Optimal. Leaf size=25 \[ -\frac {\text {Li}_2\left (1-c \left (d+e x^n\right )\right )}{c e n} \]
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Rubi [A]
time = 0.07, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2525, 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.
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Rule 2438
Rule 2440
Rule 2525
Rubi steps
\begin {align*} \int \frac {x^{-1+n} \log \left (c \left (d+e x^n\right )\right )}{-1+c d+c e x^n} \, dx &=\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*}
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Mathematica [A]
time = 0.02, 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.
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Maple [A]
time = 0.51, size = 23, normalized size = 0.92
method | result | size |
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.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs.
\(2 (24) = 48\).
time = 0.29, size = 118, normalized size = 4.72 \begin {gather*} \frac {e^{\left (-1\right )} \log \left (c x^{n} e + c d - 1\right ) \log \left ({\left (x^{n} e + d\right )} c\right )}{c n} - \frac {e^{\left (-1\right )} \log \left (c x^{n} e + c d - 1\right ) \log \left (x^{n} e + d\right )}{c n} + \frac {{\left (\log \left (-c d - c e^{\left (n \log \left (x\right ) + 1\right )} + 1\right ) \log \left (d + e^{\left (n \log \left (x\right ) + 1\right )}\right ) + {\rm Li}_2\left (c d + c e^{\left (n \log \left (x\right ) + 1\right )}\right )\right )} e^{\left (-1\right )}}{c n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, 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.
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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.
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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.
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Mupad [B]
time = 0.65, size = 21, normalized size = 0.84 \begin {gather*} -\frac {{\mathrm {Li}}_{\mathrm {2}}\left (c\,\left (d+e\,x^n\right )\right )}{c\,e\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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