Optimal. Leaf size=19 \[ -\frac {e^t}{1+t}+\frac {\text {Ei}(1+t)}{e} \]
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Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2208, 2209}
\begin {gather*} \frac {\text {ExpIntegralEi}(t+1)}{e}-\frac {e^t}{t+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 2208
Rule 2209
Rubi steps
\begin {align*} \int \frac {e^t}{(1+t)^2} \, dt &=-\frac {e^t}{1+t}+\int \frac {e^t}{1+t} \, dt\\ &=-\frac {e^t}{1+t}+\frac {\text {Ei}(1+t)}{e}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 19, normalized size = 1.00 \begin {gather*} -\frac {e^t}{1+t}+\frac {\text {Ei}(1+t)}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 22, normalized size = 1.16
method | result | size |
default | \(-\frac {{\mathrm e}^{t}}{1+t}-{\mathrm e}^{-1} \expIntegral \left (1, -1-t \right )\) | \(22\) |
risch | \(-\frac {{\mathrm e}^{t}}{1+t}-{\mathrm e}^{-1} \expIntegral \left (1, -1-t \right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.81, size = 16, normalized size = 0.84 \begin {gather*} -\frac {e^{\left (-1\right )} E_{2}\left (-t - 1\right )}{t + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.64, size = 23, normalized size = 1.21 \begin {gather*} \frac {{\left ({\left (t + 1\right )} {\rm Ei}\left (t + 1\right ) - e^{\left (t + 1\right )}\right )} e^{\left (-1\right )}}{t + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{t}}{\left (t + 1\right )^{2}}\, dt \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. 80 vs.
\(2 (17) = 34\).
time = 0.47, size = 80, normalized size = 4.21 \begin {gather*} \frac {{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} {\rm Ei}\left (-{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} + 1\right ) - {\rm Ei}\left (-{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} + 1\right ) + e^{\left (-{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} + 1\right )}}{{\left (t + 1\right )} {\left (\frac {1}{t + 1} - 1\right )} e - e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 17, normalized size = 0.89 \begin {gather*} \mathrm {ei}\left (t+1\right )\,{\mathrm {e}}^{-1}-\frac {{\mathrm {e}}^t}{t+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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