Optimal. Leaf size=25 \[ 9+x+\left (e^{2 x}-\frac {4}{1+x}\right ) \left (x+\frac {2}{\log (4)}\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(25)=50\).
time = 0.26, antiderivative size = 55, normalized size of antiderivative = 2.20, number of steps
used = 8, number of rules used = 6, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {12, 27, 6874,
2207, 2225, 697} \begin {gather*} x+\frac {e^{2 x} (x \log (16)+4+\log (4))}{2 \log (4)}-\frac {e^{2 x} \log (16)}{4 \log (4)}-\frac {2 (4-\log (16))}{(x+1) \log (4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 697
Rule 2207
Rule 2225
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {8+\left (-3+2 x+x^2\right ) \log (4)+e^{2 x} \left (4+8 x+4 x^2+\left (1+4 x+5 x^2+2 x^3\right ) \log (4)\right )}{1+2 x+x^2} \, dx}{\log (4)}\\ &=\frac {\int \frac {8+\left (-3+2 x+x^2\right ) \log (4)+e^{2 x} \left (4+8 x+4 x^2+\left (1+4 x+5 x^2+2 x^3\right ) \log (4)\right )}{(1+x)^2} \, dx}{\log (4)}\\ &=\frac {\int \left (e^{2 x} (4+\log (4)+x \log (16))+\frac {8-3 \log (4)+x^2 \log (4)+x \log (16)}{(1+x)^2}\right ) \, dx}{\log (4)}\\ &=\frac {\int e^{2 x} (4+\log (4)+x \log (16)) \, dx}{\log (4)}+\frac {\int \frac {8-3 \log (4)+x^2 \log (4)+x \log (16)}{(1+x)^2} \, dx}{\log (4)}\\ &=\frac {e^{2 x} (4+\log (4)+x \log (16))}{2 \log (4)}+\frac {\int \left (\log (4)-\frac {2 (-4+\log (16))}{(1+x)^2}\right ) \, dx}{\log (4)}-\frac {\log (16) \int e^{2 x} \, dx}{2 \log (4)}\\ &=x-\frac {2 (4-\log (16))}{(1+x) \log (4)}-\frac {e^{2 x} \log (16)}{4 \log (4)}+\frac {e^{2 x} (4+\log (4)+x \log (16))}{2 \log (4)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.24, size = 42, normalized size = 1.68 \begin {gather*} \frac {-32+x^2 \log (256)+e^{2 x} (1+x) (8+x \log (256))+x \log (65536)+\log (1099511627776)}{4 (1+x) \log (4)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.87, size = 44, normalized size = 1.76
method | result | size |
risch | \(x -\frac {4}{\ln \left (2\right ) \left (x +1\right )}+\frac {4}{x +1}+\frac {\left (2+2 x \ln \left (2\right )\right ) {\mathrm e}^{2 x}}{2 \ln \left (2\right )}\) | \(38\) |
default | \(\frac {-\frac {8}{x +1}+\frac {8 \ln \left (2\right )}{x +1}+2 x \ln \left (2\right )+2 \,{\mathrm e}^{2 x}+2 x \ln \left (2\right ) {\mathrm e}^{2 x}}{2 \ln \left (2\right )}\) | \(44\) |
norman | \(\frac {x^{2}-\frac {\left (3 \ln \left (2\right )-4\right ) x}{\ln \left (2\right )}+{\mathrm e}^{2 x} x^{2}+\frac {{\mathrm e}^{2 x}}{\ln \left (2\right )}+\frac {\left (1+\ln \left (2\right )\right ) x \,{\mathrm e}^{2 x}}{\ln \left (2\right )}}{x +1}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 37, normalized size = 1.48 \begin {gather*} \frac {{\left ({\left (x^{2} + x\right )} \log \left (2\right ) + x + 1\right )} e^{\left (2 \, x\right )} + {\left (x^{2} + x + 4\right )} \log \left (2\right ) - 4}{{\left (x + 1\right )} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 31, normalized size = 1.24 \begin {gather*} x + \frac {\left (x \log {\left (2 \right )} + 1\right ) e^{2 x}}{\log {\left (2 \right )}} + \frac {-4 + 4 \log {\left (2 \right )}}{x \log {\left (2 \right )} + \log {\left (2 \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. 54 vs.
\(2 (25) = 50\).
time = 0.40, size = 54, normalized size = 2.16 \begin {gather*} \frac {x^{2} e^{\left (2 \, x\right )} \log \left (2\right ) + x^{2} \log \left (2\right ) + x e^{\left (2 \, x\right )} \log \left (2\right ) + x e^{\left (2 \, x\right )} + x \log \left (2\right ) + e^{\left (2 \, x\right )} + 4 \, \log \left (2\right ) - 4}{{\left (x + 1\right )} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 39, normalized size = 1.56 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}+x\,\ln \left (2\right )+x\,{\mathrm {e}}^{2\,x}\,\ln \left (2\right )}{\ln \left (2\right )}+\frac {\ln \left (2\right )+\ln \left (8\right )-4}{\ln \left (2\right )\,\left (x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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