Optimal. Leaf size=27 \[ 1-\frac {\left (e^{4/x}-x\right ) \left (2+\frac {4 x}{5}\right )}{4 x^3} \]
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Rubi [A] time = 0.22, antiderivative size = 41, normalized size of antiderivative = 1.52, number of steps used = 15, number of rules used = 6, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 14, 37, 6742, 2212, 2209} \begin {gather*} -\frac {e^{4/x}}{2 x^3}+\frac {(x+5)^2}{50 x^2}-\frac {e^{4/x}}{5 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 37
Rule 2209
Rule 2212
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{10} \int \frac {-10 x^2-2 x^3+e^{4/x} \left (20+23 x+4 x^2\right )}{x^5} \, dx\\ &=\frac {1}{10} \int \left (-\frac {2 (5+x)}{x^3}+\frac {e^{4/x} \left (20+23 x+4 x^2\right )}{x^5}\right ) \, dx\\ &=\frac {1}{10} \int \frac {e^{4/x} \left (20+23 x+4 x^2\right )}{x^5} \, dx-\frac {1}{5} \int \frac {5+x}{x^3} \, dx\\ &=\frac {(5+x)^2}{50 x^2}+\frac {1}{10} \int \left (\frac {20 e^{4/x}}{x^5}+\frac {23 e^{4/x}}{x^4}+\frac {4 e^{4/x}}{x^3}\right ) \, dx\\ &=\frac {(5+x)^2}{50 x^2}+\frac {2}{5} \int \frac {e^{4/x}}{x^3} \, dx+2 \int \frac {e^{4/x}}{x^5} \, dx+\frac {23}{10} \int \frac {e^{4/x}}{x^4} \, dx\\ &=-\frac {e^{4/x}}{2 x^3}-\frac {23 e^{4/x}}{40 x^2}-\frac {e^{4/x}}{10 x}+\frac {(5+x)^2}{50 x^2}-\frac {1}{10} \int \frac {e^{4/x}}{x^2} \, dx-\frac {23}{20} \int \frac {e^{4/x}}{x^3} \, dx-\frac {3}{2} \int \frac {e^{4/x}}{x^4} \, dx\\ &=\frac {e^{4/x}}{40}-\frac {e^{4/x}}{2 x^3}-\frac {e^{4/x}}{5 x^2}+\frac {3 e^{4/x}}{16 x}+\frac {(5+x)^2}{50 x^2}+\frac {23}{80} \int \frac {e^{4/x}}{x^2} \, dx+\frac {3}{4} \int \frac {e^{4/x}}{x^3} \, dx\\ &=-\frac {3 e^{4/x}}{64}-\frac {e^{4/x}}{2 x^3}-\frac {e^{4/x}}{5 x^2}+\frac {(5+x)^2}{50 x^2}-\frac {3}{16} \int \frac {e^{4/x}}{x^2} \, dx\\ &=-\frac {e^{4/x}}{2 x^3}-\frac {e^{4/x}}{5 x^2}+\frac {(5+x)^2}{50 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 23, normalized size = 0.85 \begin {gather*} \frac {\left (-e^{4/x}+x\right ) (5+2 x)}{10 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 27, normalized size = 1.00 \begin {gather*} \frac {2 \, x^{2} - {\left (2 \, x + 5\right )} e^{\frac {4}{x}} + 5 \, x}{10 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.56, size = 33, normalized size = 1.22 \begin {gather*} \frac {1}{5 \, x} - \frac {e^{\frac {4}{x}}}{5 \, x^{2}} + \frac {1}{2 \, x^{2}} - \frac {e^{\frac {4}{x}}}{2 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 28, normalized size = 1.04
method | result | size |
risch | \(\frac {5+2 x}{10 x^{2}}-\frac {\left (5+2 x \right ) {\mathrm e}^{\frac {4}{x}}}{10 x^{3}}\) | \(28\) |
derivativedivides | \(\frac {1}{2 x^{2}}+\frac {1}{5 x}-\frac {{\mathrm e}^{\frac {4}{x}}}{5 x^{2}}-\frac {{\mathrm e}^{\frac {4}{x}}}{2 x^{3}}\) | \(38\) |
default | \(\frac {1}{2 x^{2}}+\frac {1}{5 x}-\frac {{\mathrm e}^{\frac {4}{x}}}{5 x^{2}}-\frac {{\mathrm e}^{\frac {4}{x}}}{2 x^{3}}\) | \(38\) |
norman | \(\frac {\frac {x^{2}}{2}+\frac {x^{3}}{5}-\frac {x^{2} {\mathrm e}^{\frac {4}{x}}}{5}-\frac {x \,{\mathrm e}^{\frac {4}{x}}}{2}}{x^{4}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.38, size = 38, normalized size = 1.41 \begin {gather*} \frac {1}{5 \, x} + \frac {1}{2 \, x^{2}} + \frac {1}{128} \, \Gamma \left (4, -\frac {4}{x}\right ) - \frac {23}{640} \, \Gamma \left (3, -\frac {4}{x}\right ) + \frac {1}{40} \, \Gamma \left (2, -\frac {4}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.42, size = 36, normalized size = 1.33 \begin {gather*} -\frac {x^3\,\left (\frac {{\mathrm {e}}^{4/x}}{5}-\frac {1}{2}\right )+\frac {x^2\,{\mathrm {e}}^{4/x}}{2}-\frac {x^4}{5}}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 27, normalized size = 1.00 \begin {gather*} - \frac {- 2 x - 5}{10 x^{2}} + \frac {\left (- 2 x - 5\right ) e^{\frac {4}{x}}}{10 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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