Optimal. Leaf size=533 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+1\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}+\frac {2 p \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right ) \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}} \]
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Rubi [A] time = 0.43, antiderivative size = 533, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {205, 2470, 12, 4928, 4856, 2402, 2315, 2447} \[ \frac {i p \text {PolyLog}\left (2,1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{2 \sqrt {f} \sqrt {g}}-\frac {i p \text {PolyLog}\left (2,1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (-\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (\sqrt {f}-i \sqrt {g} x\right ) \left (\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {2 p \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right ) \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 2315
Rule 2402
Rule 2447
Rule 2470
Rule 4856
Rule 4928
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}-(2 e p) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f} \sqrt {g} \left (d+e x^2\right )} \, dx\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {(2 e p) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{d+e x^2} \, dx}{\sqrt {f} \sqrt {g}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {(2 e p) \int \left (-\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{\sqrt {f} \sqrt {g}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {\left (\sqrt {e} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{\sqrt {f} \sqrt {g}}-\frac {\left (\sqrt {e} p\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{\sqrt {f} \sqrt {g}}\\ &=\frac {2 p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}-2 \frac {p \int \frac {\log \left (\frac {2}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{1+\frac {g x^2}{f}} \, dx}{f}+\frac {p \int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {f} \left (-i \sqrt {e}+\frac {\sqrt {-d} \sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{f}+\frac {p \int \frac {\log \left (\frac {2 \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {f} \left (i \sqrt {e}+\frac {\sqrt {-d} \sqrt {g}}{\sqrt {f}}\right ) \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right )}\right )}{1+\frac {g x^2}{f}} \, dx}{f}\\ &=\frac {2 p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}-2 \frac {(i p) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i \sqrt {g} x}{\sqrt {f}}}\right )}{\sqrt {f} \sqrt {g}}\\ &=\frac {2 p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}-\frac {p \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{\sqrt {f} \sqrt {g}}+\frac {\tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {f} \sqrt {g}}-\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f}}{\sqrt {f}-i \sqrt {g} x}\right )}{\sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1+\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}-\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}+\frac {i p \text {Li}_2\left (1-\frac {2 \sqrt {f} \sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (i \sqrt {e} \sqrt {f}+\sqrt {-d} \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}\right )}{2 \sqrt {f} \sqrt {g}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 564, normalized size = 1.06 \[ -\frac {i \left (2 i \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )+p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )+p \text {Li}_2\left (\frac {\sqrt {e} \left (\sqrt {f}-i \sqrt {g} x\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )-p \text {Li}_2\left (\frac {\sqrt {e} \left (i \sqrt {g} x+\sqrt {f}\right )}{\sqrt {e} \sqrt {f}-i \sqrt {-d} \sqrt {g}}\right )-p \text {Li}_2\left (\frac {\sqrt {e} \left (i \sqrt {g} x+\sqrt {f}\right )}{\sqrt {e} \sqrt {f}+i \sqrt {-d} \sqrt {g}}\right )+p \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}}\right )+p \log \left (1-\frac {i \sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {-d} \sqrt {g}-i \sqrt {e} \sqrt {f}}\right )-p \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {\sqrt {g} \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {-d} \sqrt {g}-i \sqrt {e} \sqrt {f}}\right )-p \log \left (1+\frac {i \sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {\sqrt {g} \left (\sqrt {-d}+\sqrt {e} x\right )}{\sqrt {-d} \sqrt {g}+i \sqrt {e} \sqrt {f}}\right )\right )}{2 \sqrt {f} \sqrt {g}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 504, normalized size = 0.95 \[ -\frac {i \pi \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \mathrm {csgn}\left (i c \right ) \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 \sqrt {f g}}+\frac {i \pi \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{2 \sqrt {f g}}+\frac {i \pi \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \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}}{2 \sqrt {f g}}-\frac {i \pi \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{2 \sqrt {f g}}+\frac {\arctan \left (\frac {g x}{\sqrt {f g}}\right ) \ln \relax (c )}{\sqrt {f g}}+\frac {p \left (\ln \left (-\RootOf \left (g \,\textit {\_Z}^{2}+f \right )+x \right ) \ln \left (e \,x^{2}+d \right )-\dilog \left (\frac {\RootOf \left (g \,\textit {\_Z}^{2}+f \right )-x +\RootOf \left (e \,\textit {\_Z}^{2} g +2 \RootOf \left (g \,\textit {\_Z}^{2}+f \right ) \textit {\_Z} g e +d g -e f , \mathit {index} =1\right )}{\RootOf \left (e \,\textit {\_Z}^{2} g +2 \RootOf \left (g \,\textit {\_Z}^{2}+f \right ) \textit {\_Z} g e +d g -e f , \mathit {index} =1\right )}\right )-\dilog \left (\frac {\RootOf \left (g \,\textit {\_Z}^{2}+f \right )-x +\RootOf \left (e \,\textit {\_Z}^{2} g +2 \RootOf \left (g \,\textit {\_Z}^{2}+f \right ) \textit {\_Z} g e +d g -e f , \mathit {index} =2\right )}{\RootOf \left (e \,\textit {\_Z}^{2} g +2 \RootOf \left (g \,\textit {\_Z}^{2}+f \right ) \textit {\_Z} g e +d g -e f , \mathit {index} =2\right )}\right )-\left (\ln \left (\frac {\RootOf \left (g \,\textit {\_Z}^{2}+f \right )-x +\RootOf \left (e \,\textit {\_Z}^{2} g +2 \RootOf \left (g \,\textit {\_Z}^{2}+f \right ) \textit {\_Z} g e +d g -e f , \mathit {index} =1\right )}{\RootOf \left (e \,\textit {\_Z}^{2} g +2 \RootOf \left (g \,\textit {\_Z}^{2}+f \right ) \textit {\_Z} g e +d g -e f , \mathit {index} =1\right )}\right )+\ln \left (\frac {\RootOf \left (g \,\textit {\_Z}^{2}+f \right )-x +\RootOf \left (e \,\textit {\_Z}^{2} g +2 \RootOf \left (g \,\textit {\_Z}^{2}+f \right ) \textit {\_Z} g e +d g -e f , \mathit {index} =2\right )}{\RootOf \left (e \,\textit {\_Z}^{2} g +2 \RootOf \left (g \,\textit {\_Z}^{2}+f \right ) \textit {\_Z} g e +d g -e f , \mathit {index} =2\right )}\right )\right ) \ln \left (-\RootOf \left (g \,\textit {\_Z}^{2}+f \right )+x \right )\right )}{2 g \RootOf \left (g \,\textit {\_Z}^{2}+f \right )}+\frac {\left (-p \ln \left (e \,x^{2}+d \right )+\ln \left (\left (e \,x^{2}+d \right )^{p}\right )\right ) \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{\sqrt {f g}} \]
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 \[ \int \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^2+f} \,d x \]
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 \[ \int \frac {\log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{f + g x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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