Optimal. Leaf size=35 \[ \frac {x}{\log \left (x \left (\frac {1}{5}-\frac {e^{-x^2} x+5 \left (\frac {3}{x}+x\right )}{x}\right )\right )} \]
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Rubi [F] time = 5.31, 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 {-5 x^2+10 x^4+e^{x^2} \left (75-24 x^2\right )+\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log \left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )}{\left (5 x^2+e^{x^2} \left (75+24 x^2\right )\right ) \log ^2\left (\frac {e^{-x^2} \left (-5 x^2+e^{x^2} \left (-75-24 x^2\right )\right )}{5 x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {10 x^2 \left (-25+25 x^2+8 x^4\right )}{\left (25+8 x^2\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {25-8 x^2+25 \log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )+8 x^2 \log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}\right ) \, dx\\ &=10 \int \frac {x^2 \left (-25+25 x^2+8 x^4\right )}{\left (25+8 x^2\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \frac {25-8 x^2+25 \log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )+8 x^2 \log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ &=10 \int \left (-\frac {25}{8 \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {625}{8 \left (25+8 x^2\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}\right ) \, dx+\int \frac {25-8 x^2+\left (25+8 x^2\right ) \log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ &=10 \int \frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\frac {125}{4} \int \frac {1}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {3125}{4} \int \frac {1}{\left (25+8 x^2\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \left (\frac {25-8 x^2}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {1}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}\right ) \, dx\\ &=10 \int \frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\frac {125}{4} \int \frac {1}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {3125}{4} \int \left (\frac {i}{10 \left (5 i-2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {i}{10 \left (5 i+2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}\right ) \, dx+\int \frac {25-8 x^2}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \frac {1}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ &=\frac {625}{8} i \int \frac {1}{\left (5 i-2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {625}{8} i \int \frac {1}{\left (5 i+2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+10 \int \frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\frac {125}{4} \int \frac {1}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \left (-\frac {1}{\log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {50}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}\right ) \, dx+\int \frac {1}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ &=\frac {625}{8} i \int \frac {1}{\left (5 i-2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {625}{8} i \int \frac {1}{\left (5 i+2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+10 \int \frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\frac {125}{4} \int \frac {1}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+50 \int \frac {1}{\left (25+8 x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\int \frac {1}{\log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \frac {1}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ &=\frac {625}{8} i \int \frac {1}{\left (5 i-2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {625}{8} i \int \frac {1}{\left (5 i+2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+10 \int \frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\frac {125}{4} \int \frac {1}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+50 \int \left (\frac {i}{10 \left (5 i-2 \sqrt {2} x\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}+\frac {i}{10 \left (5 i+2 \sqrt {2} x\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )}\right ) \, dx-\int \frac {1}{\log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \frac {1}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ &=5 i \int \frac {1}{\left (5 i-2 \sqrt {2} x\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+5 i \int \frac {1}{\left (5 i+2 \sqrt {2} x\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {625}{8} i \int \frac {1}{\left (5 i-2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\frac {625}{8} i \int \frac {1}{\left (5 i+2 \sqrt {2} x\right ) \left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+10 \int \frac {x^4}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\frac {125}{4} \int \frac {1}{\left (75 e^{x^2}+5 x^2+24 e^{x^2} x^2\right ) \log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx-\int \frac {1}{\log ^2\left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx+\int \frac {1}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 26, normalized size = 0.74 \begin {gather*} \frac {x}{\log \left (-\frac {15}{x}+\left (-\frac {24}{5}-e^{-x^2}\right ) x\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 35, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (-\frac {{\left (5 \, x^{2} + 3 \, {\left (8 \, x^{2} + 25\right )} e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}}{5 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 37, normalized size = 1.06 \begin {gather*} \frac {x}{\log \left (-\frac {{\left (24 \, x^{2} e^{\left (x^{2}\right )} + 5 \, x^{2} + 75 \, e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}}{5 \, x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.24, size = 486, normalized size = 13.89
method | result | size |
risch | \(\frac {2 i x}{2 \pi \mathrm {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right )-\pi \,\mathrm {csgn}\left (i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x^{2}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right )^{3}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x^{2}} \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )-\pi \mathrm {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )^{3}-\pi \mathrm {csgn}\left (\frac {i \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right ) {\mathrm e}^{-x^{2}}}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )-2 \pi +2 i \ln \left (\left ({\mathrm e}^{x^{2}}+\frac {5}{24}\right ) x^{2}+\frac {25 \,{\mathrm e}^{x^{2}}}{8}\right )-2 i \ln \relax (x )-2 i \ln \relax (5)-2 i \ln \left ({\mathrm e}^{x^{2}}\right )+2 i \ln \relax (3)+6 i \ln \relax (2)}\) | \(486\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 35, normalized size = 1.00 \begin {gather*} -\frac {x}{x^{2} + \log \relax (5) - \log \left (-5 \, x^{2} - 3 \, {\left (8 \, x^{2} + 25\right )} e^{\left (x^{2}\right )}\right ) + \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {{\mathrm {e}}^{x^2}\,\left (24\,x^2-75\right )-\ln \left (-\frac {{\mathrm {e}}^{-x^2}\,\left (\frac {{\mathrm {e}}^{x^2}\,\left (24\,x^2+75\right )}{5}+x^2\right )}{x}\right )\,\left ({\mathrm {e}}^{x^2}\,\left (24\,x^2+75\right )+5\,x^2\right )+5\,x^2-10\,x^4}{{\ln \left (-\frac {{\mathrm {e}}^{-x^2}\,\left (\frac {{\mathrm {e}}^{x^2}\,\left (24\,x^2+75\right )}{5}+x^2\right )}{x}\right )}^2\,\left ({\mathrm {e}}^{x^2}\,\left (24\,x^2+75\right )+5\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 27, normalized size = 0.77 \begin {gather*} \frac {x}{\log {\left (\frac {\left (- x^{2} + \frac {\left (- 24 x^{2} - 75\right ) e^{x^{2}}}{5}\right ) e^{- x^{2}}}{x} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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