Optimal. Leaf size=44 \[ -\frac {1}{2} \log \left (-\frac {b}{a x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )-\frac {1}{2} p \text {Li}_2\left (\frac {b}{a x^2}+1\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2454, 2394, 2315} \[ -\frac {1}{2} p \text {PolyLog}\left (2,\frac {b}{a x^2}+1\right )-\frac {1}{2} \log \left (-\frac {b}{a x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2394
Rule 2454
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {1}{2} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )+\frac {1}{2} (b p) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{2} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log \left (-\frac {b}{a x^2}\right )-\frac {1}{2} p \text {Li}_2\left (1+\frac {b}{a x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 45, normalized size = 1.02 \[ -\frac {1}{2} \log \left (-\frac {b}{a x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )-\frac {1}{2} p \text {Li}_2\left (\frac {a+\frac {b}{x^2}}{a}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (c \left (\frac {a x^{2} + b}{x^{2}}\right )^{p}\right )}{x}, x\right ) \]
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 \[ \int \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.63, size = 89, normalized size = 2.02 \[ \frac {1}{2} \, b p {\left (\frac {2 \, \log \left (a + \frac {b}{x^{2}}\right ) \log \relax (x)}{b} + \frac {2 \, \log \relax (x)^{2}}{b} - \frac {2 \, \log \left (\frac {a x^{2}}{b} + 1\right ) \log \relax (x) + {\rm Li}_2\left (-\frac {a x^{2}}{b}\right )}{b}\right )} - p \log \left (a + \frac {b}{x^{2}}\right ) \log \relax (x) + \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{x} \,d x \]
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 \[ \int \frac {\log {\left (c \left (a + \frac {b}{x^{2}}\right )^{p} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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