Integrand size = 14, antiderivative size = 15 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {\log ^q\left (c x^n\right )}{n q} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2339, 30} \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {\log ^q\left (c x^n\right )}{n q} \]
[In]
[Out]
Rule 30
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\log ^q\left (c x^n\right )}{n q} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {\log ^q\left (c x^n\right )}{n q} \]
[In]
[Out]
Time = 1.42 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(\frac {\ln \left (c \,x^{n}\right )^{q}}{n q}\) | \(16\) |
default | \(\frac {\ln \left (c \,x^{n}\right )^{q}}{n q}\) | \(16\) |
risch | \(\frac {{\left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )}^{-1+q} \left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )}{n q}\) | \(118\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {{\left (n \log \left (x\right ) + \log \left (c\right )\right )} {\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q - 1}}{n q} \]
[In]
[Out]
Time = 0.53 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.27 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=- \begin {cases} - \log {\left (c \right )}^{q - 1} \log {\left (x \right )} & \text {for}\: n = 0 \\- \frac {\begin {cases} \frac {\log {\left (c x^{n} \right )}^{q}}{q} & \text {for}\: q \neq 0 \\\log {\left (\log {\left (c x^{n} \right )} \right )} & \text {otherwise} \end {cases}}{n} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {\log \left (c x^{n}\right )^{q}}{n q} \]
[In]
[Out]
none
Time = 0.53 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q}}{n q} \]
[In]
[Out]
Time = 1.44 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {{\ln \left (c\,x^n\right )}^q}{n\,q} \]
[In]
[Out]