Integrand size = 26, antiderivative size = 189 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=-\frac {2 c^2 e (a+b \arctan (c x))}{3 x}-\frac {c^3 e (a+b \arctan (c x))^2}{3 b}+b c^3 e \log (x)-\frac {1}{3} b c^3 e \log \left (1+c^2 x^2\right )-\frac {b c \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{6 x^2}-\frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}-\frac {1}{6} b c^3 \left (d+e \log \left (1+c^2 x^2\right )\right ) \log \left (1-\frac {1}{1+c^2 x^2}\right )+\frac {1}{6} b c^3 e \operatorname {PolyLog}\left (2,\frac {1}{1+c^2 x^2}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {5137, 2525, 2458, 2389, 2379, 2438, 2351, 31, 5038, 4946, 272, 36, 29, 5004} \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=-\frac {c^3 e (a+b \arctan (c x))^2}{3 b}-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}-\frac {2 c^2 e (a+b \arctan (c x))}{3 x}+b c^3 e \log (x)-\frac {b c \left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{6 x^2}-\frac {1}{6} b c^3 \log \left (1-\frac {1}{c^2 x^2+1}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+\frac {1}{6} b c^3 e \operatorname {PolyLog}\left (2,\frac {1}{c^2 x^2+1}\right )-\frac {1}{3} b c^3 e \log \left (c^2 x^2+1\right ) \]
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 2351
Rule 2379
Rule 2389
Rule 2438
Rule 2458
Rule 2525
Rule 4946
Rule 5004
Rule 5038
Rule 5137
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}+\frac {1}{3} (b c) \int \frac {d+e \log \left (1+c^2 x^2\right )}{x^3 \left (1+c^2 x^2\right )} \, dx+\frac {1}{3} \left (2 c^2 e\right ) \int \frac {a+b \arctan (c x)}{x^2 \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}+\frac {1}{6} (b c) \text {Subst}\left (\int \frac {d+e \log \left (1+c^2 x\right )}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{3} \left (2 c^2 e\right ) \int \frac {a+b \arctan (c x)}{x^2} \, dx-\frac {1}{3} \left (2 c^4 e\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx \\ & = -\frac {2 c^2 e (a+b \arctan (c x))}{3 x}-\frac {c^3 e (a+b \arctan (c x))^2}{3 b}-\frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}+\frac {b \text {Subst}\left (\int \frac {d+e \log (x)}{x \left (-\frac {1}{c^2}+\frac {x}{c^2}\right )^2} \, dx,x,1+c^2 x^2\right )}{6 c}+\frac {1}{3} \left (2 b c^3 e\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {2 c^2 e (a+b \arctan (c x))}{3 x}-\frac {c^3 e (a+b \arctan (c x))^2}{3 b}-\frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}+\frac {b \text {Subst}\left (\int \frac {d+e \log (x)}{\left (-\frac {1}{c^2}+\frac {x}{c^2}\right )^2} \, dx,x,1+c^2 x^2\right )}{6 c}-\frac {1}{6} (b c) \text {Subst}\left (\int \frac {d+e \log (x)}{x \left (-\frac {1}{c^2}+\frac {x}{c^2}\right )} \, dx,x,1+c^2 x^2\right )+\frac {1}{3} \left (b c^3 e\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {2 c^2 e (a+b \arctan (c x))}{3 x}-\frac {c^3 e (a+b \arctan (c x))^2}{3 b}-\frac {b c \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{6 x^2}-\frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}-\frac {1}{6} b c^3 \left (d+e \log \left (1+c^2 x^2\right )\right ) \log \left (1-\frac {1}{1+c^2 x^2}\right )+\frac {1}{6} (b c e) \text {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x}{c^2}} \, dx,x,1+c^2 x^2\right )+\frac {1}{6} \left (b c^3 e\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {1}{x}\right )}{x} \, dx,x,1+c^2 x^2\right )+\frac {1}{3} \left (b c^3 e\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{3} \left (b c^5 e\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -\frac {2 c^2 e (a+b \arctan (c x))}{3 x}-\frac {c^3 e (a+b \arctan (c x))^2}{3 b}+b c^3 e \log (x)-\frac {1}{3} b c^3 e \log \left (1+c^2 x^2\right )-\frac {b c \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{6 x^2}-\frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}-\frac {1}{6} b c^3 \left (d+e \log \left (1+c^2 x^2\right )\right ) \log \left (1-\frac {1}{1+c^2 x^2}\right )+\frac {1}{6} b c^3 e \operatorname {PolyLog}\left (2,\frac {1}{1+c^2 x^2}\right ) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=\frac {1}{12} \left (-\frac {8 c^2 e (a+b \arctan (c x))}{x}-\frac {4 c^3 e (a+b \arctan (c x))^2}{b}+6 b c^3 e \left (2 \log (x)-\log \left (1+c^2 x^2\right )\right )-\frac {2 b c \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^2}-\frac {4 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^3}+\frac {b c^3 \left (d+e \log \left (1+c^2 x^2\right )\right )^2}{e}-2 b c^3 \left (\log \left (-c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )+e \operatorname {PolyLog}\left (2,1+c^2 x^2\right )\right )\right ) \]
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\[\int \frac {\left (a +b \arctan \left (c x \right )\right ) \left (d +e \ln \left (c^{2} x^{2}+1\right )\right )}{x^{4}}d x\]
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\[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (c^{2} x^{2} + 1\right ) + d\right )}}{x^{4}} \,d x } \]
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Exception generated. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (c^{2} x^{2} + 1\right ) + d\right )}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (c^2\,x^2+1\right )\right )}{x^4} \,d x \]
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