Optimal. Leaf size=52 \[ -\frac {4 e^{x/2}}{3}-\frac {4}{9} e^{3 x/2}+\frac {2}{3} e^{3 x/2} \log \left (e^x-1\right )+\frac {4}{3} \tanh ^{-1}\left (e^{x/2}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2194, 2554, 12, 2248, 302, 207} \[ -\frac {4 e^{x/2}}{3}-\frac {4}{9} e^{3 x/2}+\frac {2}{3} e^{3 x/2} \log \left (e^x-1\right )+\frac {4}{3} \tanh ^{-1}\left (e^{x/2}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 207
Rule 302
Rule 2194
Rule 2248
Rule 2554
Rubi steps
\begin {align*} \int e^{3 x/2} \log \left (-1+e^x\right ) \, dx &=\frac {2}{3} e^{3 x/2} \log \left (-1+e^x\right )-\int \frac {2 e^{5 x/2}}{3 \left (-1+e^x\right )} \, dx\\ &=\frac {2}{3} e^{3 x/2} \log \left (-1+e^x\right )-\frac {2}{3} \int \frac {e^{5 x/2}}{-1+e^x} \, dx\\ &=\frac {2}{3} e^{3 x/2} \log \left (-1+e^x\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,e^{x/2}\right )\\ &=\frac {2}{3} e^{3 x/2} \log \left (-1+e^x\right )-\frac {4}{3} \operatorname {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,e^{x/2}\right )\\ &=-\frac {4 e^{x/2}}{3}-\frac {4}{9} e^{3 x/2}+\frac {2}{3} e^{3 x/2} \log \left (-1+e^x\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,e^{x/2}\right )\\ &=-\frac {4 e^{x/2}}{3}-\frac {4}{9} e^{3 x/2}+\frac {4}{3} \tanh ^{-1}\left (e^{x/2}\right )+\frac {2}{3} e^{3 x/2} \log \left (-1+e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 42, normalized size = 0.81 \[ \frac {2}{9} \left (e^{x/2} \left (3 e^x \log \left (e^x-1\right )-2 \left (e^x+3\right )\right )+6 \tanh ^{-1}\left (e^{x/2}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{3 x/2} \log \left (-1+e^x\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.88, size = 42, normalized size = 0.81 \[ \frac {2}{3} \, e^{\left (\frac {3}{2} \, x\right )} \log \left (e^{x} - 1\right ) - \frac {4}{9} \, e^{\left (\frac {3}{2} \, x\right )} - \frac {4}{3} \, e^{\left (\frac {1}{2} \, x\right )} + \frac {2}{3} \, \log \left (e^{\left (\frac {1}{2} \, x\right )} + 1\right ) - \frac {2}{3} \, \log \left (e^{\left (\frac {1}{2} \, x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 43, normalized size = 0.83 \[ \frac {2}{3} \, e^{\left (\frac {3}{2} \, x\right )} \log \left (e^{x} - 1\right ) - \frac {4}{9} \, e^{\left (\frac {3}{2} \, x\right )} - \frac {4}{3} \, e^{\left (\frac {1}{2} \, x\right )} + \frac {2}{3} \, \log \left (e^{\left (\frac {1}{2} \, x\right )} + 1\right ) - \frac {2}{3} \, \log \left ({\left | e^{\left (\frac {1}{2} \, x\right )} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 43, normalized size = 0.83
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{\frac {3 x}{2}} \ln \left (-1+{\mathrm e}^{x}\right )}{3}-\frac {4 \,{\mathrm e}^{\frac {3 x}{2}}}{9}-\frac {4 \,{\mathrm e}^{\frac {x}{2}}}{3}-\frac {2 \ln \left (-1+{\mathrm e}^{\frac {x}{2}}\right )}{3}+\frac {2 \ln \left ({\mathrm e}^{\frac {x}{2}}+1\right )}{3}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 42, normalized size = 0.81 \[ \frac {2}{3} \, e^{\left (\frac {3}{2} \, x\right )} \log \left (e^{x} - 1\right ) - \frac {4}{9} \, e^{\left (\frac {3}{2} \, x\right )} - \frac {4}{3} \, e^{\left (\frac {1}{2} \, x\right )} + \frac {2}{3} \, \log \left (e^{\left (\frac {1}{2} \, x\right )} + 1\right ) - \frac {2}{3} \, \log \left (e^{\left (\frac {1}{2} \, x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 31, normalized size = 0.60 \[ \frac {4\,\mathrm {atanh}\left (\sqrt {{\mathrm {e}}^x}\right )}{3}-\frac {4\,{\mathrm {e}}^{\frac {3\,x}{2}}}{9}-\frac {4\,{\mathrm {e}}^{x/2}}{3}+\frac {2\,{\mathrm {e}}^{\frac {3\,x}{2}}\,\ln \left ({\mathrm {e}}^x-1\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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