Optimal. Leaf size=20 \[ -1+x+16 x^2 \left (-9+x-\frac {\log (5)}{e^2}\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(20)=40\).
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 2.45, number of steps
used = 4, number of rules used = 1, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {12}
\begin {gather*} 16 x^4-288 x^3-\frac {32 x^3 \log (5)}{e^2}+1296 x^2+\frac {16 x^2 \log ^2(5)}{e^4}+\frac {288 x^2 \log (5)}{e^2}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (e^4 \left (1+2592 x-864 x^2+64 x^3\right )+e^2 \left (576 x-96 x^2\right ) \log (5)+32 x \log ^2(5)\right ) \, dx}{e^4}\\ &=\frac {16 x^2 \log ^2(5)}{e^4}+\frac {\log (5) \int \left (576 x-96 x^2\right ) \, dx}{e^2}+\int \left (1+2592 x-864 x^2+64 x^3\right ) \, dx\\ &=x+1296 x^2-288 x^3+16 x^4+\frac {288 x^2 \log (5)}{e^2}-\frac {32 x^3 \log (5)}{e^2}+\frac {16 x^2 \log ^2(5)}{e^4}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(41\) vs. \(2(20)=40\).
time = 0.03, size = 41, normalized size = 2.05 \begin {gather*} x+16 x^4-\frac {32 x^3 \left (9 e^2+\log (5)\right )}{e^2}+\frac {16 x^2 \left (9 e^2+\log (5)\right )^2}{e^4} \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. \(66\) vs.
\(2(21)=42\).
time = 0.29, size = 67, normalized size = 3.35
method | result | size |
risch | \(16 x^{4}-288 x^{3}-32 \,{\mathrm e}^{-2} \ln \left (5\right ) x^{3}+1296 x^{2}+288 \,{\mathrm e}^{-2} \ln \left (5\right ) x^{2}+16 \,{\mathrm e}^{-4} x^{2} \ln \left (5\right )^{2}+x\) | \(47\) |
norman | \(\left (\left (-288 \,{\mathrm e}^{2}-32 \ln \left (5\right )\right ) x^{3}+{\mathrm e}^{2} x +16 x^{4} {\mathrm e}^{2}+16 \left (81 \,{\mathrm e}^{4}+18 \,{\mathrm e}^{2} \ln \left (5\right )+\ln \left (5\right )^{2}\right ) {\mathrm e}^{-2} x^{2}\right ) {\mathrm e}^{-2}\) | \(57\) |
gosper | \(x \left (16 x^{3} {\mathrm e}^{4}-288 x^{2} {\mathrm e}^{4}-32 \ln \left (5\right ) {\mathrm e}^{2} x^{2}+1296 x \,{\mathrm e}^{4}+288 x \,{\mathrm e}^{2} \ln \left (5\right )+16 x \ln \left (5\right )^{2}+{\mathrm e}^{4}\right ) {\mathrm e}^{-4}\) | \(60\) |
default | \({\mathrm e}^{-4} \left (16 x^{4} {\mathrm e}^{4}-288 x^{3} {\mathrm e}^{4}-32 \ln \left (5\right ) {\mathrm e}^{2} x^{3}+1296 x^{2} {\mathrm e}^{4}+288 \ln \left (5\right ) {\mathrm e}^{2} x^{2}+16 x^{2} \ln \left (5\right )^{2}+x \,{\mathrm e}^{4}\right )\) | \(67\) |
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. 48 vs.
\(2 (20) = 40\).
time = 0.27, size = 48, normalized size = 2.40 \begin {gather*} {\left (16 \, x^{2} \log \left (5\right )^{2} - 32 \, {\left (x^{3} - 9 \, x^{2}\right )} e^{2} \log \left (5\right ) + {\left (16 \, x^{4} - 288 \, x^{3} + 1296 \, x^{2} + x\right )} e^{4}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (20) = 40\).
time = 0.39, size = 48, normalized size = 2.40 \begin {gather*} {\left (16 \, x^{2} \log \left (5\right )^{2} - 32 \, {\left (x^{3} - 9 \, x^{2}\right )} e^{2} \log \left (5\right ) + {\left (16 \, x^{4} - 288 \, x^{3} + 1296 \, x^{2} + x\right )} e^{4}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (19) = 38\).
time = 0.01, size = 51, normalized size = 2.55 \begin {gather*} 16 x^{4} + \frac {x^{3} \left (- 288 e^{2} - 32 \log {\left (5 \right )}\right )}{e^{2}} + \frac {x^{2} \cdot \left (16 \log {\left (5 \right )}^{2} + 288 e^{2} \log {\left (5 \right )} + 1296 e^{4}\right )}{e^{4}} + x \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. 48 vs.
\(2 (20) = 40\).
time = 0.41, size = 48, normalized size = 2.40 \begin {gather*} {\left (16 \, x^{2} \log \left (5\right )^{2} - 32 \, {\left (x^{3} - 9 \, x^{2}\right )} e^{2} \log \left (5\right ) + {\left (16 \, x^{4} - 288 \, x^{3} + 1296 \, x^{2} + x\right )} e^{4}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 49, normalized size = 2.45 \begin {gather*} 16\,x^4-\frac {{\mathrm {e}}^{-4}\,\left (864\,{\mathrm {e}}^4+96\,{\mathrm {e}}^2\,\ln \left (5\right )\right )\,x^3}{3}+\frac {{\mathrm {e}}^{-4}\,\left (2592\,{\mathrm {e}}^4+576\,{\mathrm {e}}^2\,\ln \left (5\right )+32\,{\ln \left (5\right )}^2\right )\,x^2}{2}+x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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