Integrand size = 6, antiderivative size = 19 \[ \int \log (a+b x) \, dx=-x+\frac {(a+b x) \log (a+b x)}{b} \]
[Out]
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2436, 2332} \[ \int \log (a+b x) \, dx=\frac {(a+b x) \log (a+b x)}{b}-x \]
[In]
[Out]
Rule 2332
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int \log (x) \, dx,x,a+b x)}{b} \\ & = -x+\frac {(a+b x) \log (a+b x)}{b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \log (a+b x) \, dx=-x+\frac {(a+b x) \log (a+b x)}{b} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26
method | result | size |
norman | \(x \ln \left (b x +a \right )+\frac {a \ln \left (b x +a \right )}{b}-x\) | \(24\) |
risch | \(x \ln \left (b x +a \right )+\frac {a \ln \left (b x +a \right )}{b}-x\) | \(24\) |
derivativedivides | \(\frac {\left (b x +a \right ) \ln \left (b x +a \right )-b x -a}{b}\) | \(25\) |
default | \(\frac {\left (b x +a \right ) \ln \left (b x +a \right )-b x -a}{b}\) | \(25\) |
parallelrisch | \(\frac {\ln \left (b x +a \right ) x b -b x +a \ln \left (b x +a \right )+a}{b}\) | \(28\) |
parts | \(x \ln \left (b x +a \right )-b \left (\frac {x}{b}-\frac {a \ln \left (b x +a \right )}{b^{2}}\right )\) | \(31\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \log (a+b x) \, dx=-\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right )}{b} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \log (a+b x) \, dx=- b \left (- \frac {a \log {\left (a + b x \right )}}{b^{2}} + \frac {x}{b}\right ) + x \log {\left (a + b x \right )} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \log (a+b x) \, dx=-\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right ) + a}{b} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \log (a+b x) \, dx=-\frac {b x - {\left (b x + a\right )} \log \left (b x + a\right ) + a}{b} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \log (a+b x) \, dx=x\,\ln \left (a+b\,x\right )-x+\frac {a\,\ln \left (a+b\,x\right )}{b} \]
[In]
[Out]