Optimal. Leaf size=23 \[ \frac {2}{\left (\frac {1}{e^3}+x\right ) \left (\frac {x}{1-5 x}+\log (4)\right )} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(23)=46\).
time = 0.36, antiderivative size = 69, normalized size of antiderivative = 3.00, number of steps
used = 4, number of rules used = 4, integrand size = 150, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {1694, 12,
1828, 8} \begin {gather*} \frac {2 e^6 (1-5 x) (1-5 \log (4))}{e^6 x^2 (1-5 \log (4))^2+e^3 x (1-5 \log (4)) \left (1-5 \log (4)+e^3 \log (4)\right )+e^3 (1-5 \log (4)) \log (4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 1694
Rule 1828
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\text {Subst}\left (\int \frac {8 e^6 (1-5 \log (4)) \left (20 e^6 x^2 (1-5 \log (4))^2-4 e^3 x (1-5 \log (4)) \left (5+2 e^3-25 \log (4)-5 e^3 \log (4)\right )+5 \left (1-5 \log (4)-e^3 \log (4)\right )^2\right )}{\left (4 e^6 x^2 (1-5 \log (4))^2-\left (-1+5 \log (4)+e^3 \log (4)\right )^2\right )^2} \, dx,x,x+\frac {2 e^3-20 e^3 \log (4)+2 e^6 \log (4)+50 e^3 \log ^2(4)-10 e^6 \log ^2(4)}{4 \left (e^6-10 e^6 \log (4)+25 e^6 \log ^2(4)\right )}\right )\\ &=\left (8 e^6 (1-5 \log (4))\right ) \text {Subst}\left (\int \frac {20 e^6 x^2 (1-5 \log (4))^2-4 e^3 x (1-5 \log (4)) \left (5+2 e^3-25 \log (4)-5 e^3 \log (4)\right )+5 \left (1-5 \log (4)-e^3 \log (4)\right )^2}{\left (4 e^6 x^2 (1-5 \log (4))^2-\left (-1+5 \log (4)+e^3 \log (4)\right )^2\right )^2} \, dx,x,x+\frac {2 e^3-20 e^3 \log (4)+2 e^6 \log (4)+50 e^3 \log ^2(4)-10 e^6 \log ^2(4)}{4 \left (e^6-10 e^6 \log (4)+25 e^6 \log ^2(4)\right )}\right )\\ &=\frac {2 e^6 (1-5 x)}{e^6 x^2 (1-5 \log (4))+e^3 \log (4)+e^3 x \left (1-5 \log (4)+e^3 \log (4)\right )}+\frac {\left (4 e^6 (1-5 \log (4))\right ) \text {Subst}\left (\int 0 \, dx,x,x+\frac {2 e^3-20 e^3 \log (4)+2 e^6 \log (4)+50 e^3 \log ^2(4)-10 e^6 \log ^2(4)}{4 \left (e^6-10 e^6 \log (4)+25 e^6 \log ^2(4)\right )}\right )}{\left (1-5 \log (4)-e^3 \log (4)\right )^2}\\ &=\frac {2 e^6 (1-5 x)}{e^6 x^2 (1-5 \log (4))+e^3 \log (4)+e^3 x \left (1-5 \log (4)+e^3 \log (4)\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 34, normalized size = 1.48 \begin {gather*} -\frac {2 e^3 (1-5 x)}{\left (1+e^3 x\right ) (-\log (4)+x (-1+5 \log (4)))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 47, normalized size = 2.04
method | result | size |
risch | \(\frac {x \,{\mathrm e}^{3}-\frac {{\mathrm e}^{3}}{5}}{{\mathrm e}^{3} \ln \left (2\right ) x^{2}-\frac {\ln \left (2\right ) {\mathrm e}^{3} x}{5}-\frac {x^{2} {\mathrm e}^{3}}{10}+x \ln \left (2\right )-\frac {\ln \left (2\right )}{5}-\frac {x}{10}}\) | \(47\) |
gosper | \(\frac {2 \left (5 x -1\right ) {\mathrm e}^{3}}{10 \,{\mathrm e}^{3} \ln \left (2\right ) x^{2}-2 \ln \left (2\right ) {\mathrm e}^{3} x -x^{2} {\mathrm e}^{3}+10 x \ln \left (2\right )-2 \ln \left (2\right )-x}\) | \(48\) |
norman | \(\frac {\frac {\left (4 \,{\mathrm e}^{6} \ln \left (2\right )+2 \,{\mathrm e}^{3}\right ) x}{2 \ln \left (2\right )}-\frac {\left (10 \ln \left (2\right )-1\right ) {\mathrm e}^{6} x^{2}}{\ln \left (2\right )}}{\left (10 x \ln \left (2\right )-2 \ln \left (2\right )-x \right ) \left (x \,{\mathrm e}^{3}+1\right )}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 47, normalized size = 2.04 \begin {gather*} \frac {2 \, {\left (5 \, x e^{3} - e^{3}\right )}}{{\left (10 \, e^{3} \log \left (2\right ) - e^{3}\right )} x^{2} - {\left (2 \, {\left (e^{3} - 5\right )} \log \left (2\right ) + 1\right )} x - 2 \, \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 40, normalized size = 1.74 \begin {gather*} -\frac {2 \, {\left (5 \, x - 1\right )} e^{3}}{x^{2} e^{3} - 2 \, {\left ({\left (5 \, x^{2} - x\right )} e^{3} + 5 \, x - 1\right )} \log \left (2\right ) + x} \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. 49 vs.
\(2 (19) = 38\).
time = 2.87, size = 49, normalized size = 2.13 \begin {gather*} - \frac {- 10 x e^{3} + 2 e^{3}}{x^{2} \left (- e^{3} + 10 e^{3} \log {\left (2 \right )}\right ) + x \left (- 2 e^{3} \log {\left (2 \right )} - 1 + 10 \log {\left (2 \right )}\right ) - 2 \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. 50 vs.
\(2 (24) = 48\).
time = 0.41, size = 50, normalized size = 2.17 \begin {gather*} \frac {2 \, {\left (5 \, x e^{3} - e^{3}\right )}}{10 \, x^{2} e^{3} \log \left (2\right ) - x^{2} e^{3} - 2 \, x e^{3} \log \left (2\right ) + 10 \, x \log \left (2\right ) - x - 2 \, \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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