Optimal. Leaf size=62 \[ \frac {2}{3} \sqrt {-6 e^x+3 e^{2 x}-1}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-e^x\right )}{\sqrt {-6 e^x+3 e^{2 x}-1}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2282, 640, 621, 206} \[ \frac {2}{3} \sqrt {-6 e^x+3 e^{2 x}-1}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-e^x\right )}{\sqrt {-6 e^x+3 e^{2 x}-1}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
Rule 2282
Rubi steps
\begin {align*} \int \frac {-e^x+2 e^{2 x}}{\sqrt {-1-6 e^x+3 e^{2 x}}} \, dx &=\operatorname {Subst}\left (\int \frac {-1+2 x}{\sqrt {-1-6 x+3 x^2}} \, dx,x,e^x\right )\\ &=\frac {2}{3} \sqrt {-1-6 e^x+3 e^{2 x}}+\operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-6 x+3 x^2}} \, dx,x,e^x\right )\\ &=\frac {2}{3} \sqrt {-1-6 e^x+3 e^{2 x}}+2 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {-6+6 e^x}{\sqrt {-1-6 e^x+3 e^{2 x}}}\right )\\ &=\frac {2}{3} \sqrt {-1-6 e^x+3 e^{2 x}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-e^x\right )}{\sqrt {-1-6 e^x+3 e^{2 x}}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 54, normalized size = 0.87 \[ \frac {2}{3} \sqrt {-6 e^x+3 e^{2 x}-1}+\frac {\tanh ^{-1}\left (\frac {e^x-1}{\sqrt {-2 e^x+e^{2 x}-\frac {1}{3}}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 62, normalized size = 1.00 \[ \frac {1}{6} \, \sqrt {3} \log \left ({\left (\sqrt {3} e^{x} - \sqrt {3}\right )} \sqrt {3 \, e^{\left (2 \, x\right )} - 6 \, e^{x} - 1} + 3 \, e^{\left (2 \, x\right )} - 6 \, e^{x} + 1\right ) + \frac {2}{3} \, \sqrt {3 \, e^{\left (2 \, x\right )} - 6 \, e^{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 49, normalized size = 0.79 \[ -\frac {1}{3} \, \sqrt {3} \log \left ({\left | -\sqrt {3} e^{x} + \sqrt {3} + \sqrt {3 \, e^{\left (2 \, x\right )} - 6 \, e^{x} - 1} \right |}\right ) + \frac {2}{3} \, \sqrt {3 \, e^{\left (2 \, x\right )} - 6 \, e^{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 50, normalized size = 0.81 \[ \frac {\sqrt {3}\, \ln \left (\frac {\left (3 \,{\mathrm e}^{x}-3\right ) \sqrt {3}}{3}+\sqrt {-6 \,{\mathrm e}^{x}+3 \,{\mathrm e}^{2 x}-1}\right )}{3}+\frac {2 \sqrt {-6 \,{\mathrm e}^{x}+3 \,{\mathrm e}^{2 x}-1}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.97, size = 48, normalized size = 0.77 \[ \frac {1}{3} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, e^{\left (2 \, x\right )} - 6 \, e^{x} - 1} + 6 \, e^{x} - 6\right ) + \frac {2}{3} \, \sqrt {3 \, e^{\left (2 \, x\right )} - 6 \, e^{x} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.20, size = 49, normalized size = 0.79 \[ \frac {\sqrt {3}\,\ln \left (\sqrt {3\,{\mathrm {e}}^{2\,x}-6\,{\mathrm {e}}^x-1}-\sqrt {3}+\sqrt {3}\,{\mathrm {e}}^x\right )}{3}+\frac {2\,\sqrt {3\,{\mathrm {e}}^{2\,x}-6\,{\mathrm {e}}^x-1}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (2 e^{x} - 1\right ) e^{x}}{\sqrt {3 e^{2 x} - 6 e^{x} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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