Optimal. Leaf size=18 \[ e^{\left (x+\frac {x}{\frac {1}{e}-\log (2)}\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.24, antiderivative size = 27, normalized size of antiderivative = 1.50, number of steps
used = 7, number of rules used = 4, integrand size = 122, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6, 12, 2257,
2240} \begin {gather*} \exp \left (\frac {x^2 (1+e (1-\log (2)))^2}{(1-e \log (2))^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6
Rule 12
Rule 2240
Rule 2257
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {x^2+2 e x^2+e^2 x^2+\left (-2 e x^2-2 e^2 x^2\right ) \log (2)+e^2 x^2 \log ^2(2)}{1-2 e \log (2)+e^2 \log ^2(2)}\right ) \left (2 e^2 x+(2+4 e) x+\left (-4 e x-4 e^2 x\right ) \log (2)+2 e^2 x \log ^2(2)\right )}{1-2 e \log (2)+e^2 \log ^2(2)} \, dx\\ &=\int \frac {\exp \left (\frac {x^2+2 e x^2+e^2 x^2+\left (-2 e x^2-2 e^2 x^2\right ) \log (2)+e^2 x^2 \log ^2(2)}{1-2 e \log (2)+e^2 \log ^2(2)}\right ) \left (\left (2+4 e+2 e^2\right ) x+\left (-4 e x-4 e^2 x\right ) \log (2)+2 e^2 x \log ^2(2)\right )}{1-2 e \log (2)+e^2 \log ^2(2)} \, dx\\ &=\int \frac {\exp \left (\frac {x^2+2 e x^2+e^2 x^2+\left (-2 e x^2-2 e^2 x^2\right ) \log (2)+e^2 x^2 \log ^2(2)}{1-2 e \log (2)+e^2 \log ^2(2)}\right ) \left (\left (-4 e x-4 e^2 x\right ) \log (2)+x \left (2+4 e+2 e^2+2 e^2 \log ^2(2)\right )\right )}{1-2 e \log (2)+e^2 \log ^2(2)} \, dx\\ &=\frac {\int \exp \left (\frac {x^2+2 e x^2+e^2 x^2+\left (-2 e x^2-2 e^2 x^2\right ) \log (2)+e^2 x^2 \log ^2(2)}{1-2 e \log (2)+e^2 \log ^2(2)}\right ) \left (\left (-4 e x-4 e^2 x\right ) \log (2)+x \left (2+4 e+2 e^2+2 e^2 \log ^2(2)\right )\right ) \, dx}{1-2 e \log (2)+e^2 \log ^2(2)}\\ &=\frac {\int 2 \exp \left (\frac {x^2 (1+e (1-\log (2)))^2}{(1-e \log (2))^2}\right ) x (1+e (1-\log (2)))^2 \, dx}{1-2 e \log (2)+e^2 \log ^2(2)}\\ &=\frac {\left (2 (1+e (1-\log (2)))^2\right ) \int \exp \left (\frac {x^2 (1+e (1-\log (2)))^2}{(1-e \log (2))^2}\right ) x \, dx}{(1-e \log (2))^2}\\ &=\exp \left (\frac {x^2 (1+e (1-\log (2)))^2}{(1-e \log (2))^2}\right )\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(18)=36\).
time = 1.28, size = 64, normalized size = 3.56 \begin {gather*} 2^{-1+\frac {1}{(-1+e \log (2))^2}} e^{\frac {e \log ^2(2) (-2+e \log (2))+x^2 \left (1+e^2 \left (1+\log ^2(2)-\log (4)\right )-e (-2+\log (4))\right )}{(-1+e \log (2))^2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs.
\(2(16)=32\).
time = 0.45, size = 73, normalized size = 4.06
method | result | size |
risch | \({\mathrm e}^{\frac {x^{2} \left (-{\mathrm e}^{2} \ln \left (2\right )^{2}+2 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2}+2 \,{\mathrm e} \ln \left (2\right )-2 \,{\mathrm e}-1\right )}{-{\mathrm e}^{2} \ln \left (2\right )^{2}+2 \,{\mathrm e} \ln \left (2\right )-1}}\) | \(54\) |
gosper | \({\mathrm e}^{\frac {x^{2} \left ({\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e}^{2} \ln \left (2\right )-2 \,{\mathrm e} \ln \left (2\right )+{\mathrm e}^{2}+2 \,{\mathrm e}+1\right )}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}}\) | \(58\) |
derivativedivides | \({\mathrm e}^{\frac {x^{2} {\mathrm e}^{2} \ln \left (2\right )^{2}+\left (-2 x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}\right ) \ln \left (2\right )+x^{2} {\mathrm e}^{2}+2 x^{2} {\mathrm e}+x^{2}}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}}\) | \(73\) |
default | \({\mathrm e}^{\frac {x^{2} {\mathrm e}^{2} \ln \left (2\right )^{2}+\left (-2 x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}\right ) \ln \left (2\right )+x^{2} {\mathrm e}^{2}+2 x^{2} {\mathrm e}+x^{2}}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}}\) | \(73\) |
norman | \({\mathrm e}^{\frac {x^{2} {\mathrm e}^{2} \ln \left (2\right )^{2}+\left (-2 x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}\right ) \ln \left (2\right )+x^{2} {\mathrm e}^{2}+2 x^{2} {\mathrm e}+x^{2}}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}}\) | \(73\) |
meijerg | \(-\frac {\left (2 \,{\mathrm e}^{2} \ln \left (2\right )^{2}-4 \,{\mathrm e}^{2} \ln \left (2\right )+2 \,{\mathrm e}^{2}-4 \,{\mathrm e} \ln \left (2\right )+4 \,{\mathrm e}+2\right ) \left (1-{\mathrm e}^{\frac {x^{2} \left ({\mathrm e}^{2} \ln \left (2\right )^{2}+\left (-2 \,{\mathrm e}^{2}-2 \,{\mathrm e}\right ) \ln \left (2\right )+{\mathrm e}^{2}+2 \,{\mathrm e}+1\right )}{{\mathrm e}^{2} \ln \left (2\right )^{2}-2 \,{\mathrm e} \ln \left (2\right )+1}}\right )}{2 \left ({\mathrm e}^{2} \ln \left (2\right )^{2}+\left (-2 \,{\mathrm e}^{2}-2 \,{\mathrm e}\right ) \ln \left (2\right )+{\mathrm e}^{2}+2 \,{\mathrm e}+1\right )}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1721 vs.
\(2 (16) = 32\).
time = 0.28, size = 1721, normalized size = 95.61 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (16) = 32\).
time = 0.37, size = 63, normalized size = 3.50 \begin {gather*} e^{\left (\frac {x^{2} e^{2} \log \left (2\right )^{2} + x^{2} e^{2} + 2 \, x^{2} e + x^{2} - 2 \, {\left (x^{2} e^{2} + x^{2} e\right )} \log \left (2\right )}{e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (14) = 28\).
time = 0.18, size = 71, normalized size = 3.94 \begin {gather*} e^{\frac {x^{2} + x^{2} e^{2} \log {\left (2 \right )}^{2} + 2 e x^{2} + x^{2} e^{2} + \left (- 2 x^{2} e^{2} - 2 e x^{2}\right ) \log {\left (2 \right )}}{- 2 e \log {\left (2 \right )} + 1 + e^{2} \log {\left (2 \right )}^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 268 vs.
\(2 (16) = 32\).
time = 0.67, size = 268, normalized size = 14.89 \begin {gather*} \frac {{\left (e^{4} \log \left (2\right )^{4} - 2 \, e^{4} \log \left (2\right )^{3} - 4 \, e^{3} \log \left (2\right )^{3} + e^{4} \log \left (2\right )^{2} + 6 \, e^{3} \log \left (2\right )^{2} + 6 \, e^{2} \log \left (2\right )^{2} - 2 \, e^{3} \log \left (2\right ) - 6 \, e^{2} \log \left (2\right ) - 4 \, e \log \left (2\right ) + e^{2} + 2 \, e + 1\right )} e^{\left (\frac {x^{2} e^{2} \log \left (2\right )^{2}}{e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1} - \frac {2 \, x^{2} e^{2} \log \left (2\right )}{e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1} - \frac {2 \, x^{2} e \log \left (2\right )}{e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1} + \frac {x^{2} e^{2}}{e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1} + \frac {2 \, x^{2} e}{e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1} + \frac {x^{2}}{e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1}\right )}}{{\left (e^{2} \log \left (2\right )^{2} - 2 \, e^{2} \log \left (2\right ) - 2 \, e \log \left (2\right ) + e^{2} + 2 \, e + 1\right )} {\left (e^{2} \log \left (2\right )^{2} - 2 \, e \log \left (2\right ) + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.59, size = 133, normalized size = 7.39 \begin {gather*} {\left (\frac {1}{4}\right )}^{\frac {x^2\,\mathrm {e}+x^2\,{\mathrm {e}}^2}{{\mathrm {e}}^2\,{\ln \left (2\right )}^2-2\,\mathrm {e}\,\ln \left (2\right )+1}}\,{\mathrm {e}}^{\frac {x^2}{{\mathrm {e}}^2\,{\ln \left (2\right )}^2-2\,\mathrm {e}\,\ln \left (2\right )+1}}\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^2\,{\ln \left (2\right )}^2}{{\mathrm {e}}^2\,{\ln \left (2\right )}^2-2\,\mathrm {e}\,\ln \left (2\right )+1}}\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^2}{{\mathrm {e}}^2\,{\ln \left (2\right )}^2-2\,\mathrm {e}\,\ln \left (2\right )+1}}\,{\mathrm {e}}^{\frac {2\,x^2\,\mathrm {e}}{{\mathrm {e}}^2\,{\ln \left (2\right )}^2-2\,\mathrm {e}\,\ln \left (2\right )+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________