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 = 0.0271548, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 2455
Rule 263
Rule 325
Rule 205
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*}
Mathematica [A] time = 0.0139083, 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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Maple [F] time = 0.235, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\ln \left ( c \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \right ) }\, dx \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 = 2.27257, size = 261, normalized size = 5.22 \begin{align*} \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 \left (c\right )}{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 \left (c\right )}{x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 59.9988, size = 129, normalized size = 2.58 \begin{align*} \begin{cases} - \frac{\log{\left (0^{p} c \right )}}{x} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{p \log{\left (b \right )}}{x} + \frac{2 p \log{\left (x \right )}}{x} + \frac{2 p}{x} - \frac{\log{\left (c \right )}}{x} & \text{for}\: a = 0 \\- \frac{\log{\left (a^{p} c \right )}}{x} & \text{for}\: b = 0 \\- \frac{p \log{\left (a + \frac{b}{x^{2}} \right )}}{x} + \frac{2 p}{x} - \frac{\log{\left (c \right )}}{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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26026, size = 73, normalized size = 1.46 \begin{align*} \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 \left (c\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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