Optimal. Leaf size=30 \[ e^{-x+\frac {4}{-\frac {5}{x}-x+\frac {5 \log (5)}{x}}}+2 x \]
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Rubi [F]
time = 7.97, 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 {50+20 x^2+2 x^4+\left (-100-20 x^2\right ) \log (5)+50 \log ^2(5)+e^{\frac {9 x+x^3-5 x \log (5)}{-5-x^2+5 \log (5)}} \left (-45-6 x^2-x^4+\left (70+10 x^2\right ) \log (5)-25 \log ^2(5)\right )}{25+10 x^2+x^4+\left (-50-10 x^2\right ) \log (5)+25 \log ^2(5)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {20 x^2+2 x^4+\left (-100-20 x^2\right ) \log (5)+e^{\frac {9 x+x^3-5 x \log (5)}{-5-x^2+5 \log (5)}} \left (-45-6 x^2-x^4+\left (70+10 x^2\right ) \log (5)-25 \log ^2(5)\right )+50 \left (1+\log ^2(5)\right )}{x^4+10 x^2 (1-\log (5))+25 (1-\log (5))^2} \, dx\\ &=\int \frac {20 x^2+2 x^4+\left (-100-20 x^2\right ) \log (5)+e^{\frac {9 x+x^3-5 x \log (5)}{-5-x^2+5 \log (5)}} \left (-45-6 x^2-x^4+\left (70+10 x^2\right ) \log (5)-25 \log ^2(5)\right )+50 \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right )^2} \, dx\\ &=\int \left (\frac {20 x^2}{\left (5+x^2-5 \log (5)\right )^2}+\frac {2 x^4}{\left (5+x^2-5 \log (5)\right )^2}+\frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \left (-x^4-2 x^2 (3-5 \log (5))-5 (9-5 \log (5)) (1-\log (5))\right )}{\left (5+x^2-5 \log (5)\right )^2}-\frac {20 \left (5+x^2\right ) \log (5)}{\left (5+x^2-5 \log (5)\right )^2}+\frac {50 \left (1+\log ^2(5)\right )}{\left (5+x^2-5 \log (5)\right )^2}\right ) \, dx\\ &=2 \int \frac {x^4}{\left (5+x^2-5 \log (5)\right )^2} \, dx+20 \int \frac {x^2}{\left (5+x^2-5 \log (5)\right )^2} \, dx-(20 \log (5)) \int \frac {5+x^2}{\left (5+x^2-5 \log (5)\right )^2} \, dx+\left (50 \left (1+\log ^2(5)\right )\right ) \int \frac {1}{\left (5+x^2-5 \log (5)\right )^2} \, dx+\int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \left (-x^4-2 x^2 (3-5 \log (5))-5 (9-5 \log (5)) (1-\log (5))\right )}{\left (5+x^2-5 \log (5)\right )^2} \, dx\\ &=-\frac {10 x}{x^2+5 (1-\log (5))}-\frac {x^3}{x^2+5 (1-\log (5))}-\frac {10 x \log ^2(5)}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {5 x \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+3 \int \frac {x^2}{5+x^2-5 \log (5)} \, dx+10 \int \frac {1}{5+x^2-5 \log (5)} \, dx-\frac {(10 (2-\log (5)) \log (5)) \int \frac {1}{5+x^2-5 \log (5)} \, dx}{1-\log (5)}+\frac {\left (5 \left (1+\log ^2(5)\right )\right ) \int \frac {1}{5+x^2-5 \log (5)} \, dx}{1-\log (5)}+\int \left (-5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}+\frac {4\ 5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{5+x^2-5 \log (5)}+\frac {8\ 5^{1+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} (-1+\log (5))}{\left (5+x^2-5 \log (5)\right )^2}\right ) \, dx\\ &=3 x-\frac {10 x}{x^2+5 (1-\log (5))}-\frac {x^3}{x^2+5 (1-\log (5))}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {\frac {5}{-1+\log (5)}}-\frac {2 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) (2-\log (5)) \log (5)}{(-1+\log (5))^{3/2}}-\frac {10 x \log ^2(5)}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {5 x \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {\sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \left (1+\log ^2(5)\right )}{(-1+\log (5))^{3/2}}+4 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{5+x^2-5 \log (5)} \, dx-(8 (1-\log (5))) \int \frac {5^{1+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (5+x^2-5 \log (5)\right )^2} \, dx-(15 (1-\log (5))) \int \frac {1}{5+x^2-5 \log (5)} \, dx-\int 5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \, dx\\ &=3 x-\frac {10 x}{x^2+5 (1-\log (5))}-\frac {x^3}{x^2+5 (1-\log (5))}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {\frac {5}{-1+\log (5)}}-3 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {5 (-1+\log (5))}-\frac {2 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) (2-\log (5)) \log (5)}{(-1+\log (5))^{3/2}}-\frac {10 x \log ^2(5)}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {5 x \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {\sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \left (1+\log ^2(5)\right )}{(-1+\log (5))^{3/2}}+4 \int \left (\frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \sqrt {-1+\log (5)}}{2 \left (-x+\sqrt {5 (-1+\log (5))}\right ) (5-5 \log (5))}+\frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \sqrt {-1+\log (5)}}{2 \left (x+\sqrt {5 (-1+\log (5))}\right ) (5-5 \log (5))}\right ) \, dx-(8 (1-\log (5))) \int \left (\frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{2 \left (-x^2+5 (-1+\log (5))\right ) (-1+\log (5))}+\frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{4 \left (-x+\sqrt {5 (-1+\log (5))}\right )^2 (-1+\log (5))}+\frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{4 \left (x+\sqrt {5 (-1+\log (5))}\right )^2 (-1+\log (5))}\right ) \, dx-\int 5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \, dx\\ &=3 x-\frac {10 x}{x^2+5 (1-\log (5))}-\frac {x^3}{x^2+5 (1-\log (5))}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {\frac {5}{-1+\log (5)}}-3 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {5 (-1+\log (5))}-\frac {2 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) (2-\log (5)) \log (5)}{(-1+\log (5))^{3/2}}-\frac {10 x \log ^2(5)}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {5 x \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {\sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \left (1+\log ^2(5)\right )}{(-1+\log (5))^{3/2}}+2 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (-x+\sqrt {5 (-1+\log (5))}\right )^2} \, dx+2 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (x+\sqrt {5 (-1+\log (5))}\right )^2} \, dx+4 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{-x^2+5 (-1+\log (5))} \, dx-\frac {2 \int \frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{-x+\sqrt {5 (-1+\log (5))}} \, dx}{5 \sqrt {-1+\log (5)}}-\frac {2 \int \frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{x+\sqrt {5 (-1+\log (5))}} \, dx}{5 \sqrt {-1+\log (5)}}-\int 5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \, dx\\ &=3 x-\frac {10 x}{x^2+5 (1-\log (5))}-\frac {x^3}{x^2+5 (1-\log (5))}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {\frac {5}{-1+\log (5)}}-3 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {5 (-1+\log (5))}-\frac {2 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) (2-\log (5)) \log (5)}{(-1+\log (5))^{3/2}}-\frac {10 x \log ^2(5)}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {5 x \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {\sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \left (1+\log ^2(5)\right )}{(-1+\log (5))^{3/2}}+2 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (-x+\sqrt {5 (-1+\log (5))}\right )^2} \, dx+2 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (x+\sqrt {5 (-1+\log (5))}\right )^2} \, dx+4 \int \left (\frac {5^{-\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{2 \left (-x+\sqrt {5 (-1+\log (5))}\right ) \sqrt {-1+\log (5)}}+\frac {5^{-\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{2 \left (x+\sqrt {5 (-1+\log (5))}\right ) \sqrt {-1+\log (5)}}\right ) \, dx-\frac {2 \int \frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{-x+\sqrt {5 (-1+\log (5))}} \, dx}{5 \sqrt {-1+\log (5)}}-\frac {2 \int \frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{x+\sqrt {5 (-1+\log (5))}} \, dx}{5 \sqrt {-1+\log (5)}}-\int 5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \, dx\\ &=3 x-\frac {10 x}{x^2+5 (1-\log (5))}-\frac {x^3}{x^2+5 (1-\log (5))}-2 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {\frac {5}{-1+\log (5)}}-3 \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \sqrt {5 (-1+\log (5))}-\frac {2 \sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) (2-\log (5)) \log (5)}{(-1+\log (5))^{3/2}}-\frac {10 x \log ^2(5)}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {5 x \left (1+\log ^2(5)\right )}{\left (x^2+5 (1-\log (5))\right ) (1-\log (5))}+\frac {\sqrt {5} \tanh ^{-1}\left (\frac {x}{\sqrt {5 (-1+\log (5))}}\right ) \left (1+\log ^2(5)\right )}{(-1+\log (5))^{3/2}}+2 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (-x+\sqrt {5 (-1+\log (5))}\right )^2} \, dx+2 \int \frac {5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{\left (x+\sqrt {5 (-1+\log (5))}\right )^2} \, dx-\frac {2 \int \frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{-x+\sqrt {5 (-1+\log (5))}} \, dx}{5 \sqrt {-1+\log (5)}}-\frac {2 \int \frac {5^{\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{x+\sqrt {5 (-1+\log (5))}} \, dx}{5 \sqrt {-1+\log (5)}}+\frac {2 \int \frac {5^{-\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{-x+\sqrt {5 (-1+\log (5))}} \, dx}{\sqrt {-1+\log (5)}}+\frac {2 \int \frac {5^{-\frac {1}{2}+\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}}}{x+\sqrt {5 (-1+\log (5))}} \, dx}{\sqrt {-1+\log (5)}}-\int 5^{\frac {5 x}{5+x^2-5 \log (5)}} e^{-\frac {x \left (9+x^2\right )}{5+x^2-5 \log (5)}} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 24, normalized size = 0.80 \begin {gather*} e^{-x-\frac {4 x}{5+x^2-5 \log (5)}}+2 x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.00, size = 33, normalized size = 1.10
method | result | size |
risch | \(2 x +{\mathrm e}^{-\frac {x \left (-x^{2}+5 \ln \left (5\right )-9\right )}{5 \ln \left (5\right )-x^{2}-5}}\) | \(33\) |
norman | \(\frac {\left (5 \ln \left (5\right )-5\right ) {\mathrm e}^{\frac {-5 x \ln \left (5\right )+x^{3}+9 x}{5 \ln \left (5\right )-x^{2}-5}}+\left (10 \ln \left (5\right )-10\right ) x -2 x^{3}-x^{2} {\mathrm e}^{\frac {-5 x \ln \left (5\right )+x^{3}+9 x}{5 \ln \left (5\right )-x^{2}-5}}}{5 \ln \left (5\right )-x^{2}-5}\) | \(95\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 396 vs.
\(2 (27) = 54\).
time = 0.56, size = 396, normalized size = 13.20 \begin {gather*} -\frac {5}{2} \, {\left (\frac {2 \, x}{x^{2} {\left (\log \left (5\right ) - 1\right )} - 5 \, \log \left (5\right )^{2} + 10 \, \log \left (5\right ) - 5} + \frac {\log \left (\frac {x - \sqrt {5 \, \log \left (5\right ) - 5}}{x + \sqrt {5 \, \log \left (5\right ) - 5}}\right )}{\sqrt {5 \, \log \left (5\right ) - 5} {\left (\log \left (5\right ) - 1\right )}}\right )} \log \left (5\right )^{2} - 5 \, {\left (\frac {\log \left (\frac {x - \sqrt {5 \, \log \left (5\right ) - 5}}{x + \sqrt {5 \, \log \left (5\right ) - 5}}\right )}{\sqrt {5 \, \log \left (5\right ) - 5}} - \frac {2 \, x}{x^{2} - 5 \, \log \left (5\right ) + 5}\right )} \log \left (5\right ) + 5 \, {\left (\frac {2 \, x}{x^{2} {\left (\log \left (5\right ) - 1\right )} - 5 \, \log \left (5\right )^{2} + 10 \, \log \left (5\right ) - 5} + \frac {\log \left (\frac {x - \sqrt {5 \, \log \left (5\right ) - 5}}{x + \sqrt {5 \, \log \left (5\right ) - 5}}\right )}{\sqrt {5 \, \log \left (5\right ) - 5} {\left (\log \left (5\right ) - 1\right )}}\right )} \log \left (5\right ) + \frac {15 \, {\left (\log \left (5\right ) - 1\right )} \log \left (\frac {x - \sqrt {5 \, \log \left (5\right ) - 5}}{x + \sqrt {5 \, \log \left (5\right ) - 5}}\right )}{2 \, \sqrt {5 \, \log \left (5\right ) - 5}} + 2 \, x - \frac {5 \, x {\left (\log \left (5\right ) - 1\right )}}{x^{2} - 5 \, \log \left (5\right ) + 5} + \frac {5 \, \log \left (\frac {x - \sqrt {5 \, \log \left (5\right ) - 5}}{x + \sqrt {5 \, \log \left (5\right ) - 5}}\right )}{\sqrt {5 \, \log \left (5\right ) - 5}} - \frac {5 \, x}{x^{2} {\left (\log \left (5\right ) - 1\right )} - 5 \, \log \left (5\right )^{2} + 10 \, \log \left (5\right ) - 5} - \frac {10 \, x}{x^{2} - 5 \, \log \left (5\right ) + 5} - \frac {5 \, \log \left (\frac {x - \sqrt {5 \, \log \left (5\right ) - 5}}{x + \sqrt {5 \, \log \left (5\right ) - 5}}\right )}{2 \, \sqrt {5 \, \log \left (5\right ) - 5} {\left (\log \left (5\right ) - 1\right )}} + e^{\left (-x - \frac {4 \, x}{x^{2} - 5 \, \log \left (5\right ) + 5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 30, normalized size = 1.00 \begin {gather*} 2 \, x + e^{\left (-\frac {x^{3} - 5 \, x \log \left (5\right ) + 9 \, x}{x^{2} - 5 \, \log \left (5\right ) + 5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 27, normalized size = 0.90 \begin {gather*} 2 x + e^{\frac {x^{3} - 5 x \log {\left (5 \right )} + 9 x}{- x^{2} - 5 + 5 \log {\left (5 \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.53, size = 30, normalized size = 1.00 \begin {gather*} 2 \, x + e^{\left (-\frac {x^{3} - 5 \, x \log \left (5\right ) + 9 \, x}{x^{2} - 5 \, \log \left (5\right ) + 5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 53, normalized size = 1.77 \begin {gather*} 2\,x+5^{\frac {5\,x}{x^2-5\,\ln \left (5\right )+5}}\,{\mathrm {e}}^{-\frac {9\,x}{x^2-5\,\ln \left (5\right )+5}}\,{\mathrm {e}}^{-\frac {x^3}{x^2-5\,\ln \left (5\right )+5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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