Optimal. Leaf size=229 \[ \frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
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Rubi [A] time = 0.23, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2409, 2394, 2393, 2391} \[ -\frac {p \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2394
Rule 2409
Rubi steps
\begin {align*} \int \frac {\log \left (c (d+e x)^p\right )}{f+g x^2} \, dx &=\int \left (\frac {\sqrt {-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 \sqrt {-f}}-\frac {\int \frac {\log \left (c (d+e x)^p\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 \sqrt {-f}}\\ &=\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {(e p) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{2 \sqrt {-f} \sqrt {g}}+\frac {(e p) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{2 \sqrt {-f} \sqrt {g}}\\ &=\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 \sqrt {-f} \sqrt {g}}\\ &=\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c (d+e x)^p\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} \sqrt {g}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 178, normalized size = 0.78 \[ \frac {\log \left (c (d+e x)^p\right ) \left (\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right )-\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )\right )-p \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+p \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 \sqrt {-f} \sqrt {g}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (e x + d\right )}^{p} c\right )}{g x^{2} + f}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (e x + d\right )}^{p} c\right )}{g x^{2} + f}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.67, size = 419, normalized size = 1.83 \[ -\frac {i \pi \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{p}\right )}{2 \sqrt {f g}}+\frac {i \pi \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{p}\right )^{2}}{2 \sqrt {f g}}+\frac {i \pi \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \mathrm {csgn}\left (i \left (e x +d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{p}\right )^{2}}{2 \sqrt {f g}}-\frac {i \pi \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{p}\right )^{3}}{2 \sqrt {f g}}+\frac {p \ln \left (\frac {d g +\sqrt {-f g}\, e -\left (e x +d \right ) g}{d g +\sqrt {-f g}\, e}\right ) \ln \left (e x +d \right )}{2 \sqrt {-f g}}-\frac {p \ln \left (\frac {-d g +\sqrt {-f g}\, e +\left (e x +d \right ) g}{-d g +\sqrt {-f g}\, e}\right ) \ln \left (e x +d \right )}{2 \sqrt {-f g}}+\frac {p \dilog \left (\frac {d g +\sqrt {-f g}\, e -\left (e x +d \right ) g}{d g +\sqrt {-f g}\, e}\right )}{2 \sqrt {-f g}}-\frac {p \dilog \left (\frac {-d g +\sqrt {-f g}\, e +\left (e x +d \right ) g}{-d g +\sqrt {-f g}\, e}\right )}{2 \sqrt {-f g}}+\frac {\arctan \left (\frac {g x}{\sqrt {f g}}\right ) \ln \relax (c )}{\sqrt {f g}}+\frac {\left (-p \ln \left (e x +d \right )+\ln \left (\left (e x +d \right )^{p}\right )\right ) \arctan \left (\frac {-2 d g +2 \left (e x +d \right ) g}{2 \sqrt {f g}\, e}\right )}{\sqrt {f g}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.29, size = 309, normalized size = 1.35 \[ \frac {e p {\left (\frac {2 \, \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left (e x + d\right )}{e} + \frac {\arctan \left (\frac {{\left (e^{2} x + d e\right )} \sqrt {f} \sqrt {g}}{e^{2} f + d^{2} g}, \frac {d e g x + d^{2} g}{e^{2} f + d^{2} g}\right ) \log \left (g x^{2} + f\right ) - \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right ) \log \left (\frac {e^{2} g x^{2} + 2 \, d e g x + d^{2} g}{e^{2} f + d^{2} g}\right ) - i \, {\rm Li}_2\left (\frac {d e g x + e^{2} f - {\left (i \, e^{2} x - i \, d e\right )} \sqrt {f} \sqrt {g}}{e^{2} f + 2 i \, d e \sqrt {f} \sqrt {g} - d^{2} g}\right ) + i \, {\rm Li}_2\left (\frac {d e g x + e^{2} f + {\left (i \, e^{2} x - i \, d e\right )} \sqrt {f} \sqrt {g}}{e^{2} f - 2 i \, d e \sqrt {f} \sqrt {g} - d^{2} g}\right )}{e}\right )}}{2 \, \sqrt {f g}} - \frac {p \arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left (e x + d\right )}{\sqrt {f g}} + \frac {\arctan \left (\frac {g x}{\sqrt {f g}}\right ) \log \left ({\left (e x + d\right )}^{p} c\right )}{\sqrt {f g}} \]
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 \[ \int \frac {\ln \left (c\,{\left (d+e\,x\right )}^p\right )}{g\,x^2+f} \,d x \]
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 \[ \int \frac {\log {\left (c \left (d + e x\right )^{p} \right )}}{f + g x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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