Optimal. Leaf size=22 \[ \frac {2 \tan ^{-1}\left (\frac {2 \log (7 x)+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {618, 204} \[ \frac {2 \tan ^{-1}\left (\frac {2 \log (7 x)+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rubi steps
\begin {align*} \int \frac {1}{x+x \log (7 x)+x \log ^2(7 x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\log (7 x)\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \log (7 x)\right )\right )\\ &=\frac {2 \tan ^{-1}\left (\frac {1+2 \log (7 x)}{\sqrt {3}}\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 22, normalized size = 1.00 \[ \frac {2 \tan ^{-1}\left (\frac {2 \log (7 x)+1}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 21, normalized size = 0.95 \[ \frac {2}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \log \left (7 \, x\right ) + \frac {1}{3} \, \sqrt {3}\right ) \]
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 \[ \int \frac {1}{x \log \left (7 \, x\right )^{2} + x \log \left (7 \, x\right ) + x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 20, normalized size = 0.91 \[ \frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \ln \left (7 x \right )+1\right ) \sqrt {3}}{3}\right )}{3} \]
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 \[ \int \frac {1}{x \log \left (7 \, x\right )^{2} + x \log \left (7 \, x\right ) + x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 19, normalized size = 0.86 \[ \frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\left (2\,\ln \left (7\,x\right )+1\right )}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 22, normalized size = 1.00 \[ \operatorname {RootSum} {\left (3 z^{2} + 1, \left (i \mapsto i \log {\left (\frac {3 i}{2} + \log {\left (7 x \right )} + \frac {1}{2} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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