Optimal. Leaf size=33 \[ -\frac {\sqrt {1+x^2}}{x}+\sinh ^{-1}(x)-\frac {\sqrt {1+x^2} \log (x)}{x} \]
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Rubi [A]
time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2373, 283, 221}
\begin {gather*} -\frac {\sqrt {x^2+1}}{x}-\frac {\sqrt {x^2+1} \log (x)}{x}+\sinh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 283
Rule 2373
Rubi steps
\begin {align*} \int \frac {\log (x)}{x^2 \sqrt {1+x^2}} \, dx &=-\frac {\sqrt {1+x^2} \log (x)}{x}+\int \frac {\sqrt {1+x^2}}{x^2} \, dx\\ &=-\frac {\sqrt {1+x^2}}{x}-\frac {\sqrt {1+x^2} \log (x)}{x}+\int \frac {1}{\sqrt {1+x^2}} \, dx\\ &=-\frac {\sqrt {1+x^2}}{x}+\sinh ^{-1}(x)-\frac {\sqrt {1+x^2} \log (x)}{x}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 21, normalized size = 0.64 \begin {gather*} \sinh ^{-1}(x)-\frac {\sqrt {1+x^2} (1+\log (x))}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 29, normalized size = 0.88
method | result | size |
meijerg | \(\arcsinh \left (x \right )+\frac {-\ln \left (x \right ) \sqrt {x^{2}+1}-\sqrt {x^{2}+1}}{x}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.66, size = 29, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {x^{2} + 1} \log \left (x\right )}{x} - \frac {\sqrt {x^{2} + 1}}{x} + \operatorname {arsinh}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 33, normalized size = 1.00 \begin {gather*} -\frac {x \log \left (-x + \sqrt {x^{2} + 1}\right ) + \sqrt {x^{2} + 1} {\left (\log \left (x\right ) + 1\right )} + x}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.78, size = 26, normalized size = 0.79 \begin {gather*} \operatorname {asinh}{\left (x \right )} - \frac {\sqrt {x^{2} + 1} \log {\left (x \right )}}{x} - \frac {\sqrt {x^{2} + 1}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 58, normalized size = 1.76 \begin {gather*} \frac {2 \, \log \left (x\right )}{{\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 1} + \frac {2}{{\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 1} - \log \left (-x + \sqrt {x^{2} + 1}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (x\right )}{x^2\,\sqrt {x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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