Optimal. Leaf size=68 \[ -\frac{2 a^{3/2} p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{3 b^{3/2}}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{3 x^3}-\frac{2 a p}{3 b x}+\frac{2 p}{9 x^3} \]
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Rubi [A] time = 0.0355895, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2455, 263, 325, 205} \[ -\frac{2 a^{3/2} p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{3 b^{3/2}}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{3 x^3}-\frac{2 a p}{3 b x}+\frac{2 p}{9 x^3} \]
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^4} \, dx &=-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{3 x^3}-\frac{1}{3} (2 b p) \int \frac{1}{\left (a+\frac{b}{x^2}\right ) x^6} \, dx\\ &=-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{3 x^3}-\frac{1}{3} (2 b p) \int \frac{1}{x^4 \left (b+a x^2\right )} \, dx\\ &=\frac{2 p}{9 x^3}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{3 x^3}+\frac{1}{3} (2 a p) \int \frac{1}{x^2 \left (b+a x^2\right )} \, dx\\ &=\frac{2 p}{9 x^3}-\frac{2 a p}{3 b x}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{3 x^3}-\frac{\left (2 a^2 p\right ) \int \frac{1}{b+a x^2} \, dx}{3 b}\\ &=\frac{2 p}{9 x^3}-\frac{2 a p}{3 b x}-\frac{2 a^{3/2} p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{3 b^{3/2}}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0212296, size = 70, normalized size = 1.03 \[ \frac{2 a^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} x}\right )}{3 b^{3/2}}-\frac{\log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{3 x^3}-\frac{2 a p}{3 b x}+\frac{2 p}{9 x^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.273, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}}\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 = 2.31024, size = 362, normalized size = 5.32 \begin{align*} \left [\frac{3 \, a p x^{3} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) - 6 \, a p x^{2} - 3 \, b p \log \left (\frac{a x^{2} + b}{x^{2}}\right ) + 2 \, b p - 3 \, b \log \left (c\right )}{9 \, b x^{3}}, -\frac{6 \, a p x^{3} \sqrt{\frac{a}{b}} \arctan \left (x \sqrt{\frac{a}{b}}\right ) + 6 \, a p x^{2} + 3 \, b p \log \left (\frac{a x^{2} + b}{x^{2}}\right ) - 2 \, b p + 3 \, b \log \left (c\right )}{9 \, b x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 126.087, size = 177, normalized size = 2.6 \begin{align*} \begin{cases} - \frac{\log{\left (0^{p} c \right )}}{3 x^{3}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{p \log{\left (b \right )}}{3 x^{3}} + \frac{2 p \log{\left (x \right )}}{3 x^{3}} + \frac{2 p}{9 x^{3}} - \frac{\log{\left (c \right )}}{3 x^{3}} & \text{for}\: a = 0 \\- \frac{\log{\left (a^{p} c \right )}}{3 x^{3}} & \text{for}\: b = 0 \\- \frac{2 a p}{3 b x} + \frac{i a p \log{\left (- i \sqrt{b} \sqrt{\frac{1}{a}} + x \right )}}{3 b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{i a p \log{\left (i \sqrt{b} \sqrt{\frac{1}{a}} + x \right )}}{3 b^{\frac{3}{2}} \sqrt{\frac{1}{a}}} - \frac{p \log{\left (a + \frac{b}{x^{2}} \right )}}{3 x^{3}} + \frac{2 p}{9 x^{3}} - \frac{\log{\left (c \right )}}{3 x^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28948, size = 99, normalized size = 1.46 \begin{align*} -\frac{2 \, a^{2} p \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} b} - \frac{p \log \left (a x^{2} + b\right )}{3 \, x^{3}} + \frac{p \log \left (x^{2}\right )}{3 \, x^{3}} - \frac{6 \, a p x^{2} - 2 \, b p + 3 \, b \log \left (c\right )}{9 \, b x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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