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 = 0.0148343, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 2448
Rule 263
Rule 205
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*}
Mathematica [A] time = 0.0075675, 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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Maple [A] time = 0.063, size = 38, normalized size = 0.9 \begin{align*} x\ln \left ( c \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{p} \right ) +2\,{\frac{bp}{\sqrt{ab}}\arctan \left ({\frac{ax}{\sqrt{ab}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19322, size = 244, normalized size = 5.95 \begin{align*} \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 \left (c\right ), 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 \left (c\right )\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 24.0015, size = 109, normalized size = 2.66 \begin{align*} \begin{cases} p x \log{\left (a + \frac{b}{x^{2}} \right )} + x \log{\left (c \right )} - \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{\left (b \right )} - 2 p x \log{\left (x \right )} + 2 p x + x \log{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24949, size = 57, normalized size = 1.39 \begin{align*} 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 \left (c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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