Integrand size = 24, antiderivative size = 528 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}-\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2}{1-i c x}\right )}{f}+\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{2 f}+\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{2 f}-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \arctan (c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {i b e g \operatorname {PolyLog}(2,-i c x)}{2 f}-\frac {i b e g \operatorname {PolyLog}(2,i c x)}{2 f}+\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 f}-\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{4 f}-\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{4 f} \]
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Time = 0.52 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4946, 331, 209, 5141, 815, 649, 211, 266, 457, 78, 6857, 4940, 2438, 5048, 4966, 2449, 2352, 2497} \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {a e g \log \left (f+g x^2\right )}{2 f}+\frac {a e g \log (x)}{f}-\frac {1}{2} b c^2 \arctan (c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {b e \arctan (c x) \left (c^2 f-g\right ) \log \left (\frac {2}{1-i c x}\right )}{f}+\frac {b e \arctan (c x) \left (c^2 f-g\right ) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(1-i c x) \left (c \sqrt {-f}-i \sqrt {g}\right )}\right )}{2 f}+\frac {b e \arctan (c x) \left (c^2 f-g\right ) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(1-i c x) \left (c \sqrt {-f}+i \sqrt {g}\right )}\right )}{2 f}+\frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 f}-\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{4 f}-\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {g} x+\sqrt {-f}\right )}{\left (\sqrt {-f} c+i \sqrt {g}\right ) (1-i c x)}\right )}{4 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {i b e g \operatorname {PolyLog}(2,-i c x)}{2 f}-\frac {i b e g \operatorname {PolyLog}(2,i c x)}{2 f} \]
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Rule 78
Rule 209
Rule 211
Rule 266
Rule 331
Rule 457
Rule 649
Rule 815
Rule 2352
Rule 2438
Rule 2449
Rule 2497
Rule 4940
Rule 4946
Rule 4966
Rule 5048
Rule 5141
Rule 6857
Rubi steps \begin{align*} \text {integral}& = -\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \arctan (c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-(2 e g) \int \left (\frac {-a-b c x}{2 x \left (f+g x^2\right )}-\frac {b \left (1+c^2 x^2\right ) \arctan (c x)}{2 x \left (f+g x^2\right )}\right ) \, dx \\ & = -\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \arctan (c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-(e g) \int \frac {-a-b c x}{x \left (f+g x^2\right )} \, dx+(b e g) \int \frac {\left (1+c^2 x^2\right ) \arctan (c x)}{x \left (f+g x^2\right )} \, dx \\ & = -\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \arctan (c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-(e g) \int \left (-\frac {a}{f x}+\frac {-b c f+a g x}{f \left (f+g x^2\right )}\right ) \, dx+(b e g) \int \left (\frac {\arctan (c x)}{f x}+\frac {\left (c^2 f-g\right ) x \arctan (c x)}{f \left (f+g x^2\right )}\right ) \, dx \\ & = \frac {a e g \log (x)}{f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \arctan (c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {(e g) \int \frac {-b c f+a g x}{f+g x^2} \, dx}{f}+\frac {(b e g) \int \frac {\arctan (c x)}{x} \, dx}{f}+\frac {\left (b e \left (c^2 f-g\right ) g\right ) \int \frac {x \arctan (c x)}{f+g x^2} \, dx}{f} \\ & = \frac {a e g \log (x)}{f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \arctan (c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+(b c e g) \int \frac {1}{f+g x^2} \, dx+\frac {(i b e g) \int \frac {\log (1-i c x)}{x} \, dx}{2 f}-\frac {(i b e g) \int \frac {\log (1+i c x)}{x} \, dx}{2 f}+\frac {\left (b e \left (c^2 f-g\right ) g\right ) \int \left (-\frac {\arctan (c x)}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\arctan (c x)}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{f}-\frac {\left (a e g^2\right ) \int \frac {x}{f+g x^2} \, dx}{f} \\ & = \frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \arctan (c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {i b e g \operatorname {PolyLog}(2,-i c x)}{2 f}-\frac {i b e g \operatorname {PolyLog}(2,i c x)}{2 f}-\frac {\left (b e \left (c^2 f-g\right ) \sqrt {g}\right ) \int \frac {\arctan (c x)}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 f}+\frac {\left (b e \left (c^2 f-g\right ) \sqrt {g}\right ) \int \frac {\arctan (c x)}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 f} \\ & = \frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}-\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2}{1-i c x}\right )}{f}+\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{2 f}+\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{2 f}-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \arctan (c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {i b e g \operatorname {PolyLog}(2,-i c x)}{2 f}-\frac {i b e g \operatorname {PolyLog}(2,i c x)}{2 f}+2 \frac {\left (b c e \left (c^2 f-g\right )\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 f}-\frac {\left (b c e \left (c^2 f-g\right )\right ) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 f}-\frac {\left (b c e \left (c^2 f-g\right )\right ) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 f} \\ & = \frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}-\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2}{1-i c x}\right )}{f}+\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{2 f}+\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{2 f}-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \arctan (c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {i b e g \operatorname {PolyLog}(2,-i c x)}{2 f}-\frac {i b e g \operatorname {PolyLog}(2,i c x)}{2 f}-\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{4 f}-\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{4 f}+2 \frac {\left (i b e \left (c^2 f-g\right )\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{2 f} \\ & = \frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}-\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2}{1-i c x}\right )}{f}+\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{2 f}+\frac {b e \left (c^2 f-g\right ) \arctan (c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{2 f}-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {1}{2} b c^2 \arctan (c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {i b e g \operatorname {PolyLog}(2,-i c x)}{2 f}-\frac {i b e g \operatorname {PolyLog}(2,i c x)}{2 f}+\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 f}-\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-i \sqrt {g}\right ) (1-i c x)}\right )}{4 f}-\frac {i b e \left (c^2 f-g\right ) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+i \sqrt {g}\right ) (1-i c x)}\right )}{4 f} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1217\) vs. \(2(528)=1056\).
Time = 5.20 (sec) , antiderivative size = 1217, normalized size of antiderivative = 2.30 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=-\frac {2 a d f+2 b c d f x+2 b d f \arctan (c x)+2 b c^2 d f x^2 \arctan (c x)-4 b c e \sqrt {f} \sqrt {g} x^2 \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )-4 i b c^2 e f x^2 \arcsin \left (\sqrt {\frac {c^2 f}{c^2 f-g}}\right ) \arctan \left (\frac {c g x}{\sqrt {c^2 f g}}\right )+4 i b e g x^2 \arcsin \left (\sqrt {\frac {c^2 f}{c^2 f-g}}\right ) \arctan \left (\frac {c g x}{\sqrt {c^2 f g}}\right )-4 b e g x^2 \arctan (c x) \log \left (1-e^{2 i \arctan (c x)}\right )+4 b c^2 e f x^2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-2 b c^2 e f x^2 \arcsin \left (\sqrt {\frac {c^2 f}{c^2 f-g}}\right ) \log \left (\frac {c^2 \left (1+e^{2 i \arctan (c x)}\right ) f+\left (-1+e^{2 i \arctan (c x)}\right ) g-2 e^{2 i \arctan (c x)} \sqrt {c^2 f g}}{c^2 f-g}\right )+2 b e g x^2 \arcsin \left (\sqrt {\frac {c^2 f}{c^2 f-g}}\right ) \log \left (\frac {c^2 \left (1+e^{2 i \arctan (c x)}\right ) f+\left (-1+e^{2 i \arctan (c x)}\right ) g-2 e^{2 i \arctan (c x)} \sqrt {c^2 f g}}{c^2 f-g}\right )-2 b c^2 e f x^2 \arctan (c x) \log \left (\frac {c^2 \left (1+e^{2 i \arctan (c x)}\right ) f+\left (-1+e^{2 i \arctan (c x)}\right ) g-2 e^{2 i \arctan (c x)} \sqrt {c^2 f g}}{c^2 f-g}\right )+2 b e g x^2 \arctan (c x) \log \left (\frac {c^2 \left (1+e^{2 i \arctan (c x)}\right ) f+\left (-1+e^{2 i \arctan (c x)}\right ) g-2 e^{2 i \arctan (c x)} \sqrt {c^2 f g}}{c^2 f-g}\right )+2 b c^2 e f x^2 \arcsin \left (\sqrt {\frac {c^2 f}{c^2 f-g}}\right ) \log \left (1+\frac {e^{2 i \arctan (c x)} \left (c^2 f+g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )-2 b e g x^2 \arcsin \left (\sqrt {\frac {c^2 f}{c^2 f-g}}\right ) \log \left (1+\frac {e^{2 i \arctan (c x)} \left (c^2 f+g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )-2 b c^2 e f x^2 \arctan (c x) \log \left (1+\frac {e^{2 i \arctan (c x)} \left (c^2 f+g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )+2 b e g x^2 \arctan (c x) \log \left (1+\frac {e^{2 i \arctan (c x)} \left (c^2 f+g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )-4 a e g x^2 \log (x)+2 a e f \log \left (f+g x^2\right )+2 b c e f x \log \left (f+g x^2\right )+2 a e g x^2 \log \left (f+g x^2\right )+2 b e f \arctan (c x) \log \left (f+g x^2\right )+2 b c^2 e f x^2 \arctan (c x) \log \left (f+g x^2\right )-2 i b c^2 e f x^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+2 i b e g x^2 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )+i b c^2 e f x^2 \operatorname {PolyLog}\left (2,\frac {e^{2 i \arctan (c x)} \left (-c^2 f-g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )-i b e g x^2 \operatorname {PolyLog}\left (2,\frac {e^{2 i \arctan (c x)} \left (-c^2 f-g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )+i b c^2 e f x^2 \operatorname {PolyLog}\left (2,-\frac {e^{2 i \arctan (c x)} \left (c^2 f+g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )-i b e g x^2 \operatorname {PolyLog}\left (2,-\frac {e^{2 i \arctan (c x)} \left (c^2 f+g+2 \sqrt {c^2 f g}\right )}{c^2 f-g}\right )}{4 f x^2} \]
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\[\int \frac {\left (a +b \arctan \left (c x \right )\right ) \left (d +e \ln \left (g \,x^{2}+f \right )\right )}{x^{3}}d x\]
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\[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right )}{x^3} \,d x \]
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