Integrand size = 23, antiderivative size = 1181 \[ \int \frac {(a+b \arctan (c x))^2}{x^3 \left (d+e x^2\right )^2} \, dx=-\frac {b c (a+b \arctan (c x))}{d^2 x}-\frac {c^2 (a+b \arctan (c x))^2}{2 d^2}+\frac {c^2 e (a+b \arctan (c x))^2}{2 d^2 \left (c^2 d-e\right )}-\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}-\frac {e (a+b \arctan (c x))^2}{4 d^3 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {e (a+b \arctan (c x))^2}{4 d^3 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {4 e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {2 e (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{d^3}-\frac {b c e^{3/2} (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 (-d)^{5/2} \left (c^2 d-e\right )}+\frac {e (a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{d^3}+\frac {b c e^{3/2} (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 (-d)^{5/2} \left (c^2 d-e\right )}+\frac {e (a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{d^3}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {2 i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{d^3}+\frac {2 i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^3}-\frac {2 i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {i b^2 c e^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 (-d)^{5/2} \left (c^2 d-e\right )}-\frac {i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{d^3}-\frac {i b^2 c e^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 (-d)^{5/2} \left (c^2 d-e\right )}-\frac {i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{d^3}-\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{d^3}+\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{d^3}-\frac {b^2 e \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )}{d^3}+\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^3}+\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^3} \]
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Time = 1.43 (sec) , antiderivative size = 1181, normalized size of antiderivative = 1.00, number of steps used = 47, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.957, Rules used = {5100, 4946, 5038, 272, 36, 29, 31, 5004, 4942, 5108, 5114, 6745, 5098, 4974, 4966, 2449, 2352, 2497, 5104, 5040, 4964, 4968} \[ \int \frac {(a+b \arctan (c x))^2}{x^3 \left (d+e x^2\right )^2} \, dx=\frac {c^2 \log (x) b^2}{d^2}-\frac {c^2 \log \left (c^2 x^2+1\right ) b^2}{2 d^2}+\frac {i c e^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 (-d)^{5/2} \left (c^2 d-e\right )}-\frac {i c e^{3/2} \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 (-d)^{5/2} \left (c^2 d-e\right )}-\frac {e \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right ) b^2}{d^3}+\frac {e \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) b^2}{d^3}-\frac {e \operatorname {PolyLog}\left (3,\frac {2}{i c x+1}-1\right ) b^2}{d^3}+\frac {e \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{2 d^3}+\frac {e \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{2 d^3}-\frac {c (a+b \arctan (c x)) b}{d^2 x}-\frac {c e^{3/2} (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 (-d)^{5/2} \left (c^2 d-e\right )}+\frac {c e^{3/2} (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 (-d)^{5/2} \left (c^2 d-e\right )}+\frac {2 i e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) b}{d^3}+\frac {2 i e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) b}{d^3}-\frac {2 i e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,\frac {2}{i c x+1}-1\right ) b}{d^3}-\frac {i e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{d^3}-\frac {i e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{d^3}+\frac {c^2 e (a+b \arctan (c x))^2}{2 d^2 \left (c^2 d-e\right )}-\frac {e (a+b \arctan (c x))^2}{4 d^3 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {e (a+b \arctan (c x))^2}{4 d^3 \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )}-\frac {c^2 (a+b \arctan (c x))^2}{2 d^2}-\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}-\frac {4 e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{i c x+1}\right )}{d^3}-\frac {2 e (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{d^3}+\frac {e (a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{d^3}+\frac {e (a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{d^3} \]
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 2352
Rule 2449
Rule 2497
Rule 4942
Rule 4946
Rule 4964
Rule 4966
Rule 4968
Rule 4974
Rule 5004
Rule 5038
Rule 5040
Rule 5098
Rule 5100
Rule 5104
Rule 5108
Rule 5114
Rule 6745
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(a+b \arctan (c x))^2}{d^2 x^3}-\frac {2 e (a+b \arctan (c x))^2}{d^3 x}+\frac {e^2 x (a+b \arctan (c x))^2}{d^2 \left (d+e x^2\right )^2}+\frac {2 e^2 x (a+b \arctan (c x))^2}{d^3 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {(a+b \arctan (c x))^2}{x^3} \, dx}{d^2}-\frac {(2 e) \int \frac {(a+b \arctan (c x))^2}{x} \, dx}{d^3}+\frac {\left (2 e^2\right ) \int \frac {x (a+b \arctan (c x))^2}{d+e x^2} \, dx}{d^3}+\frac {e^2 \int \frac {x (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx}{d^2} \\ & = -\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}-\frac {4 e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {(b c) \int \frac {a+b \arctan (c x)}{x^2 \left (1+c^2 x^2\right )} \, dx}{d^2}+\frac {(8 b c e) \int \frac {(a+b \arctan (c x)) \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}+\frac {e^{3/2} \int \frac {(a+b \arctan (c x))^2}{\left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )^2} \, dx}{4 (-d)^{7/2}}-\frac {e^{3/2} \int \frac {(a+b \arctan (c x))^2}{\left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )^2} \, dx}{4 (-d)^{7/2}}+\frac {\left (2 e^2\right ) \int \left (-\frac {(a+b \arctan (c x))^2}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {(a+b \arctan (c x))^2}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^3} \\ & = -\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}-\frac {e (a+b \arctan (c x))^2}{4 d^3 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {e (a+b \arctan (c x))^2}{4 d^3 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {4 e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}+\frac {(b c) \int \frac {a+b \arctan (c x)}{x^2} \, dx}{d^2}-\frac {\left (b c^3\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{d^2}+\frac {(b c e) \int \left (\frac {\sqrt {-d} e (a+b \arctan (c x))}{\left (c^2 d-e\right ) \left (-\sqrt {-d}+\sqrt {e} x\right )}+\frac {c^2 d \left (\sqrt {-d}+\sqrt {e} x\right ) (a+b \arctan (c x))}{\sqrt {-d} \left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d^3}+\frac {(b c e) \int \left (\frac {\sqrt {-d} e (a+b \arctan (c x))}{\left (-c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {c^2 \left (d+\sqrt {-d} \sqrt {e} x\right ) (a+b \arctan (c x))}{\left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 d^3}-\frac {(4 b c e) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}+\frac {(4 b c e) \int \frac {(a+b \arctan (c x)) \log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}-\frac {e^{3/2} \int \frac {(a+b \arctan (c x))^2}{\sqrt {-d}-\sqrt {e} x} \, dx}{d^3}+\frac {e^{3/2} \int \frac {(a+b \arctan (c x))^2}{\sqrt {-d}+\sqrt {e} x} \, dx}{d^3} \\ & = -\frac {b c (a+b \arctan (c x))}{d^2 x}-\frac {c^2 (a+b \arctan (c x))^2}{2 d^2}-\frac {(a+b \arctan (c x))^2}{2 d^2 x^2}-\frac {e (a+b \arctan (c x))^2}{4 d^3 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {e (a+b \arctan (c x))^2}{4 d^3 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {4 e (a+b \arctan (c x))^2 \text {arctanh}\left (1-\frac {2}{1+i c x}\right )}{d^3}-\frac {2 e (a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{d^3}+\frac {e (a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{d^3}+\frac {e (a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{d^3}+\frac {2 i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{d^3}+\frac {2 i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{d^3}-\frac {2 i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{d^3}-\frac {i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{d^3}-\frac {i b e (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{d^3}-\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{d^3}+\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^3}+\frac {b^2 e \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^3}+\frac {\left (b^2 c^2\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d^2}-\frac {\left (2 i b^2 c e\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}+\frac {\left (2 i b^2 c e\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^3}+\frac {\left (b c^3 e\right ) \int \frac {\left (\sqrt {-d}+\sqrt {e} x\right ) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{2 (-d)^{5/2} \left (c^2 d-e\right )}+\frac {\left (b c^3 e\right ) \int \frac {\left (d+\sqrt {-d} \sqrt {e} x\right ) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{2 d^3 \left (c^2 d-e\right )}-\frac {\left (b c e^2\right ) \int \frac {a+b \arctan (c x)}{-\sqrt {-d}+\sqrt {e} x} \, dx}{2 (-d)^{5/2} \left (c^2 d-e\right )}+\frac {\left (b c e^2\right ) \int \frac {a+b \arctan (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 (-d)^{5/2} \left (c^2 d-e\right )} \\ & = \text {Too large to display} \\ \end{align*}
\[ \int \frac {(a+b \arctan (c x))^2}{x^3 \left (d+e x^2\right )^2} \, dx=\int \frac {(a+b \arctan (c x))^2}{x^3 \left (d+e x^2\right )^2} \, dx \]
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\[\int \frac {\left (a +b \arctan \left (c x \right )\right )^{2}}{x^{3} \left (e \,x^{2}+d \right )^{2}}d x\]
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\[ \int \frac {(a+b \arctan (c x))^2}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{x^3 \left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \arctan (c x))^2}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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\[ \int \frac {(a+b \arctan (c x))^2}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{x^3 \left (d+e x^2\right )^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{x^3\,{\left (e\,x^2+d\right )}^2} \,d x \]
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