Integrand size = 22, antiderivative size = 447 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=-\frac {13}{30} a c^3 x-\frac {1}{60} a^3 c^3 x^3+\frac {13}{30} c^3 \arctan (a x)+\frac {29}{60} a^2 c^3 x^2 \arctan (a x)+\frac {1}{20} a^4 c^3 x^4 \arctan (a x)-\frac {34}{15} i c^3 \arctan (a x)^2-\frac {11}{4} a c^3 x \arctan (a x)^2-\frac {7}{12} a^3 c^3 x^3 \arctan (a x)^2-\frac {1}{10} a^5 c^3 x^5 \arctan (a x)^2+\frac {11}{12} c^3 \arctan (a x)^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {68}{15} c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {34}{15} i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \]
[Out]
Time = 1.25 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 69, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {5068, 4942, 5108, 5004, 5114, 5118, 6745, 4946, 5036, 4930, 5040, 4964, 2449, 2352, 327, 209, 308} \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3-\frac {1}{10} a^5 c^3 x^5 \arctan (a x)^2+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{20} a^4 c^3 x^4 \arctan (a x)-\frac {7}{12} a^3 c^3 x^3 \arctan (a x)^2-\frac {1}{60} a^3 c^3 x^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {29}{60} a^2 c^3 x^2 \arctan (a x)+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{i a x+1}-1\right )-\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,\frac {2}{i a x+1}-1\right )-\frac {11}{4} a c^3 x \arctan (a x)^2+\frac {11}{12} c^3 \arctan (a x)^3-\frac {34}{15} i c^3 \arctan (a x)^2+\frac {13}{30} c^3 \arctan (a x)-\frac {68}{15} c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {34}{15} i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )+\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{i a x+1}\right )-\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,\frac {2}{i a x+1}-1\right )-\frac {13}{30} a c^3 x \]
[In]
[Out]
Rule 209
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 4930
Rule 4942
Rule 4946
Rule 4964
Rule 5004
Rule 5036
Rule 5040
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 a^2 c^3 x \arctan (a x)^3+3 a^4 c^3 x^3 \arctan (a x)^3+a^6 c^3 x^5 \arctan (a x)^3\right ) \, dx \\ & = c^3 \int \frac {\arctan (a x)^3}{x} \, dx+\left (3 a^2 c^3\right ) \int x \arctan (a x)^3 \, dx+\left (3 a^4 c^3\right ) \int x^3 \arctan (a x)^3 \, dx+\left (a^6 c^3\right ) \int x^5 \arctan (a x)^3 \, dx \\ & = \frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\left (6 a 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^3 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{4} \left (9 a^5 c^3\right ) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{2} \left (a^7 c^3\right ) \int \frac {x^6 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = \frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )+\left (3 a 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 (3 a 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}{2} \left (9 a c^3\right ) \int \arctan (a x)^2 \, dx+\frac {1}{2} \left (9 a c^3\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{4} \left (9 a^3 c^3\right ) \int x^2 \arctan (a x)^2 \, dx+\frac {1}{4} \left (9 a^3 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{2} \left (a^5 c^3\right ) \int x^4 \arctan (a x)^2 \, dx+\frac {1}{2} \left (a^5 c^3\right ) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {9}{2} a c^3 x \arctan (a x)^2-\frac {3}{4} a^3 c^3 x^3 \arctan (a x)^2-\frac {1}{10} a^5 c^3 x^5 \arctan (a x)^2+\frac {3}{2} c^3 \arctan (a x)^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )+\left (3 i a 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 (3 i a 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 (9 a c^3\right ) \int \arctan (a x)^2 \, dx-\frac {1}{4} \left (9 a c^3\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx+\left (9 a^2 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{2} \left (a^3 c^3\right ) \int x^2 \arctan (a x)^2 \, dx-\frac {1}{2} \left (a^3 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx+\frac {1}{2} \left (3 a^4 c^3\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^6 c^3\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {9}{2} i c^3 \arctan (a x)^2-\frac {9}{4} a c^3 x \arctan (a x)^2-\frac {7}{12} a^3 c^3 x^3 \arctan (a x)^2-\frac {1}{10} a^5 c^3 x^5 \arctan (a x)^2+\frac {3}{4} c^3 \arctan (a x)^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} \left (a c^3\right ) \int \arctan (a x)^2 \, dx+\frac {1}{2} \left (a c^3\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx+\frac {1}{2} \left (3 a 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 (3 a 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 c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx+\frac {1}{2} \left (3 a^2 c^3\right ) \int x \arctan (a x) \, dx-\frac {1}{2} \left (3 a^2 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (9 a^2 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^4 c^3\right ) \int x^3 \arctan (a x) \, dx-\frac {1}{5} \left (a^4 c^3\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^4 c^3\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {3}{4} a^2 c^3 x^2 \arctan (a x)+\frac {1}{20} a^4 c^3 x^4 \arctan (a x)-\frac {3}{2} i c^3 \arctan (a x)^2-\frac {11}{4} a c^3 x \arctan (a x)^2-\frac {7}{12} a^3 c^3 x^3 \arctan (a x)^2-\frac {1}{10} a^5 c^3 x^5 \arctan (a x)^2+\frac {11}{12} c^3 \arctan (a x)^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-9 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (3 a c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx+\frac {1}{2} \left (9 a c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx+\left (9 a c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{5} \left (a^2 c^3\right ) \int x \arctan (a x) \, dx+\frac {1}{5} \left (a^2 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^2 c^3\right ) \int x \arctan (a x) \, dx+\frac {1}{3} \left (a^2 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx+\left (a^2 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{4} \left (3 a^3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {1}{20} \left (a^5 c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx \\ & = -\frac {3}{4} a c^3 x+\frac {29}{60} a^2 c^3 x^2 \arctan (a x)+\frac {1}{20} a^4 c^3 x^4 \arctan (a x)-\frac {34}{15} i c^3 \arctan (a x)^2-\frac {11}{4} a c^3 x \arctan (a x)^2-\frac {7}{12} a^3 c^3 x^3 \arctan (a x)^2-\frac {1}{10} a^5 c^3 x^5 \arctan (a x)^2+\frac {11}{12} c^3 \arctan (a x)^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-3 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )-\left (9 i 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}{5} \left (a c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx-\frac {1}{3} \left (a c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx+\frac {1}{4} \left (3 a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\left (a c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx-\frac {1}{2} \left (3 a c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{2} \left (9 a c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{10} \left (a^3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx+\frac {1}{6} \left (a^3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {1}{20} \left (a^5 c^3\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = -\frac {13}{30} a c^3 x-\frac {1}{60} a^3 c^3 x^3+\frac {3}{4} c^3 \arctan (a x)+\frac {29}{60} a^2 c^3 x^2 \arctan (a x)+\frac {1}{20} a^4 c^3 x^4 \arctan (a x)-\frac {34}{15} i c^3 \arctan (a x)^2-\frac {11}{4} a c^3 x \arctan (a x)^2-\frac {7}{12} a^3 c^3 x^3 \arctan (a x)^2-\frac {1}{10} a^5 c^3 x^5 \arctan (a x)^2+\frac {11}{12} c^3 \arctan (a x)^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {68}{15} c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {9}{2} i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} \left (3 i 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 (9 i 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}{20} \left (a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{10} \left (a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{6} \left (a c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{3} \left (a c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (a c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {13}{30} a c^3 x-\frac {1}{60} a^3 c^3 x^3+\frac {13}{30} c^3 \arctan (a x)+\frac {29}{60} a^2 c^3 x^2 \arctan (a x)+\frac {1}{20} a^4 c^3 x^4 \arctan (a x)-\frac {34}{15} i c^3 \arctan (a x)^2-\frac {11}{4} a c^3 x \arctan (a x)^2-\frac {7}{12} a^3 c^3 x^3 \arctan (a x)^2-\frac {1}{10} a^5 c^3 x^5 \arctan (a x)^2+\frac {11}{12} c^3 \arctan (a x)^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {68}{15} c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right )-\frac {1}{5} \left (i 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}{3} \left (i c^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )-\left (i c^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right ) \\ & = -\frac {13}{30} a c^3 x-\frac {1}{60} a^3 c^3 x^3+\frac {13}{30} c^3 \arctan (a x)+\frac {29}{60} a^2 c^3 x^2 \arctan (a x)+\frac {1}{20} a^4 c^3 x^4 \arctan (a x)-\frac {34}{15} i c^3 \arctan (a x)^2-\frac {11}{4} a c^3 x \arctan (a x)^2-\frac {7}{12} a^3 c^3 x^3 \arctan (a x)^2-\frac {1}{10} a^5 c^3 x^5 \arctan (a x)^2+\frac {11}{12} c^3 \arctan (a x)^3+\frac {3}{2} a^2 c^3 x^2 \arctan (a x)^3+\frac {3}{4} a^4 c^3 x^4 \arctan (a x)^3+\frac {1}{6} a^6 c^3 x^6 \arctan (a x)^3+2 c^3 \arctan (a x)^3 \text {arctanh}\left (1-\frac {2}{1+i a x}\right )-\frac {68}{15} c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )-\frac {34}{15} i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} i c^3 \arctan (a x)^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \arctan (a x) \operatorname {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )+\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )-\frac {3}{4} i c^3 \operatorname {PolyLog}\left (4,-1+\frac {2}{1+i a x}\right ) \\ \end{align*}
Time = 0.82 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.78 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=\frac {1}{960} c^3 \left (-15 i \pi ^4-416 a x-16 a^3 x^3+416 \arctan (a x)+464 a^2 x^2 \arctan (a x)+48 a^4 x^4 \arctan (a x)+2176 i \arctan (a x)^2-2640 a x \arctan (a x)^2-560 a^3 x^3 \arctan (a x)^2-96 a^5 x^5 \arctan (a x)^2+880 \arctan (a x)^3+1440 a^2 x^2 \arctan (a x)^3+720 a^4 x^4 \arctan (a x)^3+160 a^6 x^6 \arctan (a x)^3+480 i \arctan (a x)^4+960 \arctan (a x)^3 \log \left (1-e^{-2 i \arctan (a x)}\right )-4352 \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-960 \arctan (a x)^3 \log \left (1+e^{2 i \arctan (a x)}\right )+1440 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+32 i \left (68+45 \arctan (a x)^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+1440 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )-1440 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )-720 i \operatorname {PolyLog}\left (4,e^{-2 i \arctan (a x)}\right )-720 i \operatorname {PolyLog}\left (4,-e^{2 i \arctan (a x)}\right )\right ) \]
[In]
[Out]
Time = 56.76 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.49
method | result | size |
derivativedivides | \(\frac {c^{3} \left (-136 \arctan \left (a x \right )^{2}-55 i \arctan \left (a x \right )^{3}+i a x +55 \arctan \left (a x \right )^{3} a x -29 x^{2} \arctan \left (a x \right )^{2} a^{2}-26 i \arctan \left (a x \right )-3 i \arctan \left (a x \right ) a^{2} x^{2}+35 \arctan \left (a x \right )^{3} a^{3} x^{3}-6 a^{4} \arctan \left (a x \right )^{2} x^{4}-35 i \arctan \left (a x \right )^{3} a^{2} x^{2}+10 \arctan \left (a x \right )^{3} a^{5} x^{5}-25+29 i \arctan \left (a x \right )^{2} a x -10 i \arctan \left (a x \right )^{3} a^{4} x^{4}+26 x \arctan \left (a x \right ) a -a^{2} x^{2}+6 i \arctan \left (a x \right )^{2} a^{3} x^{3}+3 \arctan \left (a x \right ) x^{3} a^{3}\right ) \left (a x +i\right )}{60}+\frac {3 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}-\frac {68 c^{3} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{15}+6 i c^{3} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+\frac {34 i c^{3} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{15}+6 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} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-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 {68 i c^{3} \arctan \left (a x \right )^{2}}{15}-\frac {3 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}-\frac {3 i c^{3} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+c^{3} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 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 )+6 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c^{3} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\) | \(664\) |
default | \(\frac {c^{3} \left (-136 \arctan \left (a x \right )^{2}-55 i \arctan \left (a x \right )^{3}+i a x +55 \arctan \left (a x \right )^{3} a x -29 x^{2} \arctan \left (a x \right )^{2} a^{2}-26 i \arctan \left (a x \right )-3 i \arctan \left (a x \right ) a^{2} x^{2}+35 \arctan \left (a x \right )^{3} a^{3} x^{3}-6 a^{4} \arctan \left (a x \right )^{2} x^{4}-35 i \arctan \left (a x \right )^{3} a^{2} x^{2}+10 \arctan \left (a x \right )^{3} a^{5} x^{5}-25+29 i \arctan \left (a x \right )^{2} a x -10 i \arctan \left (a x \right )^{3} a^{4} x^{4}+26 x \arctan \left (a x \right ) a -a^{2} x^{2}+6 i \arctan \left (a x \right )^{2} a^{3} x^{3}+3 \arctan \left (a x \right ) x^{3} a^{3}\right ) \left (a x +i\right )}{60}+\frac {3 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}-\frac {68 c^{3} \arctan \left (a x \right ) \ln \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )}{15}+6 i c^{3} \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+c^{3} \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+\frac {34 i c^{3} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{15}+6 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} \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-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 {68 i c^{3} \arctan \left (a x \right )^{2}}{15}-\frac {3 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}-\frac {3 i c^{3} \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{4}+c^{3} \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 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 )+6 c^{3} \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i c^{3} \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\) | \(664\) |
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=c^{3} \left (\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx + \int 3 a^{2} x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3}{x} \,d x \]
[In]
[Out]