Optimal. Leaf size=70 \[ \frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g}+\frac {p \text {Li}_2\left (-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g} \]
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Rubi [A]
time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2525, 2441,
2440, 2438} \begin {gather*} \frac {p \text {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g}+\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g} \end {gather*}
Antiderivative was successfully verified.
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Rule 2438
Rule 2440
Rule 2441
Rule 2525
Rubi steps
\begin {align*} \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )\\ &=\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g}-\frac {(e p) \text {Subst}\left (\int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx,x,x^2\right )}{2 g}\\ &=\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g}-\frac {p \text {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x^2\right )}{2 g}\\ &=\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g}+\frac {p \text {Li}_2\left (-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 64, normalized size = 0.91 \begin {gather*} \frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )+p \text {Li}_2\left (\frac {g \left (d+e x^2\right )}{-e f+d g}\right )}{2 g} \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 = 0.37, size = 472, normalized size = 6.74
method | result | size |
risch | \(\frac {\ln \left (g \,x^{2}+f \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{2 g}-\frac {p \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (e \,\textit {\_Z}^{2}+d \right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (g \,x^{2}+f \right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\RootOf \left (e \,\textit {\_Z}^{2} g +2 \textit {\_Z} \underline {\hspace {1.25 ex}}\alpha g e -d g +e f , \mathit {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\RootOf \left (e \,\textit {\_Z}^{2} g +2 \textit {\_Z} \underline {\hspace {1.25 ex}}\alpha g e -d g +e f , \mathit {index} =1\right )}\right )+\ln \left (\frac {\RootOf \left (e \,\textit {\_Z}^{2} g +2 \textit {\_Z} \underline {\hspace {1.25 ex}}\alpha g e -d g +e f , \mathit {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\RootOf \left (e \,\textit {\_Z}^{2} g +2 \textit {\_Z} \underline {\hspace {1.25 ex}}\alpha g e -d g +e f , \mathit {index} =2\right )}\right )\right )-\dilog \left (\frac {\RootOf \left (e \,\textit {\_Z}^{2} g +2 \textit {\_Z} \underline {\hspace {1.25 ex}}\alpha g e -d g +e f , \mathit {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\RootOf \left (e \,\textit {\_Z}^{2} g +2 \textit {\_Z} \underline {\hspace {1.25 ex}}\alpha g e -d g +e f , \mathit {index} =1\right )}\right )-\dilog \left (\frac {\RootOf \left (e \,\textit {\_Z}^{2} g +2 \textit {\_Z} \underline {\hspace {1.25 ex}}\alpha g e -d g +e f , \mathit {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\RootOf \left (e \,\textit {\_Z}^{2} g +2 \textit {\_Z} \underline {\hspace {1.25 ex}}\alpha g e -d g +e f , \mathit {index} =2\right )}\right )\right )\right )}{2 g}+\frac {i \ln \left (g \,x^{2}+f \right ) \pi \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{4 g}-\frac {i \ln \left (g \,x^{2}+f \right ) \pi \,\mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{4 g}-\frac {i \ln \left (g \,x^{2}+f \right ) \pi \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{4 g}+\frac {i \ln \left (g \,x^{2}+f \right ) \pi \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )}{4 g}+\frac {\ln \left (g \,x^{2}+f \right ) \ln \left (c \right )}{2 g}\) | \(472\) |
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. 146 vs.
\(2 (70) = 140\).
time = 0.32, size = 146, normalized size = 2.09 \begin {gather*} \frac {{\left (e^{\left (-1\right )} \log \left (g x^{2} + f\right ) \log \left (x^{2} e + d\right ) - {\left (\log \left (g x^{2} + f\right ) \log \left (\frac {g x^{2} e + f e}{d g - f e} + 1\right ) + {\rm Li}_2\left (-\frac {g x^{2} e + f e}{d g - f e}\right )\right )} e^{\left (-1\right )}\right )} p e}{2 \, g} - \frac {p \log \left (g x^{2} + f\right ) \log \left (x^{2} e + d\right )}{2 \, g} + \frac {\log \left (g x^{2} + f\right ) \log \left ({\left (x^{2} e + d\right )}^{p} c\right )}{2 \, g} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{f + g x^{2}}\, dx \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 {x\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^2+f} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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