Optimal. Leaf size=26 \[ 2+x+\left (3+x^2\right )^4-\frac {\log (5)}{-x+\log (5)}+\frac {1}{\log (x)} \]
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Rubi [A]
time = 0.68, antiderivative size = 35, normalized size of antiderivative = 1.35, number of steps
used = 8, number of rules used = 6, integrand size = 129, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {1608, 27,
6874, 1864, 2339, 30} \begin {gather*} x^8+12 x^6+54 x^4+108 x^2+x+\frac {1}{\log (x)}+\frac {\log (5)}{x-\log (5)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 30
Rule 1608
Rule 1864
Rule 2339
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x^2+2 x \log (5)-\log ^2(5)+\left (x^3+216 x^4+216 x^6+72 x^8+8 x^{10}+\left (-x-2 x^2-432 x^3-432 x^5-144 x^7-16 x^9\right ) \log (5)+\left (x+216 x^2+216 x^4+72 x^6+8 x^8\right ) \log ^2(5)\right ) \log ^2(x)}{x \left (x^2-2 x \log (5)+\log ^2(5)\right ) \log ^2(x)} \, dx\\ &=\int \frac {-x^2+2 x \log (5)-\log ^2(5)+\left (x^3+216 x^4+216 x^6+72 x^8+8 x^{10}+\left (-x-2 x^2-432 x^3-432 x^5-144 x^7-16 x^9\right ) \log (5)+\left (x+216 x^2+216 x^4+72 x^6+8 x^8\right ) \log ^2(5)\right ) \log ^2(x)}{x (x-\log (5))^2 \log ^2(x)} \, dx\\ &=\int \left (\frac {8 x^9+x^2 (1-432 \log (5))-432 x^4 \log (5)-144 x^6 \log (5)-16 x^8 \log (5)-2 x (1-108 \log (5)) \log (5)-(1-\log (5)) \log (5)+72 x^7 \left (1+\frac {\log ^2(5)}{9}\right )+216 x^5 \left (1+\frac {\log ^2(5)}{3}\right )+216 x^3 \left (1+\log ^2(5)\right )}{(x-\log (5))^2}-\frac {1}{x \log ^2(x)}\right ) \, dx\\ &=\int \frac {8 x^9+x^2 (1-432 \log (5))-432 x^4 \log (5)-144 x^6 \log (5)-16 x^8 \log (5)-2 x (1-108 \log (5)) \log (5)-(1-\log (5)) \log (5)+72 x^7 \left (1+\frac {\log ^2(5)}{9}\right )+216 x^5 \left (1+\frac {\log ^2(5)}{3}\right )+216 x^3 \left (1+\log ^2(5)\right )}{(x-\log (5))^2} \, dx-\int \frac {1}{x \log ^2(x)} \, dx\\ &=\int \left (1+216 x+216 x^3+72 x^5+8 x^7-\frac {\log (5)}{(x-\log (5))^2}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )\\ &=x+108 x^2+54 x^4+12 x^6+x^8+\frac {\log (5)}{x-\log (5)}+\frac {1}{\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(128\) vs. \(2(26)=52\).
time = 0.05, size = 128, normalized size = 4.92 \begin {gather*} \frac {x-\log (5)+\left (108 x^3+54 x^5+12 x^7+x^9+x^2 (1-108 \log (5))-54 x^4 \log (5)-12 x^6 \log (5)-x^8 \log (5)-x \log (5) \left (2+108 \log (5)+54 \log ^3(5)+12 \log ^5(5)+\log ^7(5)\right )+\log (5) \left (1+\log (5)+108 \log ^2(5)+54 \log ^4(5)+12 \log ^6(5)+\log ^8(5)\right )\right ) \log (x)}{(x-\log (5)) \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 36, normalized size = 1.38
method | result | size |
default | \(x^{8}+12 x^{6}+54 x^{4}+108 x^{2}+x +\frac {\ln \left (5\right )}{-\ln \left (5\right )+x}+\frac {1}{\ln \left (x \right )}\) | \(36\) |
risch | \(\frac {x^{8} \ln \left (5\right )-x^{9}+12 x^{6} \ln \left (5\right )-12 x^{7}+54 x^{4} \ln \left (5\right )-54 x^{5}+108 x^{2} \ln \left (5\right )-108 x^{3}+x \ln \left (5\right )-x^{2}-\ln \left (5\right )}{\ln \left (5\right )-x}+\frac {1}{\ln \left (x \right )}\) | \(76\) |
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. 80 vs.
\(2 (25) = 50\).
time = 0.53, size = 80, normalized size = 3.08 \begin {gather*} \frac {{\left (x^{9} - x^{8} \log \left (5\right ) + 12 \, x^{7} - 12 \, x^{6} \log \left (5\right ) + 54 \, x^{5} - 54 \, x^{4} \log \left (5\right ) + 108 \, x^{3} - x^{2} {\left (108 \, \log \left (5\right ) - 1\right )} - x \log \left (5\right ) + \log \left (5\right )\right )} \log \left (x\right ) + x - \log \left (5\right )}{{\left (x - \log \left (5\right )\right )} \log \left (x\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. 69 vs.
\(2 (25) = 50\).
time = 0.37, size = 69, normalized size = 2.65 \begin {gather*} \frac {{\left (x^{9} + 12 \, x^{7} + 54 \, x^{5} + 108 \, x^{3} + x^{2} - {\left (x^{8} + 12 \, x^{6} + 54 \, x^{4} + 108 \, x^{2} + x - 1\right )} \log \left (5\right )\right )} \log \left (x\right ) + x - \log \left (5\right )}{{\left (x - \log \left (5\right )\right )} \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 32, normalized size = 1.23 \begin {gather*} x^{8} + 12 x^{6} + 54 x^{4} + 108 x^{2} + x + \frac {1}{\log {\left (x \right )}} + \frac {\log {\left (5 \right )}}{x - \log {\left (5 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 35, normalized size = 1.35 \begin {gather*} x^{8} + 12 \, x^{6} + 54 \, x^{4} + 108 \, x^{2} + x + \frac {\log \left (5\right )}{x - \log \left (5\right )} + \frac {1}{\log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.57, size = 35, normalized size = 1.35 \begin {gather*} x+\frac {1}{\ln \left (x\right )}+\frac {\ln \left (5\right )}{x-\ln \left (5\right )}+108\,x^2+54\,x^4+12\,x^6+x^8 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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