Optimal. Leaf size=70 \[ \frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f n}+\frac {p \text {Li}_2\left (\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2525, 2459,
2441, 2440, 2438} \begin {gather*} \frac {p \text {PolyLog}\left (2,\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f n}+\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (f x^n+g\right )}{d f-e g}\right )}{f n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2438
Rule 2440
Rule 2441
Rule 2459
Rule 2525
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{\left (f+\frac {g}{x}\right ) x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{g+f x} \, dx,x,x^n\right )}{n}\\ &=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f n}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e (g+f x)}{-d f+e g}\right )}{d+e x} \, dx,x,x^n\right )}{f n}\\ &=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f n}-\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {f x}{-d f+e g}\right )}{x} \, dx,x,d+e x^n\right )}{f n}\\ &=\frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac {e \left (g+f x^n\right )}{d f-e g}\right )}{f n}+\frac {p \text {Li}_2\left (\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 64, normalized size = 0.91 \begin {gather*} \frac {\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac {e \left (g+f x^n\right )}{-d f+e g}\right )+p \text {Li}_2\left (\frac {f \left (d+e x^n\right )}{d f-e g}\right )}{f n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.92, size = 298, normalized size = 4.26
method | result | size |
risch | \(\frac {\ln \left (g +f \,x^{n}\right ) \ln \left (\left (d +e \,x^{n}\right )^{p}\right )}{n f}-\frac {p \dilog \left (\frac {\left (g +f \,x^{n}\right ) e +d f -e g}{d f -e g}\right )}{n f}-\frac {p \ln \left (g +f \,x^{n}\right ) \ln \left (\frac {\left (g +f \,x^{n}\right ) e +d f -e g}{d f -e g}\right )}{n f}+\frac {i \ln \left (g +f \,x^{n}\right ) \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2}}{2 n f}-\frac {i \ln \left (g +f \,x^{n}\right ) \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{2 n f}-\frac {i \ln \left (g +f \,x^{n}\right ) \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{3}}{2 n f}+\frac {i \ln \left (g +f \,x^{n}\right ) \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{2 n f}+\frac {\ln \left (g +f \,x^{n}\right ) \ln \left (c \right )}{n f}\) | \(298\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.65, size = 123, normalized size = 1.76 \begin {gather*} {\left (\frac {\log \left (f + \frac {g}{x^{n}}\right )}{f n} - \frac {\log \left (\frac {1}{x^{n}}\right )}{f n}\right )} \log \left ({\left (x^{n} e + d\right )}^{p} c\right ) - \frac {{\left (\log \left (f x^{n} + g\right ) \log \left (\frac {g e + f e^{\left (n \log \left (x\right ) + 1\right )}}{d f - g e} + 1\right ) + {\rm Li}_2\left (-\frac {g e + f e^{\left (n \log \left (x\right ) + 1\right )}}{d f - g e}\right )\right )} p}{f n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x\,\left (f+\frac {g}{x^n}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________