Optimal. Leaf size=24 \[ 1+\frac {9}{64 x^4}-\log \left (\frac {4 x}{5+e^{2 x}}\right ) \]
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Rubi [A]
time = 0.25, antiderivative size = 20, normalized size of antiderivative = 0.83, number of steps
used = 10, number of rules used = 8, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6873, 12,
6874, 2320, 36, 29, 31, 14} \begin {gather*} \frac {9}{64 x^4}+\log \left (e^{2 x}+5\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 29
Rule 31
Rule 36
Rule 2320
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-45-80 x^4+e^{2 x} \left (-9-16 x^4+32 x^5\right )}{16 \left (5+e^{2 x}\right ) x^5} \, dx\\ &=\frac {1}{16} \int \frac {-45-80 x^4+e^{2 x} \left (-9-16 x^4+32 x^5\right )}{\left (5+e^{2 x}\right ) x^5} \, dx\\ &=\frac {1}{16} \int \left (-\frac {160}{5+e^{2 x}}+\frac {-9-16 x^4+32 x^5}{x^5}\right ) \, dx\\ &=\frac {1}{16} \int \frac {-9-16 x^4+32 x^5}{x^5} \, dx-10 \int \frac {1}{5+e^{2 x}} \, dx\\ &=\frac {1}{16} \int \left (32-\frac {9}{x^5}-\frac {16}{x}\right ) \, dx-5 \text {Subst}\left (\int \frac {1}{x (5+x)} \, dx,x,e^{2 x}\right )\\ &=\frac {9}{64 x^4}+2 x-\log (x)-\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{2 x}\right )+\text {Subst}\left (\int \frac {1}{5+x} \, dx,x,e^{2 x}\right )\\ &=\frac {9}{64 x^4}+\log \left (5+e^{2 x}\right )-\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 1.14, size = 33, normalized size = 1.38 \begin {gather*} \frac {1}{16} \left (\frac {9}{4 x^4}+32 x+32 \tanh ^{-1}\left (1+\frac {2 e^{2 x}}{5}\right )-16 \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.93, size = 18, normalized size = 0.75
method | result | size |
norman | \(\frac {9}{64 x^{4}}-\ln \left (x \right )+\ln \left (5+{\mathrm e}^{2 x}\right )\) | \(18\) |
risch | \(\frac {9}{64 x^{4}}-\ln \left (x \right )+\ln \left (5+{\mathrm e}^{2 x}\right )\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 17, normalized size = 0.71 \begin {gather*} \frac {9}{64 \, x^{4}} - \log \left (x\right ) + \log \left (e^{\left (2 \, x\right )} + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 26, normalized size = 1.08 \begin {gather*} -\frac {64 \, x^{4} \log \left (x\right ) - 64 \, x^{4} \log \left (e^{\left (2 \, x\right )} + 5\right ) - 9}{64 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 17, normalized size = 0.71 \begin {gather*} - \log {\left (x \right )} + \log {\left (e^{2 x} + 5 \right )} + \frac {9}{64 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 26, normalized size = 1.08 \begin {gather*} -\frac {64 \, x^{4} \log \left (x\right ) - 64 \, x^{4} \log \left (e^{\left (2 \, x\right )} + 5\right ) - 9}{64 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 17, normalized size = 0.71 \begin {gather*} \ln \left ({\mathrm {e}}^{2\,x}+5\right )-\ln \left (x\right )+\frac {9}{64\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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