Integrand size = 22, antiderivative size = 503 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=-\frac {1}{4} a^3 c^3 x+\frac {1}{4} a^2 c^3 \arctan (a x)+\frac {1}{4} a^4 c^3 x^2 \arctan (a x)-5 i a^2 c^3 \arctan (a x)^2-\frac {3 a c^3 \arctan (a x)^2}{2 x}-\frac {15}{4} a^3 c^3 x \arctan (a x)^2-\frac {1}{4} a^5 c^3 x^3 \arctan (a x)^2+\frac {3}{4} a^2 c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^3+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^3+6 a^2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-7 a^2 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {7}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \]
[Out]
Time = 0.88 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {5068, 4946, 5038, 5044, 4988, 2497, 5004, 4942, 5108, 5114, 5118, 6745, 5036, 4930, 5040, 4964, 2449, 2352, 327, 209} \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^3-\frac {1}{4} a^5 c^3 x^3 \arctan (a x)^2+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^3+\frac {1}{4} a^4 c^3 x^2 \arctan (a x)-\frac {15}{4} a^3 c^3 x \arctan (a x)^2-\frac {1}{4} a^3 c^3 x+6 a^2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )+\frac {3}{4} a^2 c^3 \arctan (a x)^3-5 i a^2 c^3 \arctan (a x)^2+\frac {1}{4} a^2 c^3 \arctan (a x)-7 a^2 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )-\frac {7}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{i a x+1}\right )-\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,\frac {2}{i a x+1}-1\right )-\frac {c^3 \arctan (a x)^3}{2 x^2}-\frac {3 a c^3 \arctan (a x)^2}{2 x} \]
[In]
[Out]
Rule 209
Rule 327
Rule 2352
Rule 2449
Rule 2497
Rule 4930
Rule 4942
Rule 4946
Rule 4964
Rule 4988
Rule 5004
Rule 5036
Rule 5038
Rule 5040
Rule 5044
Rule 5068
Rule 5108
Rule 5114
Rule 5118
Rule 6745
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^3 \arctan (a x)^3}{x^3}+\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+a^6 c^3 x^3 \arctan (a x)^3\right ) \, dx \\ & = c^3 \int \frac {\arctan (a x)^3}{x^3} \, dx+\left (3 a^2 c^3\right ) \int \frac {\arctan (a x)^3}{x} \, dx+\left (3 a^4 c^3\right ) \int x \arctan (a x)^3 \, dx+\left (a^6 c^3\right ) \int x^3 \arctan (a x)^3 \, dx \\ & = -\frac {c^3 \arctan (a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^3+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^3+6 a^2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} \left (3 a c^3\right ) \int \frac {\arctan (a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (18 a^3 c^3\right ) \int \frac {\arctan (a x)^2 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (9 a^5 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^7 c^3\right ) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {c^3 \arctan (a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^3+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^3+6 a^2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} \left (3 a c^3\right ) \int \frac {\arctan (a x)^2}{x^2} \, dx-\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{2} \left (9 a^3 c^3\right ) \int \arctan (a x)^2 \, dx+\frac {1}{2} \left (9 a^3 c^3\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx+\left (9 a^3 c^3\right ) \int \frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (9 a^3 c^3\right ) \int \frac {\arctan (a x)^2 \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^5 c^3\right ) \int x^2 \arctan (a x)^2 \, dx+\frac {1}{4} \left (3 a^5 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {3 a c^3 \arctan (a x)^2}{2 x}-\frac {9}{2} a^3 c^3 x \arctan (a x)^2-\frac {1}{4} a^5 c^3 x^3 \arctan (a x)^2+a^2 c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^3+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^3+6 a^2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )+\left (3 a^2 c^3\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx+\left (9 i a^3 c^3\right ) \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (9 i a^3 c^3\right ) \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{4} \left (3 a^3 c^3\right ) \int \arctan (a x)^2 \, dx-\frac {1}{4} \left (3 a^3 c^3\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx+\left (9 a^4 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{2} \left (a^6 c^3\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -6 i a^2 c^3 \arctan (a x)^2-\frac {3 a c^3 \arctan (a x)^2}{2 x}-\frac {15}{4} a^3 c^3 x \arctan (a x)^2-\frac {1}{4} a^5 c^3 x^3 \arctan (a x)^2+\frac {3}{4} a^2 c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^3+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^3+6 a^2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\left (3 i a^2 c^3\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx+\frac {1}{2} \left (9 a^3 c^3\right ) \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (9 a^3 c^3\right ) \int \frac {\operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (9 a^3 c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx+\frac {1}{2} \left (a^4 c^3\right ) \int x \arctan (a x) \, dx-\frac {1}{2} \left (a^4 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^4 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {1}{4} a^4 c^3 x^2 \arctan (a x)-5 i a^2 c^3 \arctan (a x)^2-\frac {3 a c^3 \arctan (a x)^2}{2 x}-\frac {15}{4} a^3 c^3 x \arctan (a x)^2-\frac {1}{4} a^5 c^3 x^3 \arctan (a x)^2+\frac {3}{4} a^2 c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^3+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^3+6 a^2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-9 a^2 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^3 c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx+\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx-\left (3 a^3 c^3\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx+\left (9 a^3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{4} \left (a^5 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx \\ & = -\frac {1}{4} a^3 c^3 x+\frac {1}{4} a^4 c^3 x^2 \arctan (a x)-5 i a^2 c^3 \arctan (a x)^2-\frac {3 a c^3 \arctan (a x)^2}{2 x}-\frac {15}{4} a^3 c^3 x \arctan (a x)^2-\frac {1}{4} a^5 c^3 x^3 \arctan (a x)^2+\frac {3}{4} a^2 c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^3+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^3+6 a^2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-7 a^2 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )-\left (9 i a^2 c^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )+\frac {1}{4} \left (a^3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{2} \left (a^3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {1}{4} a^3 c^3 x+\frac {1}{4} a^2 c^3 \arctan (a x)+\frac {1}{4} a^4 c^3 x^2 \arctan (a x)-5 i a^2 c^3 \arctan (a x)^2-\frac {3 a c^3 \arctan (a x)^2}{2 x}-\frac {15}{4} a^3 c^3 x \arctan (a x)^2-\frac {1}{4} a^5 c^3 x^3 \arctan (a x)^2+\frac {3}{4} a^2 c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^3+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^3+6 a^2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-7 a^2 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {9}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (i a^2 c^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )+\frac {1}{2} \left (3 i a^2 c^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right ) \\ & = -\frac {1}{4} a^3 c^3 x+\frac {1}{4} a^2 c^3 \arctan (a x)+\frac {1}{4} a^4 c^3 x^2 \arctan (a x)-5 i a^2 c^3 \arctan (a x)^2-\frac {3 a c^3 \arctan (a x)^2}{2 x}-\frac {15}{4} a^3 c^3 x \arctan (a x)^2-\frac {1}{4} a^5 c^3 x^3 \arctan (a x)^2+\frac {3}{4} a^2 c^3 \arctan (a x)^3-\frac {c^3 \arctan (a x)^3}{2 x^2}+\frac {3}{2} a^4 c^3 x^2 \arctan (a x)^3+\frac {1}{4} a^6 c^3 x^4 \arctan (a x)^3+6 a^2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-7 a^2 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+3 a^2 c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {3}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )-\frac {7}{2} i a^2 c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {9}{2} i a^2 c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^2 c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {9}{4} i a^2 c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 464, normalized size of antiderivative = 0.92 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=\frac {c^3 \left (-3 i a^2 \pi ^4 x^2-16 a^3 x^3+16 a^2 x^2 \arctan (a x)+16 a^4 x^4 \arctan (a x)-96 a x \arctan (a x)^2+128 i a^2 x^2 \arctan (a x)^2-240 a^3 x^3 \arctan (a x)^2-16 a^5 x^5 \arctan (a x)^2-32 \arctan (a x)^3+48 a^2 x^2 \arctan (a x)^3+96 a^4 x^4 \arctan (a x)^3+16 a^6 x^6 \arctan (a x)^3+96 i a^2 x^2 \arctan (a x)^4+192 a^2 x^2 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )+192 a^2 x^2 \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )-448 a^2 x^2 \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-192 a^2 x^2 \arctan (a x)^3 \log \left (1+e^{2 i \arctan (a x)}\right )+288 i a^2 x^2 \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+32 i a^2 x^2 \left (7+9 \arctan (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-96 i a^2 x^2 \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )+288 a^2 x^2 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-288 a^2 x^2 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )-144 i a^2 x^2 \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )-144 i a^2 x^2 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (a x)}\right )\right )}{64 x^2} \]
[In]
[Out]
Time = 77.00 (sec) , antiderivative size = 763, normalized size of antiderivative = 1.52
method | result | size |
derivativedivides | \(a^{2} \left (\frac {c^{3} \left (2 i \arctan \left (a x \right )^{3}+6 i \arctan \left (a x \right )^{2} a x -2 \arctan \left (a x \right )^{3} a x -14 x^{2} \arctan \left (a x \right )^{2} a^{2}-5 i \arctan \left (a x \right )^{3} a^{2} x^{2}+i \arctan \left (a x \right )^{2} a^{3} x^{3}+5 \arctan \left (a x \right )^{3} a^{3} x^{3}-a^{4} \arctan \left (a x \right )^{2} x^{4}-i \arctan \left (a x \right )^{3} a^{4} x^{4}+\arctan \left (a x \right )^{3} a^{5} x^{5}-a^{2} x^{2}-i \arctan \left (a x \right ) a^{2} x^{2}+\arctan \left (a x \right ) x^{3} a^{3}\right ) \left (a x +i\right )}{4 a^{2} x^{2}}-9 i c^{3} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-7 c^{3} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-3 i c^{3} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{3} \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {7 i c^{3} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+3 c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+18 i c^{3} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+18 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{3} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{3} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+18 i c^{3} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {9 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+4 i c^{3} \arctan \left (a x \right )^{2}+18 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-9 i c^{3} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-\frac {9 i c^{3} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+3 c^{3} \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+\frac {9 i c^{3} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}\right )\) | \(763\) |
default | \(a^{2} \left (\frac {c^{3} \left (2 i \arctan \left (a x \right )^{3}+6 i \arctan \left (a x \right )^{2} a x -2 \arctan \left (a x \right )^{3} a x -14 x^{2} \arctan \left (a x \right )^{2} a^{2}-5 i \arctan \left (a x \right )^{3} a^{2} x^{2}+i \arctan \left (a x \right )^{2} a^{3} x^{3}+5 \arctan \left (a x \right )^{3} a^{3} x^{3}-a^{4} \arctan \left (a x \right )^{2} x^{4}-i \arctan \left (a x \right )^{3} a^{4} x^{4}+\arctan \left (a x \right )^{3} a^{5} x^{5}-a^{2} x^{2}-i \arctan \left (a x \right ) a^{2} x^{2}+\arctan \left (a x \right ) x^{3} a^{3}\right ) \left (a x +i\right )}{4 a^{2} x^{2}}-9 i c^{3} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-7 c^{3} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-3 i c^{3} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{3} \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {7 i c^{3} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+3 c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+18 i c^{3} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+18 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i c^{3} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 c^{3} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+18 i c^{3} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {9 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}+4 i c^{3} \arctan \left (a x \right )^{2}+18 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-9 i c^{3} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )-\frac {9 i c^{3} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+3 c^{3} \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+\frac {9 i c^{3} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}\right )\) | \(763\) |
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\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=c^{3} \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{3}}\, dx + \int \frac {3 a^{2} \operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx + \int 3 a^{4} x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3}{x^3} \,d x \]
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