Optimal. Leaf size=16 \[ \frac {\text {Li}_2\left (1-\frac {x^n}{d}\right )}{n} \]
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Rubi [A]
time = 0.04, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2374, 2352}
\begin {gather*} \frac {\text {PolyLog}\left (2,1-\frac {x^n}{d}\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2374
Rubi steps
\begin {align*} \int \frac {x^{-1+n} \log \left (\frac {x^n}{d}\right )}{d-x^n} \, dx &=\frac {\text {Subst}\left (\int \frac {\log \left (\frac {x}{d}\right )}{d-x} \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Li}_2\left (1-\frac {x^n}{d}\right )}{n}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.06 \begin {gather*} \frac {\text {Li}_2\left (\frac {d-x^n}{d}\right )}{n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 13, normalized size = 0.81
method | result | size |
default | \(\frac {\dilog \left (\frac {x^{n}}{d}\right )}{n}\) | \(13\) |
risch | \(-\frac {\ln \left (x^{n}\right ) \ln \left (-\frac {-d +x^{n}}{d}\right )}{n}-\frac {\dilog \left (-\frac {-d +x^{n}}{d}\right )}{n}-\frac {i \ln \left (-d +x^{n}\right ) \pi \,\mathrm {csgn}\left (\frac {i}{d}\right ) \mathrm {csgn}\left (\frac {i x^{n}}{d}\right )^{2}}{2 n}+\frac {i \ln \left (-d +x^{n}\right ) \pi \,\mathrm {csgn}\left (\frac {i}{d}\right ) \mathrm {csgn}\left (\frac {i x^{n}}{d}\right ) \mathrm {csgn}\left (i x^{n}\right )}{2 n}+\frac {i \ln \left (-d +x^{n}\right ) \pi \mathrm {csgn}\left (\frac {i x^{n}}{d}\right )^{3}}{2 n}-\frac {i \ln \left (-d +x^{n}\right ) \pi \mathrm {csgn}\left (\frac {i x^{n}}{d}\right )^{2} \mathrm {csgn}\left (i x^{n}\right )}{2 n}+\frac {\ln \left (-d +x^{n}\right ) \ln \left (d \right )}{n}\) | \(190\) |
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. 45 vs.
\(2 (15) = 30\).
time = 0.38, size = 45, normalized size = 2.81 \begin {gather*} \frac {\log \left (d\right ) \log \left (-d + x^{n}\right )}{n} - \frac {\log \left (x^{n}\right ) \log \left (-\frac {x^{n}}{d} + 1\right ) + {\rm Li}_2\left (\frac {x^{n}}{d}\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 50 vs.
\(2 (15) = 30\).
time = 0.37, size = 50, normalized size = 3.12 \begin {gather*} -\frac {n \log \left (x\right ) \log \left (\frac {d - x^{n}}{d}\right ) + \log \left (-d + x^{n}\right ) \log \left (\frac {1}{d}\right ) + {\rm Li}_2\left (-\frac {d - x^{n}}{d} + 1\right )}{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 = 3.48, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {x^n}{d}\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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