Optimal. Leaf size=27 \[ \left (-3+\frac {5^{4 x}-x}{x}+\frac {x-4 \log (x)}{x}\right )^2 \]
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Rubi [A] time = 0.23, antiderivative size = 53, normalized size of antiderivative = 1.96, number of steps used = 6, number of rules used = 4, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {14, 2197, 6712, 2288} \begin {gather*} \frac {5^{8 x}}{x^2}+\frac {16 \log ^2(x)}{x^2}-\frac {8\ 625^x \left (3 x^2 \log (5)+2 x \log (25) \log (x)\right )}{x^3 \log (625)}+\frac {24 \log (x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2197
Rule 2288
Rule 6712
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2\ 5^{8 x} (-1+4 x \log (5))}{x^3}-\frac {8 (-1+\log (x)) (3 x+4 \log (x))}{x^3}-\frac {2\ 625^x \left (4-3 x+12 x^2 \log (5)-8 \log (x)+8 x \log (25) \log (x)\right )}{x^3}\right ) \, dx\\ &=2 \int \frac {5^{8 x} (-1+4 x \log (5))}{x^3} \, dx-2 \int \frac {625^x \left (4-3 x+12 x^2 \log (5)-8 \log (x)+8 x \log (25) \log (x)\right )}{x^3} \, dx-8 \int \frac {(-1+\log (x)) (3 x+4 \log (x))}{x^3} \, dx\\ &=\frac {5^{8 x}}{x^2}-\frac {8\ 625^x \left (3 x^2 \log (5)+2 x \log (25) \log (x)\right )}{x^3 \log (625)}+8 \operatorname {Subst}\left (\int (3+4 x) \, dx,x,\frac {\log (x)}{x}\right )\\ &=\frac {5^{8 x}}{x^2}+\frac {24 \log (x)}{x}+\frac {16 \log ^2(x)}{x^2}-\frac {8\ 625^x \left (3 x^2 \log (5)+2 x \log (25) \log (x)\right )}{x^3 \log (625)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.17, size = 47, normalized size = 1.74 \begin {gather*} \frac {4 \left (625^x \left (625^x-6 x\right ) \log (5)+\left (-4 625^x \log (25)+6 x \log (625)\right ) \log (x)+4 \log (625) \log ^2(x)\right )}{x^2 \log (625)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 40, normalized size = 1.48 \begin {gather*} -\frac {6 \cdot 5^{4 \, x} x + 8 \, {\left (5^{4 \, x} - 3 \, x\right )} \log \relax (x) - 16 \, \log \relax (x)^{2} - 5^{8 \, x}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left ({\left (4 \, x \log \relax (5) - 1\right )} 5^{8 \, x} - {\left (12 \, x^{2} \log \relax (5) - 3 \, x + 4\right )} 5^{4 \, x} - 4 \, {\left (2 \, {\left (2 \, x \log \relax (5) - 1\right )} 5^{4 \, x} + 3 \, x - 4\right )} \log \relax (x) - 16 \, \log \relax (x)^{2} + 12 \, x\right )}}{x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 44, normalized size = 1.63
method | result | size |
risch | \(\frac {16 \ln \relax (x )^{2}}{x^{2}}+\frac {8 \left (3 x -625^{x}\right ) \ln \relax (x )}{x^{2}}-\frac {625^{x} \left (6 x -625^{x}\right )}{x^{2}}\) | \(44\) |
default | \(\frac {-6 \,{\mathrm e}^{4 x \ln \relax (5)} x -8 \ln \relax (x ) {\mathrm e}^{4 x \ln \relax (5)}}{x^{2}}+\frac {16 \ln \relax (x )^{2}}{x^{2}}+\frac {24 \ln \relax (x )}{x}+\frac {{\mathrm e}^{8 x \ln \relax (5)}}{x^{2}}\) | \(54\) |
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*} 64 \, \Gamma \left (-1, -8 \, x \log \relax (5)\right ) \log \relax (5)^{2} + 128 \, \Gamma \left (-2, -4 \, x \log \relax (5)\right ) \log \relax (5)^{2} + 128 \, \Gamma \left (-2, -8 \, x \log \relax (5)\right ) \log \relax (5)^{2} - 24 \, {\rm Ei}\left (4 \, x \log \relax (5)\right ) \log \relax (5) + 24 \, \Gamma \left (-1, -4 \, x \log \relax (5)\right ) \log \relax (5) + \frac {24 \, \log \relax (x)}{x} - \frac {8 \, {\left (5^{4 \, x} \log \relax (x) - 2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) - 1\right )}}{x^{2}} - \frac {16 \, \log \relax (x)}{x^{2}} - \frac {8}{x^{2}} + 8 \, \int \frac {5^{4 \, x}}{x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.12, size = 31, normalized size = 1.15 \begin {gather*} \frac {\left (4\,\ln \relax (x)-5^{4\,x}\right )\,\left (6\,x+4\,\ln \relax (x)-5^{4\,x}\right )}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.43, size = 54, normalized size = 2.00 \begin {gather*} \frac {24 \log {\relax (x )}}{x} + \frac {16 \log {\relax (x )}^{2}}{x^{2}} + \frac {x^{2} e^{8 x \log {\relax (5 )}} + \left (- 6 x^{3} - 8 x^{2} \log {\relax (x )}\right ) e^{4 x \log {\relax (5 )}}}{x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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