Integrand size = 21, antiderivative size = 489 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=-\frac {b c}{2 d^2 x}-\frac {b c^2 \arctan (c x)}{2 d^2}+\frac {b c^2 e \arctan (c x)}{2 d^2 \left (c^2 d-e\right )}-\frac {a+b \arctan (c x)}{2 d^2 x^2}-\frac {e (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {b c e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}-\frac {2 a e \log (x)}{d^3}-\frac {2 e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^3}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{d^3}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{d^3}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^3}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^3} \]
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Time = 0.37 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {5100, 4946, 331, 209, 4940, 2438, 5094, 400, 211, 4966, 2449, 2352, 2497} \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=-\frac {2 e \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d^3}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{d^3}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{d^3}-\frac {e (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {a+b \arctan (c x)}{2 d^2 x^2}-\frac {2 a e \log (x)}{d^3}-\frac {b c e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}+\frac {b c^2 e \arctan (c x)}{2 d^2 \left (c^2 d-e\right )}-\frac {b c^2 \arctan (c x)}{2 d^2}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{d^3}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^3}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^3}-\frac {b c}{2 d^2 x} \]
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Rule 209
Rule 211
Rule 331
Rule 400
Rule 2352
Rule 2438
Rule 2449
Rule 2497
Rule 4940
Rule 4946
Rule 4966
Rule 5094
Rule 5100
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \arctan (c x)}{d^2 x^3}-\frac {2 e (a+b \arctan (c x))}{d^3 x}+\frac {e^2 x (a+b \arctan (c x))}{d^2 \left (d+e x^2\right )^2}+\frac {2 e^2 x (a+b \arctan (c x))}{d^3 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \arctan (c x)}{x^3} \, dx}{d^2}-\frac {(2 e) \int \frac {a+b \arctan (c x)}{x} \, dx}{d^3}+\frac {\left (2 e^2\right ) \int \frac {x (a+b \arctan (c x))}{d+e x^2} \, dx}{d^3}+\frac {e^2 \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx}{d^2} \\ & = -\frac {a+b \arctan (c x)}{2 d^2 x^2}-\frac {e (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {2 a e \log (x)}{d^3}+\frac {(b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d^2}-\frac {(i b e) \int \frac {\log (1-i c x)}{x} \, dx}{d^3}+\frac {(i b e) \int \frac {\log (1+i c x)}{x} \, dx}{d^3}+\frac {(b c e) \int \frac {1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 d^2}+\frac {\left (2 e^2\right ) \int \left (-\frac {a+b \arctan (c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arctan (c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^3} \\ & = -\frac {b c}{2 d^2 x}-\frac {a+b \arctan (c x)}{2 d^2 x^2}-\frac {e (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {2 a e \log (x)}{d^3}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{d^3}-\frac {\left (b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^2}+\frac {\left (b c^3 e\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^2 \left (c^2 d-e\right )}-\frac {e^{3/2} \int \frac {a+b \arctan (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{d^3}+\frac {e^{3/2} \int \frac {a+b \arctan (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{d^3}-\frac {\left (b c e^2\right ) \int \frac {1}{d+e x^2} \, dx}{2 d^2 \left (c^2 d-e\right )} \\ & = -\frac {b c}{2 d^2 x}-\frac {b c^2 \arctan (c x)}{2 d^2}+\frac {b c^2 e \arctan (c x)}{2 d^2 \left (c^2 d-e\right )}-\frac {a+b \arctan (c x)}{2 d^2 x^2}-\frac {e (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {b c e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}-\frac {2 a e \log (x)}{d^3}-\frac {2 e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^3}+\frac {e (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 )}{d^3}+\frac {e (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 )}{d^3}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{d^3}+2 \frac {(b c e) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d^3}-\frac {(b c e) \int \frac {\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 )}{1+c^2 x^2} \, dx}{d^3}-\frac {(b c e) \int \frac {\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 )}{1+c^2 x^2} \, dx}{d^3} \\ & = -\frac {b c}{2 d^2 x}-\frac {b c^2 \arctan (c x)}{2 d^2}+\frac {b c^2 e \arctan (c x)}{2 d^2 \left (c^2 d-e\right )}-\frac {a+b \arctan (c x)}{2 d^2 x^2}-\frac {e (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {b c e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}-\frac {2 a e \log (x)}{d^3}-\frac {2 e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^3}+\frac {e (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 )}{d^3}+\frac {e (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 )}{d^3}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{d^3}-\frac {i b e \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 )}{2 d^3}-\frac {i b e \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 )}{2 d^3}+2 \frac {(i b e) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{d^3} \\ & = -\frac {b c}{2 d^2 x}-\frac {b c^2 \arctan (c x)}{2 d^2}+\frac {b c^2 e \arctan (c x)}{2 d^2 \left (c^2 d-e\right )}-\frac {a+b \arctan (c x)}{2 d^2 x^2}-\frac {e (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {b c e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2} \left (c^2 d-e\right )}-\frac {2 a e \log (x)}{d^3}-\frac {2 e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^3}+\frac {e (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 )}{d^3}+\frac {e (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 )}{d^3}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{d^3}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{d^3}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{d^3}-\frac {i b e \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 )}{2 d^3}-\frac {i b e \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 )}{2 d^3} \\ \end{align*}
Time = 9.89 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.31 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=-\frac {a \left (d \left (\frac {1}{x^2}+\frac {e}{d+e x^2}\right )+4 e \log (x)-2 e \log \left (d+e x^2\right )\right )+b \left (\frac {c d}{x}+\frac {c^2 d \left (c^2 d-2 e\right ) \arctan (c x)}{c^2 d-e}+d \left (\frac {1}{x^2}+\frac {e}{d+e x^2}\right ) \arctan (c x)+\frac {c \sqrt {d} e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{c^2 d-e}+4 e \arctan (c x) \log (x)-2 e \arctan (c x) \log \left (d+e x^2\right )-2 i e (\log (x) (\log (1-i c x)-\log (1+i c x))-\operatorname {PolyLog}(2,-i c x)+\operatorname {PolyLog}(2,i c x))-e \left (2 \arctan (c x) \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right )+2 \arctan (c x) \log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )+i \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (-1-i c x)}{c \sqrt {d}-\sqrt {e}}\right )-i \log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (1-i c x)}{c \sqrt {d}+\sqrt {e}}\right )-i \log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (-1+i c x)}{c \sqrt {d}-\sqrt {e}}\right )+i \log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right ) \log \left (\frac {\sqrt {e} (1+i c x)}{c \sqrt {d}+\sqrt {e}}\right )-2 \arctan (c x) \log \left (d+e x^2\right )-i \operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{c \sqrt {d}-\sqrt {e}}\right )+i \operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{c \sqrt {d}+\sqrt {e}}\right )+i \operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{c \sqrt {d}-\sqrt {e}}\right )-i \operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{c \sqrt {d}+\sqrt {e}}\right )\right )\right )}{2 d^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.61 (sec) , antiderivative size = 851, normalized size of antiderivative = 1.74
method | result | size |
parts | \(\text {Expression too large to display}\) | \(851\) |
derivativedivides | \(\text {Expression too large to display}\) | \(877\) |
default | \(\text {Expression too large to display}\) | \(877\) |
risch | \(-\frac {b c}{2 d^{2} x}-\frac {i c^{4} b \,e^{2} \ln \left (-i c x +1\right ) x^{2}}{4 d^{2} \left (c^{2} d -e \right ) \left (-e \,c^{2} x^{2}-c^{2} d \right )}+\frac {i b \,c^{4} e^{2} \ln \left (i c x +1\right ) x^{2}}{4 d^{2} \left (c^{2} d -e \right ) \left (-e \,c^{2} x^{2}-c^{2} d \right )}-\frac {a}{2 d^{2} x^{2}}+\frac {c^{2} a e}{2 d^{2} \left (-e \,c^{2} x^{2}-c^{2} d \right )}-\frac {i c^{2} b \,e^{2} \ln \left (-i c x +1\right )}{4 d^{2} \left (c^{2} d -e \right ) \left (-e \,c^{2} x^{2}-c^{2} d \right )}-\frac {i c b \,e^{2} \operatorname {arctanh}\left (\frac {2 \left (-i c x +1\right ) e -2 e}{2 c \sqrt {e d}}\right )}{4 d^{2} \left (c^{2} d -e \right ) \sqrt {e d}}+\frac {i b \,c^{2} e^{2} \ln \left (i c x +1\right )}{4 d^{2} \left (c^{2} d -e \right ) \left (-e \,c^{2} x^{2}-c^{2} d \right )}+\frac {i b c \,e^{2} \operatorname {arctanh}\left (\frac {2 \left (i c x +1\right ) e -2 e}{2 c \sqrt {e d}}\right )}{4 d^{2} \left (c^{2} d -e \right ) \sqrt {e d}}+\frac {i c^{2} b \ln \left (-i c x \right )}{4 d^{2}}+\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {e d}+\left (-i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{2 d^{3}}-\frac {i c^{2} b \ln \left (-i c x +1\right )}{4 d^{2}}+\frac {i b e \operatorname {dilog}\left (-i c x +1\right )}{d^{3}}-\frac {i b \ln \left (-i c x +1\right )}{4 d^{2} x^{2}}+\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {e d}-\left (-i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{2 d^{3}}-\frac {i b \,c^{2} \ln \left (i c x \right )}{4 d^{2}}+\frac {i b \,c^{2} \ln \left (i c x +1\right )}{4 d^{2}}+\frac {i b \ln \left (i c x +1\right )}{4 d^{2} x^{2}}-\frac {i b e \operatorname {dilog}\left (i c x +1\right )}{d^{3}}-\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {e d}-\left (i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{2 d^{3}}-\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {e d}+\left (i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{2 d^{3}}-\frac {i b e \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {e d}-\left (i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{2 d^{3}}-\frac {i b e \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {e d}+\left (i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{2 d^{3}}+\frac {i b e \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {e d}+\left (-i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{2 d^{3}}+\frac {i b e \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {e d}-\left (-i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{2 d^{3}}+\frac {a e \ln \left (\left (-i c x +1\right )^{2} e -c^{2} d -2 \left (-i c x +1\right ) e +e \right )}{d^{3}}-\frac {2 a e \ln \left (-i c x \right )}{d^{3}}+\frac {i b \,c^{2} e \ln \left (\left (i c x +1\right )^{2} e -c^{2} d -2 \left (i c x +1\right ) e +e \right )}{8 d^{2} \left (c^{2} d -e \right )}-\frac {i c^{2} b e \ln \left (\left (-i c x +1\right )^{2} e -c^{2} d -2 \left (-i c x +1\right ) e +e \right )}{8 d^{2} \left (c^{2} d -e \right )}\) | \(1012\) |
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\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^3\,{\left (e\,x^2+d\right )}^2} \,d x \]
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