Optimal. Leaf size=24 \[ \log \left (3+\frac {4 x^2}{x^2-e^{-5 x} x^2}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 23, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {12, 2282, 616, 31} \begin {gather*} \log \left (3-7 e^{5 x}\right )-\log \left (1-e^{5 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 616
Rule 2282
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-\left (20 \int \frac {e^{5 x}}{3-10 e^{5 x}+7 e^{10 x}} \, dx\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{3-10 x+7 x^2} \, dx,x,e^{5 x}\right )\right )\\ &=-\left (7 \operatorname {Subst}\left (\int \frac {1}{-7+7 x} \, dx,x,e^{5 x}\right )\right )+7 \operatorname {Subst}\left (\int \frac {1}{-3+7 x} \, dx,x,e^{5 x}\right )\\ &=\log \left (3-7 e^{5 x}\right )-\log \left (1-e^{5 x}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 0.67 \begin {gather*} 2 \tanh ^{-1}\left (\frac {1}{4} \left (-10+14 e^{5 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 19, normalized size = 0.79 \begin {gather*} \log \left (7 \, e^{\left (5 \, x\right )} - 3\right ) - \log \left (e^{\left (5 \, x\right )} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 21, normalized size = 0.88 \begin {gather*} \log \left ({\left | 7 \, e^{\left (5 \, x\right )} - 3 \right |}\right ) - \log \left ({\left | e^{\left (5 \, x\right )} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 18, normalized size = 0.75
method | result | size |
risch | \(-\ln \left ({\mathrm e}^{5 x}-1\right )+\ln \left ({\mathrm e}^{5 x}-\frac {3}{7}\right )\) | \(18\) |
derivativedivides | \(\ln \left (7 \,{\mathrm e}^{5 x}-3\right )-\ln \left ({\mathrm e}^{5 x}-1\right )\) | \(20\) |
default | \(\ln \left (7 \,{\mathrm e}^{5 x}-3\right )-\ln \left ({\mathrm e}^{5 x}-1\right )\) | \(20\) |
norman | \(\ln \left (7 \,{\mathrm e}^{5 x}-3\right )-\ln \left ({\mathrm e}^{5 x}-1\right )\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 19, normalized size = 0.79 \begin {gather*} \log \left (7 \, e^{\left (5 \, x\right )} - 3\right ) - \log \left (e^{\left (5 \, x\right )} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.20, size = 19, normalized size = 0.79 \begin {gather*} \ln \left (7\,{\mathrm {e}}^{5\,x}-3\right )-\ln \left ({\mathrm {e}}^{5\,x}-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 17, normalized size = 0.71 \begin {gather*} - \log {\left (e^{5 x} - 1 \right )} + \log {\left (e^{5 x} - \frac {3}{7} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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