\,x^2-64\,x^3+4\,x^4+8\,{\mathrm {e}}^x\,\left (x-8\right )\,\left (8\,x+{\mathrm {e}}^{1/x}-x^2\right )-8\,x\,{\mathrm {e}}^{1/x}\,\left (x-8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*exp(2/x) + exp(1/x)*(64*x - 72*x^2 + 16*x^3) - exp(2*x)*(448*x^2 - 120*x^3 + 8*x^4) + exp(x)*(exp(1/x)
*(8*x + 56*x^2 - 8*x^3 - 64) + 512*x^2 + 256*x^3 - 104*x^4 + 8*x^5) - x^2 - 512*x^3 + 192*x^4 - 16*x^5)/x^2,x)

[Out]

x + 4*exp(2/x) + 4*exp(2*x)*(x - 8)^2 + 256*x^2 - 64*x^3 + 4*x^4 + 8*exp(x)*(x - 8)*(8*x + exp(1/x) - x^2) - 8
*x*exp(1/x)*(x - 8)

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sympy [B]  time = 8.92, size = 83, normalized size = 3.07 \begin {gather*} 4 x^{4} - 64 x^{3} + 256 x^{2} + x + \left (- 8 x^{2} + 64 x\right ) e^{\frac {1}{x}} + \left (4 x^{2} - 64 x + 256\right ) e^{2 x} + \left (- 8 x^{3} + 128 x^{2} + 8 x e^{\frac {1}{x}} - 512 x - 64 e^{\frac {1}{x}}\right ) e^{x} + 4 e^{\frac {2}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**4-120*x**3+448*x**2)*exp(x)**2+((8*x**3-56*x**2-8*x+64)*exp(1/x)-8*x**5+104*x**4-256*x**3-512
*x**2)*exp(x)-8*exp(1/x)**2+(-16*x**3+72*x**2-64*x)*exp(1/x)+16*x**5-192*x**4+512*x**3+x**2)/x**2,x)

[Out]

4*x**4 - 64*x**3 + 256*x**2 + x + (-8*x**2 + 64*x)*exp(1/x) + (4*x**2 - 64*x + 256)*exp(2*x) + (-8*x**3 + 128*
x**2 + 8*x*exp(1/x) - 512*x - 64*exp(1/x))*exp(x) + 4*exp(2/x)

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