3.102.35 \(\int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} (8-32 x-12 x^2+e^{-1-x+x^2} (2+x^2-2 x^3))}{160+240 x^2+120 x^4+20 x^6} \, dx\)

Optimal. Leaf size=32 \[ -10+e^{\frac {2-\frac {1}{4} e^{-1-x+x^2}+x}{5 \left (2+x^2\right )^2}} \]

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Rubi [F]  time = 33.54, 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 {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} \left (8-32 x-12 x^2+e^{-1-x+x^2} \left (2+x^2-2 x^3\right )\right )}{160+240 x^2+120 x^4+20 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((8 - E^(-1 - x + x^2) + 4*x)/(80 + 80*x^2 + 20*x^4))*(8 - 32*x - 12*x^2 + E^(-1 - x + x^2)*(2 + x^2 -
2*x^3)))/(160 + 240*x^2 + 120*x^4 + 20*x^6),x]

[Out]

((-1/10*I)*Defer[Int][E^((8 - E^(-1 - x + x^2) + 4*x)/(80 + 80*x^2 + 20*x^4))/(I*Sqrt[2] - x)^3, x])/Sqrt[2] -
Defer[Int][E^(-1 - x + x^2 + (8 - E^(-1 - x + x^2) + 4*x)/(80 + 80*x^2 + 20*x^4))/(I*Sqrt[2] - x)^2, x]/160 +
((I/160)*Defer[Int][E^(-1 - x + x^2 + (8 - E^(-1 - x + x^2) + 4*x)/(80 + 80*x^2 + 20*x^4))/(I*Sqrt[2] - x), x
])/Sqrt[2] - ((I/10)*Defer[Int][E^((8 - E^(-1 - x + x^2) + 4*x)/(80 + 80*x^2 + 20*x^4))/(I*Sqrt[2] + x)^3, x])
/Sqrt[2] - Defer[Int][E^(-1 - x + x^2 + (8 - E^(-1 - x + x^2) + 4*x)/(80 + 80*x^2 + 20*x^4))/(I*Sqrt[2] + x)^2
, x]/160 + ((I/160)*Defer[Int][E^(-1 - x + x^2 + (8 - E^(-1 - x + x^2) + 4*x)/(80 + 80*x^2 + 20*x^4))/(I*Sqrt[
2] + x), x])/Sqrt[2] - (8*Defer[Int][(E^((8 - E^(-1 - x + x^2) + 4*x)/(80 + 80*x^2 + 20*x^4))*x)/(2 + x^2)^3,
x])/5 + Defer[Int][(E^(-1 - x + x^2 + (8 - E^(-1 - x + x^2) + 4*x)/(80 + 80*x^2 + 20*x^4))*x)/(2 + x^2)^3, x]/
5 - Defer[Int][(E^(-1 - x + x^2 + (8 - E^(-1 - x + x^2) + 4*x)/(80 + 80*x^2 + 20*x^4))*x)/(2 + x^2)^2, x]/10

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} \left (8-32 x-12 x^2+e^{-1-x+x^2} \left (2+x^2-2 x^3\right )\right )}{20 \left (2+x^2\right )^3} \, dx\\ &=\frac {1}{20} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} \left (8-32 x-12 x^2+e^{-1-x+x^2} \left (2+x^2-2 x^3\right )\right )}{\left (2+x^2\right )^3} \, dx\\ &=\frac {1}{20} \int \left (\frac {8 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^3}-\frac {32 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3}-\frac {12 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x^2}{\left (2+x^2\right )^3}-\frac {\exp \left (-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}\right ) \left (-2-x^2+2 x^3\right )}{\left (2+x^2\right )^3}\right ) \, dx\\ &=-\left (\frac {1}{20} \int \frac {\exp \left (-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}\right ) \left (-2-x^2+2 x^3\right )}{\left (2+x^2\right )^3} \, dx\right )+\frac {2}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^3} \, dx-\frac {3}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x^2}{\left (2+x^2\right )^3} \, dx-\frac {8}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx\\ &=-\left (\frac {1}{20} \int \left (-\frac {4 \exp \left (-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}\right ) x}{\left (2+x^2\right )^3}+\frac {\exp \left (-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}\right ) (-1+2 x)}{\left (2+x^2\right )^2}\right ) \, dx\right )+\frac {2}{5} \int \left (-\frac {i e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{16 \sqrt {2} \left (i \sqrt {2}-x\right )^3}-\frac {3 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{64 \left (i \sqrt {2}-x\right )^2}-\frac {i e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{16 \sqrt {2} \left (i \sqrt {2}+x\right )^3}-\frac {3 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{64 \left (i \sqrt {2}+x\right )^2}-\frac {3 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{32 \left (-2-x^2\right )}\right ) \, dx-\frac {3}{5} \int \left (-\frac {2 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^3}+\frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^2}\right ) \, dx-\frac {8}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx\\ &=-\left (\frac {3}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^2} \, dx\right )-\frac {3}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^2} \, dx-\frac {3}{80} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{-2-x^2} \, dx-\frac {1}{20} \int \frac {\exp \left (-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}\right ) (-1+2 x)}{\left (2+x^2\right )^2} \, dx+\frac {1}{5} \int \frac {\exp \left (-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}\right ) x}{\left (2+x^2\right )^3} \, dx-\frac {3}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^2} \, dx+\frac {6}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^3} \, dx-\frac {8}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx-\frac {i \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^3} \, dx}{40 \sqrt {2}}-\frac {i \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^3} \, dx}{40 \sqrt {2}}\\ &=-\left (\frac {3}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^2} \, dx\right )-\frac {3}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^2} \, dx-\frac {3}{80} \int \left (-\frac {i e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}-\frac {i e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx-\frac {1}{20} \int \left (-\frac {e^{-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^2}+\frac {2 e^{-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^2}\right ) \, dx+\frac {1}{5} \int \frac {e^{-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx-\frac {3}{5} \int \left (-\frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{8 \left (i \sqrt {2}-x\right )^2}-\frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{8 \left (i \sqrt {2}+x\right )^2}-\frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{4 \left (-2-x^2\right )}\right ) \, dx+\frac {6}{5} \int \left (-\frac {i e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{16 \sqrt {2} \left (i \sqrt {2}-x\right )^3}-\frac {3 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{64 \left (i \sqrt {2}-x\right )^2}-\frac {i e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{16 \sqrt {2} \left (i \sqrt {2}+x\right )^3}-\frac {3 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{64 \left (i \sqrt {2}+x\right )^2}-\frac {3 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{32 \left (-2-x^2\right )}\right ) \, dx-\frac {8}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx-\frac {i \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^3} \, dx}{40 \sqrt {2}}-\frac {i \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^3} \, dx}{40 \sqrt {2}}\\ &=-\left (\frac {3}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^2} \, dx\right )-\frac {3}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^2} \, dx+\frac {1}{20} \int \frac {e^{-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^2} \, dx-\frac {9}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^2} \, dx-\frac {9}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^2} \, dx+\frac {3}{40} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^2} \, dx+\frac {3}{40} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^2} \, dx-\frac {1}{10} \int \frac {e^{-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^2} \, dx-\frac {9}{80} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{-2-x^2} \, dx+\frac {3}{20} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{-2-x^2} \, dx+\frac {1}{5} \int \frac {e^{-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx-\frac {8}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx+\frac {(3 i) \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{i \sqrt {2}-x} \, dx}{160 \sqrt {2}}+\frac {(3 i) \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{i \sqrt {2}+x} \, dx}{160 \sqrt {2}}-\frac {i \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^3} \, dx}{40 \sqrt {2}}-\frac {i \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^3} \, dx}{40 \sqrt {2}}-\frac {(3 i) \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^3} \, dx}{40 \sqrt {2}}-\frac {(3 i) \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^3} \, dx}{40 \sqrt {2}}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 4.54, size = 30, normalized size = 0.94 \begin {gather*} e^{\frac {8-e^{-1-x+x^2}+4 x}{20 \left (2+x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((8 - E^(-1 - x + x^2) + 4*x)/(80 + 80*x^2 + 20*x^4))*(8 - 32*x - 12*x^2 + E^(-1 - x + x^2)*(2 +
x^2 - 2*x^3)))/(160 + 240*x^2 + 120*x^4 + 20*x^6),x]

[Out]

E^((8 - E^(-1 - x + x^2) + 4*x)/(20*(2 + x^2)^2))

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fricas [A]  time = 0.61, size = 31, normalized size = 0.97 \begin {gather*} e^{\left (\frac {4 \, x - e^{\left (x^{2} - x - 1\right )} + 8}{20 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+x^2+2)*exp(x^2-x-1)-12*x^2-32*x+8)*exp((-exp(x^2-x-1)+4*x+8)/(20*x^4+80*x^2+80))/(20*x^6+12
0*x^4+240*x^2+160),x, algorithm="fricas")

[Out]

e^(1/20*(4*x - e^(x^2 - x - 1) + 8)/(x^4 + 4*x^2 + 4))

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giac [A]  time = 0.50, size = 54, normalized size = 1.69 \begin {gather*} e^{\left (\frac {x}{5 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}} - \frac {e^{\left (x^{2} - x - 1\right )}}{20 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}} + \frac {2}{5 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+x^2+2)*exp(x^2-x-1)-12*x^2-32*x+8)*exp((-exp(x^2-x-1)+4*x+8)/(20*x^4+80*x^2+80))/(20*x^6+12
0*x^4+240*x^2+160),x, algorithm="giac")

[Out]

e^(1/5*x/(x^4 + 4*x^2 + 4) - 1/20*e^(x^2 - x - 1)/(x^4 + 4*x^2 + 4) + 2/5/(x^4 + 4*x^2 + 4))

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maple [A]  time = 0.14, size = 27, normalized size = 0.84

method	result	size
risch	\({\mathrm e}^{\frac {-{\mathrm e}^{x^{2}-x -1}+4 x +8}{20 \left (x^{2}+2\right )^{2}}}\)	\(27\)
norman	\(\frac {x^{4} {\mathrm e}^{\frac {-{\mathrm e}^{x^{2}-x -1}+4 x +8}{20 x^{4}+80 x^{2}+80}}+4 x^{2} {\mathrm e}^{\frac {-{\mathrm e}^{x^{2}-x -1}+4 x +8}{20 x^{4}+80 x^{2}+80}}+4 \,{\mathrm e}^{\frac {-{\mathrm e}^{x^{2}-x -1}+4 x +8}{20 x^{4}+80 x^{2}+80}}}{\left (x^{2}+2\right )^{2}}\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3+x^2+2)*exp(x^2-x-1)-12*x^2-32*x+8)*exp((-exp(x^2-x-1)+4*x+8)/(20*x^4+80*x^2+80))/(20*x^6+120*x^4+
240*x^2+160),x,method=_RETURNVERBOSE)

[Out]

exp(1/20*(-exp(x^2-x-1)+4*x+8)/(x^2+2)^2)

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maxima [B]  time = 0.54, size = 61, normalized size = 1.91 \begin {gather*} e^{\left (\frac {x}{5 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}} - \frac {e^{\left (x^{2} - x\right )}}{20 \, {\left (x^{4} e + 4 \, x^{2} e + 4 \, e\right )}} + \frac {2}{5 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+x^2+2)*exp(x^2-x-1)-12*x^2-32*x+8)*exp((-exp(x^2-x-1)+4*x+8)/(20*x^4+80*x^2+80))/(20*x^6+12
0*x^4+240*x^2+160),x, algorithm="maxima")

[Out]

e^(1/5*x/(x^4 + 4*x^2 + 4) - 1/20*e^(x^2 - x)/(x^4*e + 4*x^2*e + 4*e) + 2/5/(x^4 + 4*x^2 + 4))

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mupad [B]  time = 8.98, size = 63, normalized size = 1.97 \begin {gather*} {\mathrm {e}}^{\frac {4\,x}{20\,x^4+80\,x^2+80}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-1}}{20\,x^4+80\,x^2+80}}\,{\mathrm {e}}^{\frac {8}{20\,x^4+80\,x^2+80}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((4*x - exp(x^2 - x - 1) + 8)/(80*x^2 + 20*x^4 + 80))*(32*x - exp(x^2 - x - 1)*(x^2 - 2*x^3 + 2) + 12
*x^2 - 8))/(240*x^2 + 120*x^4 + 20*x^6 + 160),x)

[Out]

exp((4*x)/(80*x^2 + 20*x^4 + 80))*exp(-(exp(-x)*exp(x^2)*exp(-1))/(80*x^2 + 20*x^4 + 80))*exp(8/(80*x^2 + 20*x
^4 + 80))

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sympy [A]  time = 0.59, size = 26, normalized size = 0.81 \begin {gather*} e^{\frac {4 x - e^{x^{2} - x - 1} + 8}{20 x^{4} + 80 x^{2} + 80}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3+x**2+2)*exp(x**2-x-1)-12*x**2-32*x+8)*exp((-exp(x**2-x-1)+4*x+8)/(20*x**4+80*x**2+80))/(20
*x**6+120*x**4+240*x**2+160),x)

[Out]

exp((4*x - exp(x**2 - x - 1) + 8)/(20*x**4 + 80*x**2 + 80))

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