Integrand size = 16, antiderivative size = 68 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^4} \, dx=\frac {2 p}{9 x^3}-\frac {2 a p}{3 b x}-\frac {2 a^{3/2} p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 b^{3/2}}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 x^3} \]
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Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2505, 269, 331, 211} \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^4} \, dx=-\frac {2 a^{3/2} p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 b^{3/2}}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 x^3}-\frac {2 a p}{3 b x}+\frac {2 p}{9 x^3} \]
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Rule 211
Rule 269
Rule 331
Rule 2505
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.03 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^4} \, dx=\frac {2 p}{9 x^3}-\frac {2 a p}{3 b x}+\frac {2 a^{3/2} p \arctan \left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{3 b^{3/2}}-\frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 x^3} \]
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Time = 0.43 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90
method | result | size |
parts | \(-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{3 x^{3}}-\frac {2 p b \left (-\frac {1}{3 b \,x^{3}}+\frac {a}{b^{2} x}+\frac {a^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\right )}{3}\) | \(61\) |
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Time = 0.35 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.26 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^4} \, dx=\left [\frac {3 \, a p x^{3} \sqrt {-\frac {a}{b}} \log \left (\frac {a x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - b}{a x^{2} + b}\right ) - 6 \, a p x^{2} - 3 \, b p \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + 2 \, b p - 3 \, b \log \left (c\right )}{9 \, b x^{3}}, -\frac {6 \, a p x^{3} \sqrt {\frac {a}{b}} \arctan \left (x \sqrt {\frac {a}{b}}\right ) + 6 \, a p x^{2} + 3 \, b p \log \left (\frac {a x^{2} + b}{x^{2}}\right ) - 2 \, b p + 3 \, b \log \left (c\right )}{9 \, b x^{3}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (63) = 126\).
Time = 24.50 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.03 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^4} \, dx=\begin {cases} - \frac {\log {\left (0^{p} c \right )}}{3 x^{3}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 p}{9 x^{3}} - \frac {\log {\left (c \left (\frac {b}{x^{2}}\right )^{p} \right )}}{3 x^{3}} & \text {for}\: a = 0 \\- \frac {\log {\left (a^{p} c \right )}}{3 x^{3}} & \text {for}\: b = 0 \\- \frac {a p \log {\left (x - \sqrt {- \frac {b}{a}} \right )}}{3 b \sqrt {- \frac {b}{a}}} + \frac {a p \log {\left (x + \sqrt {- \frac {b}{a}} \right )}}{3 b \sqrt {- \frac {b}{a}}} - \frac {2 a p}{3 b x} + \frac {2 p}{9 x^{3}} - \frac {\log {\left (c \left (a + \frac {b}{x^{2}}\right )^{p} \right )}}{3 x^{3}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^4} \, dx=-\frac {2}{9} \, b p {\left (\frac {3 \, a^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {3 \, a x^{2} - b}{b^{2} x^{3}}\right )} - \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{3 \, x^{3}} \]
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Time = 0.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.07 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^4} \, dx=-\frac {2 \, a^{2} p \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{3 \, \sqrt {a b} b} - \frac {p \log \left (a x^{2} + b\right )}{3 \, x^{3}} + \frac {p \log \left (x^{2}\right )}{3 \, x^{3}} - \frac {6 \, a p x^{2} - 2 \, b p + 3 \, b \log \left (c\right )}{9 \, b x^{3}} \]
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Time = 1.34 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.81 \[ \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^4} \, dx=\frac {\frac {2\,p}{3}-\frac {2\,a\,p\,x^2}{b}}{3\,x^3}-\frac {\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{3\,x^3}-\frac {2\,a^{3/2}\,p\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{3\,b^{3/2}} \]
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