Optimal. Leaf size=194 \[ 4 b m n x-\frac {2 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-b n x \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {i b \sqrt {e} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {i b \sqrt {e} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}} \]
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Rubi [A]
time = 0.09, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2498, 327, 211,
2417, 4940, 2438} \begin {gather*} -\frac {i b \sqrt {e} m n \text {PolyLog}\left (2,-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {i b \sqrt {e} m n \text {PolyLog}\left (2,\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {2 \sqrt {e} m \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-2 m x \left (a+b \log \left (c x^n\right )\right )-\frac {2 b \sqrt {e} m n \text {ArcTan}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-b n x \log \left (d \left (e+f x^2\right )^m\right )+4 b m n x \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 327
Rule 2417
Rule 2438
Rule 2498
Rule 4940
Rubi steps
\begin {align*} \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right ) \, dx &=-2 m x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-(b n) \int \left (-2 m+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f} x}+\log \left (d \left (e+f x^2\right )^m\right )\right ) \, dx\\ &=2 b m n x-2 m x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-(b n) \int \log \left (d \left (e+f x^2\right )^m\right ) \, dx-\frac {\left (2 b \sqrt {e} m n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {f}}\\ &=2 b m n x-2 m x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-b n x \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {\left (i b \sqrt {e} m n\right ) \int \frac {\log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {f}}+\frac {\left (i b \sqrt {e} m n\right ) \int \frac {\log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{x} \, dx}{\sqrt {f}}+(2 b f m n) \int \frac {x^2}{e+f x^2} \, dx\\ &=4 b m n x-2 m x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-b n x \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {i b \sqrt {e} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {i b \sqrt {e} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-(2 b e m n) \int \frac {1}{e+f x^2} \, dx\\ &=4 b m n x-\frac {2 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}-2 m x \left (a+b \log \left (c x^n\right )\right )+\frac {2 \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {f}}-b n x \log \left (d \left (e+f x^2\right )^m\right )+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^2\right )^m\right )-\frac {i b \sqrt {e} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}+\frac {i b \sqrt {e} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 332, normalized size = 1.71 \begin {gather*} \frac {-2 a \sqrt {f} m x+4 b \sqrt {f} m n x+2 a \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-2 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right )-2 b \sqrt {e} m n \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log (x)-2 b \sqrt {f} m x \log \left (c x^n\right )+2 b \sqrt {e} m \tan ^{-1}\left (\frac {\sqrt {f} x}{\sqrt {e}}\right ) \log \left (c x^n\right )+i b \sqrt {e} m n \log (x) \log \left (1-\frac {i \sqrt {f} x}{\sqrt {e}}\right )-i b \sqrt {e} m n \log (x) \log \left (1+\frac {i \sqrt {f} x}{\sqrt {e}}\right )+a \sqrt {f} x \log \left (d \left (e+f x^2\right )^m\right )-b \sqrt {f} n x \log \left (d \left (e+f x^2\right )^m\right )+b \sqrt {f} x \log \left (c x^n\right ) \log \left (d \left (e+f x^2\right )^m\right )-i b \sqrt {e} m n \text {Li}_2\left (-\frac {i \sqrt {f} x}{\sqrt {e}}\right )+i b \sqrt {e} m n \text {Li}_2\left (\frac {i \sqrt {f} x}{\sqrt {e}}\right )}{\sqrt {f}} \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 = 0.16, size = 2001, normalized size = 10.31
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2001\) |
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 \ln \left (d\,{\left (f\,x^2+e\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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