Optimal. Leaf size=18 \[ \log (x)-\log (1+x)-\frac {\log (1+x)}{x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2442, 36, 29, 31}
\begin {gather*} \log (x)-\frac {\log (x+1)}{x}-\log (x+1) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 2442
Rubi steps
\begin {align*} \int \frac {\log (1+x)}{x^2} \, dx &=-\frac {\log (1+x)}{x}+\int \frac {1}{x (1+x)} \, dx\\ &=-\frac {\log (1+x)}{x}+\int \frac {1}{x} \, dx-\int \frac {1}{1+x} \, dx\\ &=\log (x)-\log (1+x)-\frac {\log (1+x)}{x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \log (x)-\log (1+x)-\frac {\log (1+x)}{x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [A]
time = 1.77, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\text {Log}\left [1+x\right ]}{x}+\text {Log}\left [x\right ]-\text {Log}\left [1+x\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 16, normalized size = 0.89
method | result | size |
derivativedivides | \(\ln \left (x \right )-\frac {\ln \left (1+x \right ) \left (1+x \right )}{x}\) | \(16\) |
default | \(\ln \left (x \right )-\frac {\ln \left (1+x \right ) \left (1+x \right )}{x}\) | \(16\) |
meijerg | \(\ln \left (x \right )-\frac {\left (2+2 x \right ) \ln \left (1+x \right )}{2 x}\) | \(18\) |
risch | \(\ln \left (x \right )-\ln \left (1+x \right )-\frac {\ln \left (1+x \right )}{x}\) | \(19\) |
norman | \(\frac {-\ln \left (1+x \right ) x -\ln \left (1+x \right )}{x}+\ln \left (x \right )\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\log \left (x + 1\right )}{x} - \log \left (x + 1\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 19, normalized size = 1.06 \begin {gather*} -\frac {{\left (x + 1\right )} \log \left (x + 1\right ) - x \log \left (x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.06, size = 14, normalized size = 0.78 \begin {gather*} \log {\left (x \right )} - \log {\left (x + 1 \right )} - \frac {\log {\left (x + 1 \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \ln \left |x\right |-\ln \left |x+1\right |-\frac {\ln \left (1+x\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.17, size = 18, normalized size = 1.00 \begin {gather*} -\ln \left (\frac {1}{x}+1\right )-\frac {\ln \left (x+1\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________