Optimal. Leaf size=25 \[ -\frac {\text {Li}_2\left (1-c \left (d+e x^n\right )\right )}{c e n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2525, 2459,
2440, 2438} \begin {gather*} -\frac {\text {PolyLog}\left (2,1-c \left (d+e x^n\right )\right )}{c e n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2438
Rule 2440
Rule 2459
Rule 2525
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^n\right )\right )}{x \left (c e-(1-c d) x^{-n}\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {\log (c (d+e x))}{\left (c e+\frac {-1+c d}{x}\right ) x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \frac {\log (c (d+e x))}{-1+c d+c e x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,-1+c d+c e x^n\right )}{c e n}\\ &=-\frac {\text {Li}_2\left (1-c \left (d+e x^n\right )\right )}{c e n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 25, normalized size = 1.00 \begin {gather*} -\frac {\text {Li}_2\left (1-c \left (d+e x^n\right )\right )}{c e n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.18, size = 23, normalized size = 0.92
method | result | size |
derivativedivides | \(-\frac {\dilog \left (c e \,x^{n}+c d \right )}{n c e}\) | \(23\) |
default | \(-\frac {\dilog \left (c e \,x^{n}+c d \right )}{n c e}\) | \(23\) |
risch | \(\frac {\ln \left (1-c \left (d +e \,x^{n}\right )\right ) \ln \left (d +e \,x^{n}\right )}{e n c}-\frac {\ln \left (1-c \left (d +e \,x^{n}\right )\right ) \ln \left (c \left (d +e \,x^{n}\right )\right )}{e n c}-\frac {\dilog \left (c \left (d +e \,x^{n}\right )\right )}{e n c}+\frac {i \ln \left (-1+c d +c e \,x^{n}\right ) \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )\right )^{2}}{2 n c e}-\frac {i \ln \left (-1+c d +c e \,x^{n}\right ) \pi \,\mathrm {csgn}\left (i \left (d +e \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )\right ) \mathrm {csgn}\left (i c \right )}{2 n c e}-\frac {i \ln \left (-1+c d +c e \,x^{n}\right ) \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )\right )^{3}}{2 n c e}+\frac {i \ln \left (-1+c d +c e \,x^{n}\right ) \pi \mathrm {csgn}\left (i c \left (d +e \,x^{n}\right )\right )^{2} \mathrm {csgn}\left (i c \right )}{2 n c e}+\frac {\ln \left (-1+c d +c e \,x^{n}\right ) \ln \left (c \right )}{n c e}\) | \(298\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs.
\(2 (24) = 48\).
time = 0.37, size = 114, normalized size = 4.56 \begin {gather*} {\left (\frac {e^{\left (-1\right )} \log \left (c e + \frac {c d - 1}{x^{n}}\right )}{c n} - \frac {e^{\left (-1\right )} \log \left (\frac {1}{x^{n}}\right )}{c n}\right )} \log \left ({\left (x^{n} e + d\right )} c\right ) - \frac {{\left (\log \left (c d + c e^{\left (n \log \left (x\right ) + 1\right )}\right ) \log \left (c d + c e^{\left (n \log \left (x\right ) + 1\right )} - 1\right ) + {\rm Li}_2\left (-c d - c e^{\left (n \log \left (x\right ) + 1\right )} + 1\right )\right )} e^{\left (-1\right )}}{c n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 25, normalized size = 1.00 \begin {gather*} -\frac {{\rm Li}_2\left (-c x^{n} e - c d + 1\right ) e^{\left (-1\right )}}{c n} \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: TypeError} \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 = -0.04 \begin {gather*} \int \frac {\ln \left (c\,\left (d+e\,x^n\right )\right )}{x\,\left (c\,e+\frac {c\,d-1}{x^n}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________