Optimal. Leaf size=29 \[ \left (6+e^2-2 x\right ) \left (\frac {13}{5}+\log \left (\frac {2}{3+\frac {1}{4} (1+3 x)}\right )\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 42, normalized size of antiderivative = 1.45, number of steps
used = 7, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {6873, 6874, 45,
2436, 2332} \begin {gather*} -\frac {26 x}{5}-\frac {2}{3} (3 x+13) \log \left (\frac {8}{3 x+13}\right )-\frac {1}{3} \left (44+3 e^2\right ) \log (3 x+13) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2332
Rule 2436
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-428 \left (1+\frac {15 e^2}{428}\right )-48 x+(-130-30 x) \log \left (\frac {8}{13+3 x}\right )}{65+15 x} \, dx\\ &=\int \left (\frac {-428-15 e^2-48 x}{5 (13+3 x)}-2 \log \left (\frac {8}{13+3 x}\right )\right ) \, dx\\ &=\frac {1}{5} \int \frac {-428-15 e^2-48 x}{13+3 x} \, dx-2 \int \log \left (\frac {8}{13+3 x}\right ) \, dx\\ &=\frac {1}{5} \int \left (-16-\frac {5 \left (44+3 e^2\right )}{13+3 x}\right ) \, dx-\frac {2}{3} \text {Subst}\left (\int \log \left (\frac {8}{x}\right ) \, dx,x,13+3 x\right )\\ &=-\frac {26 x}{5}-\frac {2}{3} (13+3 x) \log \left (\frac {8}{13+3 x}\right )-\frac {1}{3} \left (44+3 e^2\right ) \log (13+3 x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 40, normalized size = 1.38 \begin {gather*} -\frac {26 x}{5}-\left (\frac {26}{3}+2 x\right ) \log \left (\frac {8}{13+3 x}\right )-\left (\frac {44}{3}+e^2\right ) \log (13+3 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(47\) vs.
\(2(20)=40\).
time = 0.44, size = 48, normalized size = 1.66
method | result | size |
norman | \(\left ({\mathrm e}^{2}+6\right ) \ln \left (\frac {8}{3 x +13}\right )-\frac {26 x}{5}-2 \ln \left (\frac {8}{3 x +13}\right ) x\) | \(33\) |
risch | \(-2 \ln \left (\frac {8}{3 x +13}\right ) x -\frac {26 x}{5}-6 \ln \left (3 x +13\right )-\ln \left (3 x +13\right ) {\mathrm e}^{2}\) | \(36\) |
derivativedivides | \({\mathrm e}^{2} \ln \left (\frac {8}{3 x +13}\right )-\frac {2 \left (3 x +13\right ) \ln \left (\frac {8}{3 x +13}\right )}{3}-\frac {26 x}{5}-\frac {338}{15}+\frac {44 \ln \left (\frac {8}{3 x +13}\right )}{3}\) | \(48\) |
default | \({\mathrm e}^{2} \ln \left (\frac {8}{3 x +13}\right )-\frac {2 \left (3 x +13\right ) \ln \left (\frac {8}{3 x +13}\right )}{3}-\frac {26 x}{5}-\frac {338}{15}+\frac {44 \ln \left (\frac {8}{3 x +13}\right )}{3}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs.
\(2 (25) = 50\).
time = 0.27, size = 70, normalized size = 2.41 \begin {gather*} -e^{2} \log \left (3 \, x + 13\right ) + \frac {13}{3} \, \log \left (3 \, x + 13\right )^{2} - \frac {2}{3} \, {\left (3 \, x - 13 \, \log \left (3 \, x + 13\right )\right )} \log \left (\frac {8}{3 \, x + 13}\right ) + \frac {13}{3} \, \log \left (\frac {8}{3 \, x + 13}\right )^{2} - \frac {26}{5} \, x - 6 \, \log \left (3 \, x + 13\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 25, normalized size = 0.86 \begin {gather*} -{\left (2 \, x - e^{2} - 6\right )} \log \left (\frac {8}{3 \, x + 13}\right ) - \frac {26}{5} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 29, normalized size = 1.00 \begin {gather*} - 2 x \log {\left (\frac {8}{3 x + 13} \right )} - \frac {26 x}{5} - \left (6 + e^{2}\right ) \log {\left (3 x + 13 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (25) = 50\).
time = 0.38, size = 61, normalized size = 2.10 \begin {gather*} \frac {1}{15} \, {\left (3 \, x + 13\right )} {\left (\frac {15 \, e^{2} \log \left (\frac {8}{3 \, x + 13}\right )}{3 \, x + 13} + \frac {220 \, \log \left (\frac {8}{3 \, x + 13}\right )}{3 \, x + 13} - 10 \, \log \left (\frac {8}{3 \, x + 13}\right ) - 26\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.55, size = 38, normalized size = 1.31 \begin {gather*} 6\,\ln \left (\frac {1}{3\,x+13}\right )-\frac {26\,x}{5}-2\,x\,\ln \left (\frac {8}{3\,x+13}\right )+\ln \left (\frac {1}{3\,x+13}\right )\,{\mathrm {e}}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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