Optimal. Leaf size=20 \[ 1+5 x^2-4 x \left (2+\log \left (4 e^x\right )\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(20)=40\).
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 2.15, number of steps
used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2631, 12, 32,
2188, 30} \begin {gather*} -3 x^2+\frac {4}{3} (x+2)^3-16 x-\frac {4}{3} \log ^3\left (4 e^x\right )-4 (x+2)^2 \log \left (4 e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 32
Rule 2188
Rule 2631
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-16 x-3 x^2-4 \int \log ^2\left (4 e^x\right ) \, dx+\int (-16-8 x) \log \left (4 e^x\right ) \, dx\\ &=-16 x-3 x^2-4 (2+x)^2 \log \left (4 e^x\right )+\frac {1}{16} \int 64 (2+x)^2 \, dx-4 \text {Subst}\left (\int x^2 \, dx,x,\log \left (4 e^x\right )\right )\\ &=-16 x-3 x^2-4 (2+x)^2 \log \left (4 e^x\right )-\frac {4}{3} \log ^3\left (4 e^x\right )+4 \int (2+x)^2 \, dx\\ &=-16 x-3 x^2+\frac {4}{3} (2+x)^3-4 (2+x)^2 \log \left (4 e^x\right )-\frac {4}{3} \log ^3\left (4 e^x\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(20)=40\).
time = 0.01, size = 49, normalized size = 2.45 \begin {gather*} -16 x-3 x^2+\frac {4 x^3}{3}-4 x^2 \log \left (4 e^x\right )-8 \log ^2\left (4 e^x\right )-\frac {4}{3} \log ^3\left (4 e^x\right ) \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. \(41\) vs.
\(2(19)=38\).
time = 0.18, size = 42, normalized size = 2.10
method | result | size |
norman | \(-16 \ln \left (4 \,{\mathrm e}^{x}\right )-5 \ln \left (4 \,{\mathrm e}^{x}\right )^{2}-6 \ln \left (4 \,{\mathrm e}^{x}\right ) x -4 x \ln \left (4 \,{\mathrm e}^{x}\right )^{2}\) | \(36\) |
default | \(-16 x -4 \ln \left (4 \,{\mathrm e}^{x}\right ) x^{2}-16 \ln \left (4 \,{\mathrm e}^{x}\right ) x +\frac {4 x^{3}}{3}+5 x^{2}-\frac {4 \ln \left (4 \,{\mathrm e}^{x}\right )^{3}}{3}\) | \(42\) |
risch | \(-4 x \ln \left ({\mathrm e}^{x}\right )^{2}-16 x \ln \left ({\mathrm e}^{x}\right ) \ln \left (2\right )+\frac {32 \ln \left (2\right )^{3}}{3}-16 x \ln \left (2\right )^{2}-16 x \ln \left ({\mathrm e}^{x}\right )-32 x \ln \left (2\right )+5 x^{2}-16 x\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 37, normalized size = 1.85 \begin {gather*} \frac {4}{3} \, x^{3} - \frac {4}{3} \, \log \left (4 \, e^{x}\right )^{3} + 5 \, x^{2} - 4 \, {\left (x^{2} + 4 \, x\right )} \log \left (4 \, e^{x}\right ) - 16 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 32, normalized size = 1.60 \begin {gather*} -4 \, x^{3} - 16 \, x \log \left (2\right )^{2} - 11 \, x^{2} - 16 \, {\left (x^{2} + 2 \, x\right )} \log \left (2\right ) - 16 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 32, normalized size = 1.60 \begin {gather*} - 4 x^{3} + x^{2} \left (- 16 \log {\left (2 \right )} - 11\right ) + x \left (- 32 \log {\left (2 \right )} - 16 - 16 \log {\left (2 \right )}^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 33, normalized size = 1.65 \begin {gather*} -4 \, x^{3} - 16 \, x^{2} \log \left (2\right ) - 16 \, x \log \left (2\right )^{2} - 11 \, x^{2} - 32 \, x \log \left (2\right ) - 16 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 32, normalized size = 1.60 \begin {gather*} -4\,x^3+\left (-8\,\ln \left (4\right )-11\right )\,x^2+\left (-16\,\ln \left (4\right )-4\,{\ln \left (4\right )}^2-16\right )\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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