Integrand size = 22, antiderivative size = 25 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {d x^m}{m}+\frac {e \log ^q\left (c x^n\right )}{n q} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {14, 2339, 30} \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {e \log ^q\left (c x^n\right )}{n q}+\frac {d x^m}{m} \]
[In]
[Out]
Rule 14
Rule 30
Rule 2339
Rubi steps \begin{align*} \text {integral}& = \int \left (d x^{-1+m}+\frac {e \log ^{-1+q}\left (c x^n\right )}{x}\right ) \, dx \\ & = \frac {d x^m}{m}+e \int \frac {\log ^{-1+q}\left (c x^n\right )}{x} \, dx \\ & = \frac {d x^m}{m}+\frac {e \text {Subst}\left (\int x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {d x^m}{m}+\frac {e \log ^q\left (c x^n\right )}{n q} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {d x^m}{m}+\frac {e \log ^q\left (c x^n\right )}{n q} \]
[In]
[Out]
Time = 0.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
default | \(\frac {d \,x^{m}}{m}+\frac {e \ln \left (c \,x^{n}\right )^{q}}{n q}\) | \(26\) |
parallelrisch | \(-\frac {-d \,x^{m} n q -\ln \left (c \,x^{n}\right ) \ln \left (c \,x^{n}\right )^{-1+q} e m}{m n q}\) | \(41\) |
risch | \(\frac {d \,x^{m}}{m}+\frac {e {\left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )}^{-1+q} \left (\ln \left (c \right )+\ln \left (x^{n}\right )-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i c \right )\right ) \left (-\operatorname {csgn}\left (i c \,x^{n}\right )+\operatorname {csgn}\left (i x^{n}\right )\right )}{2}\right )}{n q}\) | \(128\) |
[In]
[Out]
none
Time = 0.34 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {d n q x^{m} + {\left (e m n \log \left (x\right ) + e m \log \left (c\right )\right )} {\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q - 1}}{m n q} \]
[In]
[Out]
Time = 1.42 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x} \, dx=- d \left (\begin {cases} - \log {\left (x \right )} & \text {for}\: m = 0 \\- \frac {x^{m}}{m} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \log {\left (c \right )}^{q - 1} \log {\left (x \right )} & \text {for}\: n = 0 \\\frac {\begin {cases} \frac {\log {\left (c x^{n} \right )}^{q}}{q} & \text {for}\: q \neq 0 \\\log {\left (\log {\left (c x^{n} \right )} \right )} & \text {otherwise} \end {cases}}{n} & \text {otherwise} \end {cases}\right ) \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {d x^{m}}{m} + \frac {e \log \left (c x^{n}\right )^{q}}{n q} \]
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {d x^{m}}{m} + \frac {{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} e}{n q} \]
[In]
[Out]
Time = 1.54 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {d x^m+e \log ^{-1+q}\left (c x^n\right )}{x} \, dx=\frac {d\,x^m}{m}+\frac {e\,{\ln \left (c\,x^n\right )}^q}{n\,q} \]
[In]
[Out]