Optimal. Leaf size=21 \[ e^3+e^x-\log \left (\frac {1}{2 x}-x\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 16, normalized size of antiderivative = 0.76, number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1593, 6688, 2194, 446, 72} \begin {gather*} -\log \left (1-2 x^2\right )+e^x+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rule 1593
Rule 2194
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1-2 x^2+e^x \left (-x+2 x^3\right )}{x \left (-1+2 x^2\right )} \, dx\\ &=\int \left (e^x+\frac {1+2 x^2}{x \left (1-2 x^2\right )}\right ) \, dx\\ &=\int e^x \, dx+\int \frac {1+2 x^2}{x \left (1-2 x^2\right )} \, dx\\ &=e^x+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+2 x}{(1-2 x) x} \, dx,x,x^2\right )\\ &=e^x+\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x}-\frac {4}{-1+2 x}\right ) \, dx,x,x^2\right )\\ &=e^x+\log (x)-\log \left (1-2 x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 16, normalized size = 0.76 \begin {gather*} e^x+\log (x)-\log \left (1-2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 15, normalized size = 0.71 \begin {gather*} e^{x} - \log \left (2 \, x^{2} - 1\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 15, normalized size = 0.71 \begin {gather*} e^{x} - \log \left (2 \, x^{2} - 1\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 16, normalized size = 0.76
method | result | size |
default | \(\ln \relax (x )-\ln \left (2 x^{2}-1\right )+{\mathrm e}^{x}\) | \(16\) |
norman | \(\ln \relax (x )-\ln \left (2 x^{2}-1\right )+{\mathrm e}^{x}\) | \(16\) |
risch | \(\ln \relax (x )-\ln \left (2 x^{2}-1\right )+{\mathrm e}^{x}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 15, normalized size = 0.71 \begin {gather*} e^{x} - \log \left (2 \, x^{2} - 1\right ) + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 13, normalized size = 0.62 \begin {gather*} {\mathrm {e}}^x-\ln \left (x^2-\frac {1}{2}\right )+\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 14, normalized size = 0.67 \begin {gather*} e^{x} + \log {\relax (x )} - \log {\left (2 x^{2} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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