Integrand size = 21, antiderivative size = 409 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=-\frac {b c}{2 d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2} \]
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Time = 0.34 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5038, 4946, 331, 209, 5048, 4940, 2438, 5100, 4966, 2449, 2352, 2497} \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=-\frac {e \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 d^2}+\frac {e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 d^2}-\frac {a+b \arctan (c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}-\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2}-\frac {b c}{2 d x} \]
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Rule 209
Rule 331
Rule 2352
Rule 2438
Rule 2449
Rule 2497
Rule 4940
Rule 4946
Rule 4966
Rule 5038
Rule 5048
Rule 5100
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {a+b \arctan (c x)}{x^3} \, dx}{d}-\frac {e \int \frac {a+b \arctan (c x)}{x \left (d+e x^2\right )} \, dx}{d} \\ & = -\frac {a+b \arctan (c x)}{2 d x^2}+\frac {(b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d}-\frac {e \int \left (\frac {a+b \arctan (c x)}{d x}-\frac {e x (a+b \arctan (c x))}{d \left (d+e x^2\right )}\right ) \, dx}{d} \\ & = -\frac {b c}{2 d x}-\frac {a+b \arctan (c x)}{2 d x^2}-\frac {\left (b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d}-\frac {e \int \frac {a+b \arctan (c x)}{x} \, dx}{d^2}+\frac {e^2 \int \frac {x (a+b \arctan (c x))}{d+e x^2} \, dx}{d^2} \\ & = -\frac {b c}{2 d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {(i b e) \int \frac {\log (1-i c x)}{x} \, dx}{2 d^2}+\frac {(i b e) \int \frac {\log (1+i c x)}{x} \, dx}{2 d^2}+\frac {e^2 \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^2} \\ & = -\frac {b c}{2 d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{2 d^2}-\frac {e^{3/2} \int \frac {a+b \arctan (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d^2}+\frac {e^{3/2} \int \frac {a+b \arctan (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d^2} \\ & = -\frac {b c}{2 d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\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 )}{2 d^2}+\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 )}{2 d^2}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{2 d^2}+2 \frac {(b c e) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 d^2}-\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}{2 d^2}-\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}{2 d^2} \\ & = -\frac {b c}{2 d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\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 )}{2 d^2}+\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 )}{2 d^2}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{2 d^2}-\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 )}{4 d^2}-\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 )}{4 d^2}+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 )}{2 d^2} \\ & = -\frac {b c}{2 d x}-\frac {b c^2 \arctan (c x)}{2 d}-\frac {a+b \arctan (c x)}{2 d x^2}-\frac {a e \log (x)}{d^2}-\frac {e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\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 )}{2 d^2}+\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 )}{2 d^2}-\frac {i b e \operatorname {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}(2,i c x)}{2 d^2}+\frac {i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^2}-\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 )}{4 d^2}-\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 )}{4 d^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.23 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.26 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=-\frac {2 a d+2 b d \arctan (c x)+2 b c d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )+4 a e x^2 \log (x)+i b e x^2 \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )-i b e x^2 \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )-i b e x^2 \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )+i b e x^2 \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )-2 a e x^2 \log \left (d+e x^2\right )+2 i b e x^2 \operatorname {PolyLog}(2,-i c x)-2 i b e x^2 \operatorname {PolyLog}(2,i c x)+i b e x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )-i b e x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )+i b e x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )-i b e x^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 d^2 x^2} \]
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Time = 0.41 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.40
method | result | size |
risch | \(-\frac {i b \ln \left (-i c x +1\right )}{4 d \,x^{2}}-\frac {b c}{2 d x}+\frac {i b e \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {e d}+\left (-i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d^{2}}+\frac {i c^{2} b \ln \left (-i c x \right )}{4 d}-\frac {i c^{2} b \ln \left (i c x \right )}{4 d}-\frac {i c^{2} b \ln \left (-i c x +1\right )}{4 d}+\frac {i b e \ln \left (-i c x +1\right ) \ln \left (\frac {c \sqrt {e d}-\left (-i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d^{2}}+\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {e d}+\left (-i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d^{2}}-\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {e d}-\left (i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d^{2}}-\frac {a}{2 d \,x^{2}}-\frac {a e \ln \left (-i c x \right )}{d^{2}}+\frac {a e \ln \left (\left (-i c x +1\right )^{2} e -c^{2} d -2 \left (-i c x +1\right ) e +e \right )}{2 d^{2}}+\frac {i b e \operatorname {dilog}\left (-i c x +1\right )}{2 d^{2}}-\frac {i b e \operatorname {dilog}\left (i c x +1\right )}{2 d^{2}}-\frac {i b e \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {e d}-\left (i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d^{2}}+\frac {i c^{2} b \ln \left (i c x +1\right )}{4 d}+\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {e d}-\left (-i c x +1\right ) e +e}{c \sqrt {e d}+e}\right )}{4 d^{2}}-\frac {i b e \ln \left (i c x +1\right ) \ln \left (\frac {c \sqrt {e d}+\left (i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d^{2}}+\frac {i b \ln \left (i c x +1\right )}{4 d \,x^{2}}-\frac {i b e \operatorname {dilog}\left (\frac {c \sqrt {e d}+\left (i c x +1\right ) e -e}{c \sqrt {e d}-e}\right )}{4 d^{2}}\) | \(571\) |
parts | \(a \left (-\frac {1}{2 d \,x^{2}}-\frac {e \ln \left (x \right )}{d^{2}}+\frac {e \ln \left (e \,x^{2}+d \right )}{2 d^{2}}\right )+b \,c^{2} \left (-\frac {\arctan \left (c x \right )}{2 d \,c^{2} x^{2}}-\frac {\arctan \left (c x \right ) e \ln \left (c x \right )}{c^{2} d^{2}}+\frac {\arctan \left (c x \right ) e \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}-\frac {c^{2} \left (\frac {e \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x -i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x +i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}\right )}{d^{2} c^{4}}-\frac {-\frac {1}{c x}-\arctan \left (c x \right )}{d \,c^{2}}-\frac {2 e \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )}{d^{2} c^{4}}\right )}{2}\right )\) | \(757\) |
derivativedivides | \(c^{2} \left (\frac {a e \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}-\frac {a}{2 d \,c^{2} x^{2}}-\frac {a e \ln \left (c x \right )}{c^{2} d^{2}}+b \,c^{2} \left (\frac {\arctan \left (c x \right ) e \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 d^{2} c^{4}}-\frac {\arctan \left (c x \right )}{2 d \,c^{4} x^{2}}-\frac {\arctan \left (c x \right ) e \ln \left (c x \right )}{d^{2} c^{4}}-\frac {e \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x -i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x +i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}\right )}{2 d^{2} c^{4}}+\frac {-\frac {1}{c x}-\arctan \left (c x \right )}{2 d \,c^{2}}+\frac {e \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )}{d^{2} c^{4}}\right )\right )\) | \(773\) |
default | \(c^{2} \left (\frac {a e \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}-\frac {a}{2 d \,c^{2} x^{2}}-\frac {a e \ln \left (c x \right )}{c^{2} d^{2}}+b \,c^{2} \left (\frac {\arctan \left (c x \right ) e \ln \left (e \,c^{2} x^{2}+c^{2} d \right )}{2 d^{2} c^{4}}-\frac {\arctan \left (c x \right )}{2 d \,c^{4} x^{2}}-\frac {\arctan \left (c x \right ) e \ln \left (c x \right )}{d^{2} c^{4}}-\frac {e \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x -i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x +i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (e \,c^{2} x^{2}+c^{2} d \right )-2 e \left (\frac {\ln \left (c x +i\right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )\right )}{2 e}+\frac {\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =1\right )}\right )+\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )-c x -i}{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}-2 i e \textit {\_Z} +c^{2} d -e , \operatorname {index} =2\right )}\right )}{2 e}\right )\right )}{2}\right )}{2 d^{2} c^{4}}+\frac {-\frac {1}{c x}-\arctan \left (c x \right )}{2 d \,c^{2}}+\frac {e \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )}{d^{2} c^{4}}\right )\right )\) | \(773\) |
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\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{3}} \,d x } \]
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\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=\int \frac {a + b \operatorname {atan}{\left (c x \right )}}{x^{3} \left (d + e x^{2}\right )}\, dx \]
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\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{3}} \,d x } \]
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\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^3\,\left (e\,x^2+d\right )} \,d x \]
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