Optimal. Leaf size=22 \[ -e^{-4-x}-\frac {2}{x}+\frac {16 x^2}{9} \]
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Rubi [A]
time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 14, 2225}
\begin {gather*} \frac {16 x^2}{9}-e^{-x-4}-\frac {2}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2225
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {18+9 e^{-4-x} x^2+32 x^3}{x^2} \, dx\\ &=\frac {1}{9} \int \left (9 e^{-4-x}+\frac {2 \left (9+16 x^3\right )}{x^2}\right ) \, dx\\ &=\frac {2}{9} \int \frac {9+16 x^3}{x^2} \, dx+\int e^{-4-x} \, dx\\ &=-e^{-4-x}+\frac {2}{9} \int \left (\frac {9}{x^2}+16 x\right ) \, dx\\ &=-e^{-4-x}-\frac {2}{x}+\frac {16 x^2}{9}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 22, normalized size = 1.00 \begin {gather*} -e^{-4-x}-\frac {2}{x}+\frac {16 x^2}{9} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.08, size = 28, normalized size = 1.27
method | result | size |
risch | \(\frac {16 x^{2}}{9}-{\mathrm e}^{-x -4}-\frac {2}{x}\) | \(20\) |
norman | \(\frac {-2+\frac {16 x^{3}}{9}-x \,{\mathrm e}^{-x -4}}{x}\) | \(21\) |
derivativedivides | \(-\frac {2}{x}-\frac {128 x}{9}-\frac {512}{9}+\frac {16 \left (-x -4\right )^{2}}{9}-{\mathrm e}^{-x -4}\) | \(28\) |
default | \(-\frac {2}{x}-\frac {128 x}{9}-\frac {512}{9}+\frac {16 \left (-x -4\right )^{2}}{9}-{\mathrm e}^{-x -4}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 19, normalized size = 0.86 \begin {gather*} \frac {16}{9} \, x^{2} - \frac {2}{x} - e^{\left (-x - 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 21, normalized size = 0.95 \begin {gather*} \frac {16 \, x^{3} - 9 \, x e^{\left (-x - 4\right )} - 18}{9 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 15, normalized size = 0.68 \begin {gather*} \frac {16 x^{2}}{9} - e^{- x - 4} - \frac {2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs.
\(2 (19) = 38\).
time = 0.40, size = 43, normalized size = 1.95 \begin {gather*} \frac {16 \, {\left (x + 4\right )}^{3} - 192 \, {\left (x + 4\right )}^{2} - 9 \, {\left (x + 4\right )} e^{\left (-x - 4\right )} + 512 \, x + 36 \, e^{\left (-x - 4\right )} + 2030}{9 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 19, normalized size = 0.86 \begin {gather*} \frac {16\,x^2}{9}-\frac {2}{x}-{\mathrm {e}}^{-x-4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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