Optimal. Leaf size=70 \[ -\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )+\frac {1}{2} b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )-\frac {1}{4} b^2 n^2 \text {Li}_4\left (-d f x^2\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2421, 2430,
6724} \begin {gather*} \frac {1}{2} b n \text {PolyLog}\left (3,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{2} \text {PolyLog}\left (2,-d f x^2\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{4} b^2 n^2 \text {PolyLog}\left (4,-d f x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 2430
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x} \, dx &=-\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )+(b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^2\right )}{x} \, dx\\ &=-\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )+\frac {1}{2} b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )-\frac {1}{2} \left (b^2 n^2\right ) \int \frac {\text {Li}_3\left (-d f x^2\right )}{x} \, dx\\ &=-\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f x^2\right )+\frac {1}{2} b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f x^2\right )-\frac {1}{4} b^2 n^2 \text {Li}_4\left (-d f x^2\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.13, size = 484, normalized size = 6.91 \begin {gather*} \frac {1}{3} \left (\log (x) \left (b^2 n^2 \log ^2(x)-3 b n \log (x) \left (a+b \log \left (c x^n\right )\right )+3 \left (a+b \log \left (c x^n\right )\right )^2\right ) \log \left (1+d f x^2\right )-3 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\log (x) \left (\log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log \left (1+i \sqrt {d} \sqrt {f} x\right )\right )+\text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+\text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )\right )+3 b n \left (-a+b n \log (x)-b \log \left (c x^n\right )\right ) \left (\log ^2(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^2(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+2 \log (x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-2 \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )-2 \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )\right )-b^2 n^2 \left (\log ^3(x) \log \left (1-i \sqrt {d} \sqrt {f} x\right )+\log ^3(x) \log \left (1+i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (-i \sqrt {d} \sqrt {f} x\right )+3 \log ^2(x) \text {Li}_2\left (i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (-i \sqrt {d} \sqrt {f} x\right )-6 \log (x) \text {Li}_3\left (i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_4\left (-i \sqrt {d} \sqrt {f} x\right )+6 \text {Li}_4\left (i \sqrt {d} \sqrt {f} x\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.22, size = 6180, normalized size = 88.29
method | result | size |
risch | \(\text {Expression too large to display}\) | \(6180\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\ln \left (d\,\left (f\,x^2+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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