Optimal. Leaf size=41 \[ x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a}} \]
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Rubi [A] time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2448, 263, 205} \[ x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 263
Rule 2448
Rubi steps
\begin {align*} \int \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx &=x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+(2 b p) \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^2} \, dx\\ &=x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+(2 b p) \int \frac {1}{b+a x^2} \, dx\\ &=\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a}}+x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 43, normalized size = 1.05 \[ x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )-\frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{\sqrt {a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 107, normalized size = 2.61 \[ \left [p x \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + p \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right ) + x \log \relax (c), p x \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + 2 \, p \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right ) + x \log \relax (c)\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 42, normalized size = 1.02 \[ p x \log \left (a x^{2} + b\right ) - p x \log \left (x^{2}\right ) + \frac {2 \, b p \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}} + x \log \relax (c) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 0.93 \[ \frac {2 b p \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}}+x \ln \left (c \left (\frac {a \,x^{2}+b}{x^{2}}\right )^{p}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.85, size = 33, normalized size = 0.80 \[ \frac {2 \, b p \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b}} + x \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 33, normalized size = 0.80 \[ x\,\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )+\frac {2\,\sqrt {b}\,p\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.32, size = 109, normalized size = 2.66 \[ \begin {cases} p x \log {\left (a + \frac {b}{x^{2}} \right )} + x \log {\relax (c )} - \frac {i \sqrt {b} p \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + x \right )}}{a \sqrt {\frac {1}{a}}} + \frac {i \sqrt {b} p \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + x \right )}}{a \sqrt {\frac {1}{a}}} & \text {for}\: a \neq 0 \\p x \log {\relax (b )} - 2 p x \log {\relax (x )} + 2 p x + x \log {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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