Integrand size = 16, antiderivative size = 44 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\frac {2 \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{x} \]
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Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2505, 211} \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\frac {2 \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{x} \]
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Rule 211
Rule 2505
Rubi steps \begin{align*} \text {integral}& = -\frac {\log \left (c \left (a+b x^2\right )^p\right )}{x}+(2 b p) \int \frac {1}{a+b x^2} \, dx \\ & = \frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{x} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\frac {2 \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{x} \]
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Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84
method | result | size |
parts | \(-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{x}+\frac {2 p b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\) | \(37\) |
risch | \(-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{x}-\frac {i \pi a \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi a \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi a {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi a {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-2 \sqrt {-a b}\, p \ln \left (-b x -\sqrt {-a b}\right ) x +2 \sqrt {-a b}\, p \ln \left (-b x +\sqrt {-a b}\right ) x +2 \ln \left (c \right ) a}{2 a x}\) | \(195\) |
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Time = 0.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.39 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\left [\frac {p x \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - p \log \left (b x^{2} + a\right ) - \log \left (c\right )}{x}, \frac {2 \, p x \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - p \log \left (b x^{2} + a\right ) - \log \left (c\right )}{x}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (39) = 78\).
Time = 7.75 (sec) , antiderivative size = 258, normalized size of antiderivative = 5.86 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\begin {cases} - \frac {\log {\left (0^{p} c \right )}}{x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {\log {\left (a^{p} c \right )}}{x} & \text {for}\: b = 0 \\- \frac {2 p}{x} - \frac {\log {\left (c \left (b x^{2}\right )^{p} \right )}}{x} & \text {for}\: a = 0 \\- \frac {\log {\left (0^{p} c \right )}}{x} & \text {for}\: a = - b x^{2} \\- \frac {a^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{a^{2} x + a b x^{3}} - \frac {2 a p x \sqrt {- \frac {a}{b}} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{\frac {a^{2} x}{b} + a x^{3}} - \frac {a x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{\frac {a^{2} x}{b} + a x^{3}} + \frac {a x \sqrt {- \frac {a}{b}} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{\frac {a^{2} x}{b} + a x^{3}} - \frac {2 b p x^{3} \sqrt {- \frac {a}{b}} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{\frac {a^{2} x}{b} + a x^{3}} + \frac {b x^{3} \sqrt {- \frac {a}{b}} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{\frac {a^{2} x}{b} + a x^{3}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\frac {2 \, b p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{x} \]
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Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.91 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\frac {2 \, b p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} - \frac {p \log \left (b x^{2} + a\right )}{x} - \frac {\log \left (c\right )}{x} \]
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Time = 1.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx=\frac {2\,\sqrt {b}\,p\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{x} \]
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