Optimal. Leaf size=25 \[ e^{\left (1+x^2\right ) \left (4-\frac {4 x}{3}+\log \left (7-x^2\right )\right )^2} \]
________________________________________________________________________________________
Rubi [F] time = 11.58, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (\frac {1}{9} \left (144-96 x+160 x^2-96 x^3+16 x^4+\left (72-24 x+72 x^2-24 x^3\right ) \log \left (7-x^2\right )+\left (9+9 x^2\right ) \log ^2\left (7-x^2\right )\right )\right ) \left (672-2096 x+1872 x^2+16 x^3-336 x^4+64 x^5+\left (168-972 x+480 x^2+180 x^3-72 x^4\right ) \log \left (7-x^2\right )+\left (-126 x+18 x^3\right ) \log ^2\left (7-x^2\right )\right )}{-63+9 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {16 \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \left (42-131 x+117 x^2+x^3-21 x^4+4 x^5\right )}{9 \left (-7+x^2\right )}-\frac {4 \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \left (-14+81 x-40 x^2-15 x^3+6 x^4\right ) \log \left (7-x^2\right )}{3 \left (-7+x^2\right )}+2 \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \log ^2\left (7-x^2\right )\right ) \, dx\\ &=-\left (\frac {4}{3} \int \frac {\exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \left (-14+81 x-40 x^2-15 x^3+6 x^4\right ) \log \left (7-x^2\right )}{-7+x^2} \, dx\right )+\frac {16}{9} \int \frac {\exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \left (42-131 x+117 x^2+x^3-21 x^4+4 x^5\right )}{-7+x^2} \, dx+2 \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \log ^2\left (7-x^2\right ) \, dx\\ &=-\left (\frac {4}{3} \int \left (2 \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \log \left (7-x^2\right )-15 \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \log \left (7-x^2\right )+6 \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x^2 \log \left (7-x^2\right )-\frac {24 \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \log \left (7-x^2\right )}{-7+x^2}\right ) \, dx\right )+\frac {16}{9} \int \left (-30 \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right )+29 \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x-21 \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x^2+4 \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x^3-\frac {24 \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) (7-3 x)}{-7+x^2}\right ) \, dx+2 \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \log ^2\left (7-x^2\right ) \, dx\\ &=2 \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \log ^2\left (7-x^2\right ) \, dx-\frac {8}{3} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \log \left (7-x^2\right ) \, dx+\frac {64}{9} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x^3 \, dx-8 \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x^2 \log \left (7-x^2\right ) \, dx+20 \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \log \left (7-x^2\right ) \, dx+32 \int \frac {\exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \log \left (7-x^2\right )}{-7+x^2} \, dx-\frac {112}{3} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x^2 \, dx-\frac {128}{3} \int \frac {\exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) (7-3 x)}{-7+x^2} \, dx+\frac {464}{9} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \, dx-\frac {160}{3} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \, dx\\ &=2 \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \log ^2\left (7-x^2\right ) \, dx-\frac {8}{3} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \log \left (7-x^2\right ) \, dx+\frac {64}{9} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x^3 \, dx-8 \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x^2 \log \left (7-x^2\right ) \, dx+20 \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \log \left (7-x^2\right ) \, dx+32 \int \left (-\frac {\exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \log \left (7-x^2\right )}{2 \left (\sqrt {7}-x\right )}+\frac {\exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \log \left (7-x^2\right )}{2 \left (\sqrt {7}+x\right )}\right ) \, dx-\frac {112}{3} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x^2 \, dx-\frac {128}{3} \int \left (-\frac {\left (-21+7 \sqrt {7}\right ) \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right )}{14 \left (\sqrt {7}-x\right )}-\frac {\left (21+7 \sqrt {7}\right ) \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right )}{14 \left (\sqrt {7}+x\right )}\right ) \, dx+\frac {464}{9} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \, dx-\frac {160}{3} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \, dx\\ &=2 \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \log ^2\left (7-x^2\right ) \, dx-\frac {8}{3} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \log \left (7-x^2\right ) \, dx+\frac {64}{9} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x^3 \, dx-8 \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x^2 \log \left (7-x^2\right ) \, dx-16 \int \frac {\exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \log \left (7-x^2\right )}{\sqrt {7}-x} \, dx+16 \int \frac {\exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \log \left (7-x^2\right )}{\sqrt {7}+x} \, dx+20 \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \log \left (7-x^2\right ) \, dx-\frac {112}{3} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x^2 \, dx+\frac {464}{9} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) x \, dx-\frac {160}{3} \int \exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right ) \, dx-\frac {1}{3} \left (64 \left (3-\sqrt {7}\right )\right ) \int \frac {\exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right )}{\sqrt {7}-x} \, dx+\frac {1}{3} \left (64 \left (3+\sqrt {7}\right )\right ) \int \frac {\exp \left (\frac {1}{9} \left (1+x^2\right ) \left (-12+4 x-3 \log \left (7-x^2\right )\right )^2\right )}{\sqrt {7}+x} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] time = 0.23, size = 55, normalized size = 2.20 \begin {gather*} e^{\frac {1}{9} \left (1+x^2\right ) \left (16 (-3+x)^2+9 \log ^2\left (7-x^2\right )\right )} \left (7-x^2\right )^{8-\frac {8}{3} x \left (1-3 x+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.65, size = 58, normalized size = 2.32 \begin {gather*} e^{\left (\frac {16}{9} \, x^{4} - \frac {32}{3} \, x^{3} + {\left (x^{2} + 1\right )} \log \left (-x^{2} + 7\right )^{2} + \frac {160}{9} \, x^{2} - \frac {8}{3} \, {\left (x^{3} - 3 \, x^{2} + x - 3\right )} \log \left (-x^{2} + 7\right ) - \frac {32}{3} \, x + 16\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 37.51, size = 92, normalized size = 3.68 \begin {gather*} e^{\left (\frac {16}{9} \, x^{4} - \frac {8}{3} \, x^{3} \log \left (-x^{2} + 7\right ) + x^{2} \log \left (-x^{2} + 7\right )^{2} - \frac {32}{3} \, x^{3} + 8 \, x^{2} \log \left (-x^{2} + 7\right ) + \frac {160}{9} \, x^{2} - \frac {8}{3} \, x \log \left (-x^{2} + 7\right ) + \log \left (-x^{2} + 7\right )^{2} - \frac {32}{3} \, x + 8 \, \log \left (-x^{2} + 7\right ) + 16\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 50, normalized size = 2.00
method | result | size |
risch | \(\left (-x^{2}+7\right )^{-\frac {8 \left (x -3\right ) \left (x^{2}+1\right )}{3}} {\mathrm e}^{\frac {\left (x^{2}+1\right ) \left (9 \ln \left (-x^{2}+7\right )^{2}+16 x^{2}-96 x +144\right )}{9}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.06, size = 142, normalized size = 5.68 \begin {gather*} {\left (x^{16} e^{16} - 56 \, x^{14} e^{16} + 1372 \, x^{12} e^{16} - 19208 \, x^{10} e^{16} + 168070 \, x^{8} e^{16} - 941192 \, x^{6} e^{16} + 3294172 \, x^{4} e^{16} - 6588344 \, x^{2} e^{16} + 5764801 \, e^{16}\right )} e^{\left (\frac {16}{9} \, x^{4} - \frac {8}{3} \, x^{3} \log \left (-x^{2} + 7\right ) + x^{2} \log \left (-x^{2} + 7\right )^{2} - \frac {32}{3} \, x^{3} + 8 \, x^{2} \log \left (-x^{2} + 7\right ) + \frac {160}{9} \, x^{2} - \frac {8}{3} \, x \log \left (-x^{2} + 7\right ) + \log \left (-x^{2} + 7\right )^{2} - \frac {32}{3} \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.34, size = 780, normalized size = 31.20 \begin {gather*} 5764801\,{\mathrm {e}}^{\frac {16\,x^4}{9}-\frac {8\,x^3\,\ln \left (7-x^2\right )}{3}-\frac {32\,x^3}{3}+x^2\,{\ln \left (7-x^2\right )}^2+\frac {160\,x^2}{9}-\frac {8\,x\,\ln \left (7-x^2\right )}{3}-\frac {32\,x}{3}+{\ln \left (7-x^2\right )}^2+16}\,{\left (7-x^2\right )}^{8\,x^2}-6588344\,x^2\,{\mathrm {e}}^{\frac {16\,x^4}{9}-\frac {8\,x^3\,\ln \left (7-x^2\right )}{3}-\frac {32\,x^3}{3}+x^2\,{\ln \left (7-x^2\right )}^2+\frac {160\,x^2}{9}-\frac {8\,x\,\ln \left (7-x^2\right )}{3}-\frac {32\,x}{3}+{\ln \left (7-x^2\right )}^2+16}\,{\left (7-x^2\right )}^{8\,x^2}+3294172\,x^4\,{\mathrm {e}}^{\frac {16\,x^4}{9}-\frac {8\,x^3\,\ln \left (7-x^2\right )}{3}-\frac {32\,x^3}{3}+x^2\,{\ln \left (7-x^2\right )}^2+\frac {160\,x^2}{9}-\frac {8\,x\,\ln \left (7-x^2\right )}{3}-\frac {32\,x}{3}+{\ln \left (7-x^2\right )}^2+16}\,{\left (7-x^2\right )}^{8\,x^2}-941192\,x^6\,{\mathrm {e}}^{\frac {16\,x^4}{9}-\frac {8\,x^3\,\ln \left (7-x^2\right )}{3}-\frac {32\,x^3}{3}+x^2\,{\ln \left (7-x^2\right )}^2+\frac {160\,x^2}{9}-\frac {8\,x\,\ln \left (7-x^2\right )}{3}-\frac {32\,x}{3}+{\ln \left (7-x^2\right )}^2+16}\,{\left (7-x^2\right )}^{8\,x^2}+168070\,x^8\,{\mathrm {e}}^{\frac {16\,x^4}{9}-\frac {8\,x^3\,\ln \left (7-x^2\right )}{3}-\frac {32\,x^3}{3}+x^2\,{\ln \left (7-x^2\right )}^2+\frac {160\,x^2}{9}-\frac {8\,x\,\ln \left (7-x^2\right )}{3}-\frac {32\,x}{3}+{\ln \left (7-x^2\right )}^2+16}\,{\left (7-x^2\right )}^{8\,x^2}-19208\,x^{10}\,{\mathrm {e}}^{\frac {16\,x^4}{9}-\frac {8\,x^3\,\ln \left (7-x^2\right )}{3}-\frac {32\,x^3}{3}+x^2\,{\ln \left (7-x^2\right )}^2+\frac {160\,x^2}{9}-\frac {8\,x\,\ln \left (7-x^2\right )}{3}-\frac {32\,x}{3}+{\ln \left (7-x^2\right )}^2+16}\,{\left (7-x^2\right )}^{8\,x^2}+1372\,x^{12}\,{\mathrm {e}}^{\frac {16\,x^4}{9}-\frac {8\,x^3\,\ln \left (7-x^2\right )}{3}-\frac {32\,x^3}{3}+x^2\,{\ln \left (7-x^2\right )}^2+\frac {160\,x^2}{9}-\frac {8\,x\,\ln \left (7-x^2\right )}{3}-\frac {32\,x}{3}+{\ln \left (7-x^2\right )}^2+16}\,{\left (7-x^2\right )}^{8\,x^2}-56\,x^{14}\,{\mathrm {e}}^{\frac {16\,x^4}{9}-\frac {8\,x^3\,\ln \left (7-x^2\right )}{3}-\frac {32\,x^3}{3}+x^2\,{\ln \left (7-x^2\right )}^2+\frac {160\,x^2}{9}-\frac {8\,x\,\ln \left (7-x^2\right )}{3}-\frac {32\,x}{3}+{\ln \left (7-x^2\right )}^2+16}\,{\left (7-x^2\right )}^{8\,x^2}+x^{16}\,{\mathrm {e}}^{\frac {16\,x^4}{9}-\frac {8\,x^3\,\ln \left (7-x^2\right )}{3}-\frac {32\,x^3}{3}+x^2\,{\ln \left (7-x^2\right )}^2+\frac {160\,x^2}{9}-\frac {8\,x\,\ln \left (7-x^2\right )}{3}-\frac {32\,x}{3}+{\ln \left (7-x^2\right )}^2+16}\,{\left (7-x^2\right )}^{8\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.07, size = 66, normalized size = 2.64 \begin {gather*} e^{\frac {16 x^{4}}{9} - \frac {32 x^{3}}{3} + \frac {160 x^{2}}{9} - \frac {32 x}{3} + \left (x^{2} + 1\right ) \log {\left (7 - x^{2} \right )}^{2} + \left (- \frac {8 x^{3}}{3} + 8 x^{2} - \frac {8 x}{3} + 8\right ) \log {\left (7 - x^{2} \right )} + 16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________