Integrand size = 21, antiderivative size = 629 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=-\frac {b c}{2 d^3 x}-\frac {b c e^2 x}{8 d^3 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {b c^2 \arctan (c x)}{2 d^3}+\frac {b c^4 e \arctan (c x)}{4 d^2 \left (c^2 d-e\right )^2}+\frac {b c^2 e \arctan (c x)}{d^3 \left (c^2 d-e\right )}-\frac {a+b \arctan (c x)}{2 d^3 x^2}-\frac {e (a+b \arctan (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arctan (c x))}{d^3 \left (d+e x^2\right )}-\frac {b c e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2} \left (c^2 d-e\right )}-\frac {b c \left (3 c^2 d-e\right ) e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2} \left (c^2 d-e\right )^2}-\frac {3 a e \log (x)}{d^4}-\frac {3 e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^4}+\frac {3 e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^4}+\frac {3 e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}(2,-i c x)}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}(2,i c x)}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^4} \]
[Out]
Time = 0.47 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5100, 4946, 331, 209, 4940, 2438, 5094, 425, 536, 211, 400, 4966, 2449, 2352, 2497} \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=-\frac {3 e \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{d^4}+\frac {3 e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}-i \sqrt {e}\right )}\right )}{2 d^4}+\frac {3 e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{(1-i c x) \left (c \sqrt {-d}+i \sqrt {e}\right )}\right )}{2 d^4}-\frac {e (a+b \arctan (c x))}{d^3 \left (d+e x^2\right )}-\frac {a+b \arctan (c x)}{2 d^3 x^2}-\frac {e (a+b \arctan (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {3 a e \log (x)}{d^4}-\frac {b c e^{3/2} \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2} \left (c^2 d-e\right )^2}-\frac {b c e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2} \left (c^2 d-e\right )}+\frac {b c^2 e \arctan (c x)}{d^3 \left (c^2 d-e\right )}-\frac {b c^2 \arctan (c x)}{2 d^3}+\frac {b c^4 e \arctan (c x)}{4 d^2 \left (c^2 d-e\right )^2}-\frac {b c e^2 x}{8 d^3 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {3 i b e \operatorname {PolyLog}(2,-i c x)}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}(2,i c x)}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^4}-\frac {3 i b e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^4}-\frac {b c}{2 d^3 x} \]
[In]
[Out]
Rule 209
Rule 211
Rule 331
Rule 400
Rule 425
Rule 536
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^3 x^3}-\frac {3 e (a+b \arctan (c x))}{d^4 x}+\frac {e^2 x (a+b \arctan (c x))}{d^2 \left (d+e x^2\right )^3}+\frac {2 e^2 x (a+b \arctan (c x))}{d^3 \left (d+e x^2\right )^2}+\frac {3 e^2 x (a+b \arctan (c x))}{d^4 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \arctan (c x)}{x^3} \, dx}{d^3}-\frac {(3 e) \int \frac {a+b \arctan (c x)}{x} \, dx}{d^4}+\frac {\left (3 e^2\right ) \int \frac {x (a+b \arctan (c x))}{d+e x^2} \, dx}{d^4}+\frac {\left (2 e^2\right ) \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^2} \, dx}{d^3}+\frac {e^2 \int \frac {x (a+b \arctan (c x))}{\left (d+e x^2\right )^3} \, dx}{d^2} \\ & = -\frac {a+b \arctan (c x)}{2 d^3 x^2}-\frac {e (a+b \arctan (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arctan (c x))}{d^3 \left (d+e x^2\right )}-\frac {3 a e \log (x)}{d^4}+\frac {(b c) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d^3}-\frac {(3 i b e) \int \frac {\log (1-i c x)}{x} \, dx}{2 d^4}+\frac {(3 i b e) \int \frac {\log (1+i c x)}{x} \, dx}{2 d^4}+\frac {(b c e) \int \frac {1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{d^3}+\frac {(b c e) \int \frac {1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^2} \, dx}{4 d^2}+\frac {\left (3 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^4} \\ & = -\frac {b c}{2 d^3 x}-\frac {b c e^2 x}{8 d^3 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \arctan (c x)}{2 d^3 x^2}-\frac {e (a+b \arctan (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arctan (c x))}{d^3 \left (d+e x^2\right )}-\frac {3 a e \log (x)}{d^4}-\frac {3 i b e \operatorname {PolyLog}(2,-i c x)}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}(2,i c x)}{2 d^4}-\frac {\left (b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^3}+\frac {(b c e) \int \frac {2 c^2 d-e-c^2 e x^2}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{8 d^3 \left (c^2 d-e\right )}+\frac {\left (b c^3 e\right ) \int \frac {1}{1+c^2 x^2} \, dx}{d^3 \left (c^2 d-e\right )}-\frac {\left (3 e^{3/2}\right ) \int \frac {a+b \arctan (c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d^4}+\frac {\left (3 e^{3/2}\right ) \int \frac {a+b \arctan (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d^4}-\frac {\left (b c e^2\right ) \int \frac {1}{d+e x^2} \, dx}{d^3 \left (c^2 d-e\right )} \\ & = -\frac {b c}{2 d^3 x}-\frac {b c e^2 x}{8 d^3 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {b c^2 \arctan (c x)}{2 d^3}+\frac {b c^2 e \arctan (c x)}{d^3 \left (c^2 d-e\right )}-\frac {a+b \arctan (c x)}{2 d^3 x^2}-\frac {e (a+b \arctan (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arctan (c x))}{d^3 \left (d+e x^2\right )}-\frac {b c e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2} \left (c^2 d-e\right )}-\frac {3 a e \log (x)}{d^4}-\frac {3 e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^4}+\frac {3 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^4}+\frac {3 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^4}-\frac {3 i b e \operatorname {PolyLog}(2,-i c x)}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}(2,i c x)}{2 d^4}+2 \frac {(3 b c e) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 d^4}-\frac {(3 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^4}-\frac {(3 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^4}+\frac {\left (b c^5 e\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 d^2 \left (c^2 d-e\right )^2}-\frac {\left (b c \left (3 c^2 d-e\right ) e^2\right ) \int \frac {1}{d+e x^2} \, dx}{8 d^3 \left (c^2 d-e\right )^2} \\ & = -\frac {b c}{2 d^3 x}-\frac {b c e^2 x}{8 d^3 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {b c^2 \arctan (c x)}{2 d^3}+\frac {b c^4 e \arctan (c x)}{4 d^2 \left (c^2 d-e\right )^2}+\frac {b c^2 e \arctan (c x)}{d^3 \left (c^2 d-e\right )}-\frac {a+b \arctan (c x)}{2 d^3 x^2}-\frac {e (a+b \arctan (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arctan (c x))}{d^3 \left (d+e x^2\right )}-\frac {b c e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2} \left (c^2 d-e\right )}-\frac {b c \left (3 c^2 d-e\right ) e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2} \left (c^2 d-e\right )^2}-\frac {3 a e \log (x)}{d^4}-\frac {3 e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^4}+\frac {3 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^4}+\frac {3 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^4}-\frac {3 i b e \operatorname {PolyLog}(2,-i c x)}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}(2,i c x)}{2 d^4}-\frac {3 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^4}-\frac {3 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^4}+2 \frac {(3 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^4} \\ & = -\frac {b c}{2 d^3 x}-\frac {b c e^2 x}{8 d^3 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {b c^2 \arctan (c x)}{2 d^3}+\frac {b c^4 e \arctan (c x)}{4 d^2 \left (c^2 d-e\right )^2}+\frac {b c^2 e \arctan (c x)}{d^3 \left (c^2 d-e\right )}-\frac {a+b \arctan (c x)}{2 d^3 x^2}-\frac {e (a+b \arctan (c x))}{4 d^2 \left (d+e x^2\right )^2}-\frac {e (a+b \arctan (c x))}{d^3 \left (d+e x^2\right )}-\frac {b c e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2} \left (c^2 d-e\right )}-\frac {b c \left (3 c^2 d-e\right ) e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2} \left (c^2 d-e\right )^2}-\frac {3 a e \log (x)}{d^4}-\frac {3 e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^4}+\frac {3 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^4}+\frac {3 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^4}-\frac {3 i b e \operatorname {PolyLog}(2,-i c x)}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}(2,i c x)}{2 d^4}+\frac {3 i b e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^4}-\frac {3 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^4}-\frac {3 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^4} \\ \end{align*}
Time = 12.59 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.15 \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {-a \left (\frac {d \left (2 d^2+9 d e x^2+6 e^2 x^4\right )}{x^2 \left (d+e x^2\right )^2}+12 e \log (x)-6 e \log \left (d+e x^2\right )\right )+b \left (-\frac {2 c d}{x}-\frac {c d e^2 x}{2 \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {c^2 d \left (-2 c^4 d^2+9 c^2 d e-6 e^2\right ) \arctan (c x)}{\left (-c^2 d+e\right )^2}-\frac {d \left (2 d^2+9 d e x^2+6 e^2 x^4\right ) \arctan (c x)}{x^2 \left (d+e x^2\right )^2}+\frac {c \sqrt {d} e^{3/2} \left (-11 c^2 d+9 e\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \left (-c^2 d+e\right )^2}-12 e \arctan (c x) \log (x)+6 e \arctan (c x) \left (\log \left (-\frac {i \sqrt {d}}{\sqrt {e}}+x\right )+\log \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )-\log \left (d+e x^2\right )\right )+6 e \arctan (c x) \log \left (d+e x^2\right )-6 i e (\log (x) \log (1+i c x)+\operatorname {PolyLog}(2,-i c x))+6 i e (\log (x) \log (1-i c x)+\operatorname {PolyLog}(2,i c x))-3 i e \left (\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 )+\operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{c \sqrt {d}-\sqrt {e}}\right )\right )+3 i e \left (\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 )+\operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{c \sqrt {d}+\sqrt {e}}\right )\right )+3 i e \left (\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 )+\operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{c \sqrt {d}-\sqrt {e}}\right )\right )-3 i e \left (\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 )+\operatorname {PolyLog}\left (2,\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{c \sqrt {d}+\sqrt {e}}\right )\right )\right )}{4 d^4} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.00 (sec) , antiderivative size = 951, normalized size of antiderivative = 1.51
method | result | size |
parts | \(\text {Expression too large to display}\) | \(951\) |
derivativedivides | \(\text {Expression too large to display}\) | \(987\) |
default | \(\text {Expression too large to display}\) | \(987\) |
risch | \(\text {Expression too large to display}\) | \(1858\) |
[In]
[Out]
\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \arctan (c x)}{x^3 \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^3\,{\left (e\,x^2+d\right )}^3} \,d x \]
[In]
[Out]