Optimal. Leaf size=23 \[ e^{-1+e^{x^2}} \left (4-2 x+\log \left (\left (5+e^x\right )^2\right )\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(23)=46\).
time = 0.46, antiderivative size = 74, normalized size of antiderivative = 3.22, number of steps
used = 3, number of rules used = 3, integrand size = 85, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {6820, 12, 2326}
\begin {gather*} \frac {e^{-x^2+e^{x^2}-1} \left (10 e^{x^2} (2-x) x+2 e^{x^2+x} (2-x) x+e^{x^2} \left (e^x+5\right ) x \log \left (\left (e^x+5\right )^2\right )\right )}{\left (e^x+5\right ) x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2326
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{-1+e^{x^2}} \left (-5-10 e^{x^2} (-2+x) x-2 e^{x+x^2} (-2+x) x+e^{x^2} \left (5+e^x\right ) x \log \left (\left (5+e^x\right )^2\right )\right )}{5+e^x} \, dx\\ &=2 \int \frac {e^{-1+e^{x^2}} \left (-5-10 e^{x^2} (-2+x) x-2 e^{x+x^2} (-2+x) x+e^{x^2} \left (5+e^x\right ) x \log \left (\left (5+e^x\right )^2\right )\right )}{5+e^x} \, dx\\ &=\frac {e^{-1+e^{x^2}-x^2} \left (10 e^{x^2} (2-x) x+2 e^{x+x^2} (2-x) x+e^{x^2} \left (5+e^x\right ) x \log \left (\left (5+e^x\right )^2\right )\right )}{\left (5+e^x\right ) x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.22, size = 23, normalized size = 1.00 \begin {gather*} e^{-1+e^{x^2}} \left (4-2 x+\log \left (\left (5+e^x\right )^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.19, size = 94, normalized size = 4.09
method | result | size |
risch | \(2 \,{\mathrm e}^{{\mathrm e}^{x^{2}}-1} \ln \left ({\mathrm e}^{x}+5\right )-\frac {\left (i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+5\right )\right )^{2} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+5\right )^{2}\right )-2 i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+5\right )\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+5\right )^{2}\right )^{2}+i \pi \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+5\right )^{2}\right )^{3}+4 x -8\right ) {\mathrm e}^{{\mathrm e}^{x^{2}}-1}}{2}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.73, size = 19, normalized size = 0.83 \begin {gather*} -2 \, {\left (x - \log \left (e^{x} + 5\right ) - 2\right )} e^{\left (e^{\left (x^{2}\right )} - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs.
\(2 (23) = 46\).
time = 0.31, size = 47, normalized size = 2.04 \begin {gather*} -{\left (2 \, {\left (x - 2\right )} e^{\left (x^{2} + e^{\left (x^{2}\right )} - 1\right )} - e^{\left (x^{2} + e^{\left (x^{2}\right )} - 1\right )} \log \left (e^{\left (2 \, x\right )} + 10 \, e^{x} + 25\right )\right )} e^{\left (-x^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.66, size = 24, normalized size = 1.04 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{x^2}-1}\,\left (\ln \left ({\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^x+25\right )-2\,x+4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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