Optimal. Leaf size=56 \[ -x+\frac {1}{10} \left (5+\sqrt {5}\right ) \log \left (2 e^x+1-\sqrt {5}\right )+\frac {1}{10} \left (5-\sqrt {5}\right ) \log \left (2 e^x+1+\sqrt {5}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2282, 705, 29, 632, 31} \[ -x+\frac {1}{10} \left (5+\sqrt {5}\right ) \log \left (2 e^x+1-\sqrt {5}\right )+\frac {1}{10} \left (5-\sqrt {5}\right ) \log \left (2 e^x+1+\sqrt {5}\right ) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 632
Rule 705
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{-1+e^x+e^{2 x}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \left (-1+x+x^2\right )} \, dx,x,e^x\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )-\operatorname {Subst}\left (\int \frac {-1-x}{-1+x+x^2} \, dx,x,e^x\right )\\ &=-x+\frac {1}{10} \left (5-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,e^x\right )+\frac {1}{10} \left (5+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,e^x\right )\\ &=-x+\frac {1}{10} \left (5+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 e^x\right )+\frac {1}{10} \left (5-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 44, normalized size = 0.79 \[ -x+\frac {1}{2} \log \left (-e^x-e^{2 x}+1\right )-\frac {\tanh ^{-1}\left (\frac {2 e^x+1}{\sqrt {5}}\right )}{\sqrt {5}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 53, normalized size = 0.95 \[ \frac {1}{10} \, \sqrt {5} \log \left (-\frac {2 \, {\left (\sqrt {5} - 1\right )} e^{x} + \sqrt {5} - 2 \, e^{\left (2 \, x\right )} - 3}{e^{\left (2 \, x\right )} + e^{x} - 1}\right ) - x + \frac {1}{2} \, \log \left (e^{\left (2 \, x\right )} + e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 46, normalized size = 0.82 \[ \frac {1}{10} \, \sqrt {5} \log \left (\frac {{\left | -\sqrt {5} + 2 \, e^{x} + 1 \right |}}{\sqrt {5} + 2 \, e^{x} + 1}\right ) - x + \frac {1}{2} \, \log \left ({\left | e^{\left (2 \, x\right )} + e^{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 35, normalized size = 0.62 \[ -\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 \,{\mathrm e}^{x}+1\right ) \sqrt {5}}{5}\right )}{5}+\frac {\ln \left ({\mathrm e}^{x}+{\mathrm e}^{2 x}-1\right )}{2}-\ln \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.86, size = 43, normalized size = 0.77 \[ \frac {1}{10} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - 2 \, e^{x} - 1}{\sqrt {5} + 2 \, e^{x} + 1}\right ) - x + \frac {1}{2} \, \log \left (e^{\left (2 \, x\right )} + e^{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.70, size = 32, normalized size = 0.57 \[ \frac {\ln \left ({\mathrm {e}}^{2\,x}+{\mathrm {e}}^x-1\right )}{2}-x-\frac {\sqrt {5}\,\mathrm {atanh}\left (\frac {\sqrt {5}\,\left (2\,{\mathrm {e}}^x+1\right )}{5}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 22, normalized size = 0.39 \[ - x + \operatorname {RootSum} {\left (5 z^{2} - 5 z + 1, \left (i \mapsto i \log {\left (- 5 i + e^{x} + 3 \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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