Optimal. Leaf size=50 \[ -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {2 p}{x} \]
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Rubi [A] time = 0.03, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2455, 263, 325, 205} \[ -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {2 p}{x} \]
Antiderivative was successfully verified.
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Rule 205
Rule 263
Rule 325
Rule 2455
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^2} \, dx &=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}-(2 b p) \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^4} \, dx\\ &=-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}-(2 b p) \int \frac {1}{x^2 \left (b+a x^2\right )} \, dx\\ &=\frac {2 p}{x}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}+(2 a p) \int \frac {1}{b+a x^2} \, dx\\ &=\frac {2 p}{x}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 52, normalized size = 1.04 \[ -\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x}-\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{\sqrt {b}}+\frac {2 p}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 119, normalized size = 2.38 \[ \left [\frac {p x \sqrt {-\frac {a}{b}} \log \left (\frac {a x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - b}{a x^{2} + b}\right ) - p \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + 2 \, p - \log \relax (c)}{x}, \frac {2 \, p x \sqrt {\frac {a}{b}} \arctan \left (x \sqrt {\frac {a}{b}}\right ) - p \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + 2 \, p - \log \relax (c)}{x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 54, normalized size = 1.08 \[ \frac {2 \, a p \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}} - \frac {p \log \left (a x^{2} + b\right )}{x} + \frac {p \log \left (x^{2}\right )}{x} + \frac {2 \, p - \log \relax (c)}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 49, normalized size = 0.98 \[ 2 \, b p {\left (\frac {a \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {1}{b x}\right )} - \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 42, normalized size = 0.84 \[ \frac {2\,p}{x}-\frac {\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{x}+\frac {2\,\sqrt {a}\,p\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 25.45, size = 129, normalized size = 2.58 \[ \begin {cases} - \frac {\log {\left (0^{p} c \right )}}{x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {\log {\left (a^{p} c \right )}}{x} & \text {for}\: b = 0 \\- \frac {p \log {\relax (b )}}{x} + \frac {2 p \log {\relax (x )}}{x} + \frac {2 p}{x} - \frac {\log {\relax (c )}}{x} & \text {for}\: a = 0 \\- \frac {p \log {\left (a + \frac {b}{x^{2}} \right )}}{x} + \frac {2 p}{x} - \frac {\log {\relax (c )}}{x} - \frac {i p \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + x \right )}}{\sqrt {b} \sqrt {\frac {1}{a}}} + \frac {i p \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + x \right )}}{\sqrt {b} \sqrt {\frac {1}{a}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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