Optimal. Leaf size=28 \[ \log \left (\frac {\left (x+e^{x^2} x\right )^2 \left (4 x+3 x^2\right )}{5 x^2}\right ) \]
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Rubi [A]
time = 0.44, antiderivative size = 19, normalized size of antiderivative = 0.68, number of steps
used = 10, number of rules used = 8, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.151, Rules used = {6873, 6874,
6847, 2320, 36, 29, 31, 1634} \begin {gather*} 2 \log \left (e^{x^2}+1\right )+\log (x)+\log (3 x+4) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 1634
Rule 2320
Rule 6847
Rule 6873
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4+6 x+e^{x^2} \left (4+6 x+16 x^2+12 x^3\right )}{\left (1+e^{x^2}\right ) x (4+3 x)} \, dx\\ &=\int \left (-\frac {4 x}{1+e^{x^2}}+\frac {2 \left (2+3 x+8 x^2+6 x^3\right )}{x (4+3 x)}\right ) \, dx\\ &=2 \int \frac {2+3 x+8 x^2+6 x^3}{x (4+3 x)} \, dx-4 \int \frac {x}{1+e^{x^2}} \, dx\\ &=2 \int \left (\frac {1}{2 x}+2 x+\frac {3}{2 (4+3 x)}\right ) \, dx-2 \text {Subst}\left (\int \frac {1}{1+e^x} \, dx,x,x^2\right )\\ &=2 x^2+\log (x)+\log (4+3 x)-2 \text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,e^{x^2}\right )\\ &=2 x^2+\log (x)+\log (4+3 x)-2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{x^2}\right )+2 \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^{x^2}\right )\\ &=2 \log \left (1+e^{x^2}\right )+\log (x)+\log (4+3 x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.69, size = 26, normalized size = 0.93 \begin {gather*} 2 x^2+4 \tanh ^{-1}\left (1+2 e^{x^2}\right )+\log (x)+\log (4+3 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 19, normalized size = 0.68
method | result | size |
norman | \(2 \ln \left ({\mathrm e}^{x^{2}}+1\right )+\ln \left (x \right )+\ln \left (4+3 x \right )\) | \(19\) |
risch | \(\ln \left (3 x^{2}+4 x \right )+2 \ln \left ({\mathrm e}^{x^{2}}+1\right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 18, normalized size = 0.64 \begin {gather*} \log \left (3 \, x + 4\right ) + \log \left (x\right ) + 2 \, \log \left (e^{\left (x^{2}\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 20, normalized size = 0.71 \begin {gather*} \log \left (3 \, x^{2} + 4 \, x\right ) + 2 \, \log \left (e^{\left (x^{2}\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 19, normalized size = 0.68 \begin {gather*} \log {\left (3 x^{2} + 4 x \right )} + 2 \log {\left (e^{x^{2}} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 18, normalized size = 0.64 \begin {gather*} \log \left (3 \, x + 4\right ) + \log \left (x\right ) + 2 \, \log \left (e^{\left (x^{2}\right )} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 18, normalized size = 0.64 \begin {gather*} \ln \left (x\right )+\ln \left (\left (3\,x+4\right )\,{\left ({\mathrm {e}}^{x^2}+1\right )}^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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