Integrand size = 19, antiderivative size = 268 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{21 a}+\frac {16 i c^3 \arctan (a x)^2}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2+\frac {32 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {16 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5000, 4930, 5040, 4964, 2449, 2352, 8, 200} \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {1}{105} a^4 c^3 x^5+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2+\frac {6}{35} c^3 x \left (a^2 x^2+1\right )^2 \arctan (a x)^2+\frac {8}{35} c^3 x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}-\frac {3 c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)}{35 a}-\frac {8 c^3 \left (a^2 x^2+1\right ) \arctan (a x)}{35 a}+\frac {19}{315} a^2 c^3 x^3+\frac {16}{35} c^3 x \arctan (a x)^2+\frac {16 i c^3 \arctan (a x)^2}{35 a}+\frac {32 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {16 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{35 a}+\frac {38 c^3 x}{105} \]
[In]
[Out]
Rule 8
Rule 200
Rule 2352
Rule 2449
Rule 4930
Rule 4964
Rule 5000
Rule 5040
Rubi steps \begin{align*} \text {integral}& = -\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{21 a}+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2+\frac {1}{21} c \int \left (c+a^2 c x^2\right )^2 \, dx+\frac {1}{7} (6 c) \int \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx \\ & = -\frac {3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{21 a}+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2+\frac {1}{21} c \int \left (c^2+2 a^2 c^2 x^2+a^4 c^2 x^4\right ) \, dx+\frac {1}{35} \left (3 c^2\right ) \int \left (c+a^2 c x^2\right ) \, dx+\frac {1}{35} \left (24 c^2\right ) \int \left (c+a^2 c x^2\right ) \arctan (a x)^2 \, dx \\ & = \frac {2 c^3 x}{15}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{21 a}+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2+\frac {1}{35} \left (8 c^3\right ) \int 1 \, dx+\frac {1}{35} \left (16 c^3\right ) \int \arctan (a x)^2 \, dx \\ & = \frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{21 a}+\frac {16}{35} c^3 x \arctan (a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2-\frac {1}{35} \left (32 a c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{21 a}+\frac {16 i c^3 \arctan (a x)^2}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2+\frac {1}{35} \left (32 c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx \\ & = \frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{21 a}+\frac {16 i c^3 \arctan (a x)^2}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2+\frac {32 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}-\frac {1}{35} \left (32 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = \frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{21 a}+\frac {16 i c^3 \arctan (a x)^2}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2+\frac {32 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {\left (32 i c^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{35 a} \\ & = \frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{21 a}+\frac {16 i c^3 \arctan (a x)^2}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2+\frac {32 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {16 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a} \\ \end{align*}
Time = 1.01 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.51 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {c^3 \left (a x \left (114+19 a^2 x^2+3 a^4 x^4\right )+9 \left (-16 i+35 a x+35 a^3 x^3+21 a^5 x^5+5 a^7 x^7\right ) \arctan (a x)^2-3 \arctan (a x) \left (38+57 a^2 x^2+24 a^4 x^4+5 a^6 x^6-96 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-144 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{315 a} \]
[In]
[Out]
Time = 3.13 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}+a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {2 c^{3} \left (\frac {5 a^{6} \arctan \left (a x \right ) x^{6}}{6}+4 \arctan \left (a x \right ) a^{4} x^{4}+\frac {19 a^{2} \arctan \left (a x \right ) x^{2}}{2}+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{5} x^{5}}{6}-\frac {19 a^{3} x^{3}}{18}-\frac {19 a x}{3}+\frac {19 \arctan \left (a x \right )}{3}+4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{35}}{a}\) | \(280\) |
default | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}+a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {2 c^{3} \left (\frac {5 a^{6} \arctan \left (a x \right ) x^{6}}{6}+4 \arctan \left (a x \right ) a^{4} x^{4}+\frac {19 a^{2} \arctan \left (a x \right ) x^{2}}{2}+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{5} x^{5}}{6}-\frac {19 a^{3} x^{3}}{18}-\frac {19 a x}{3}+\frac {19 \arctan \left (a x \right )}{3}+4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{35}}{a}\) | \(280\) |
parts | \(\frac {c^{3} \arctan \left (a x \right )^{2} a^{6} x^{7}}{7}+\frac {3 c^{3} \arctan \left (a x \right )^{2} a^{4} x^{5}}{5}+c^{3} \arctan \left (a x \right )^{2} a^{2} x^{3}+c^{3} x \arctan \left (a x \right )^{2}-\frac {2 c^{3} \left (\frac {5 a^{5} \arctan \left (a x \right ) x^{6}}{6}+4 a^{3} \arctan \left (a x \right ) x^{4}+\frac {19 a \arctan \left (a x \right ) x^{2}}{2}+\frac {8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{a}-\frac {a^{5} x^{5}+\frac {19 a^{3} x^{3}}{3}+38 a x -38 \arctan \left (a x \right )-24 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+24 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{6 a}\right )}{35}\) | \(281\) |
risch | \(-\frac {38 c^{3} \arctan \left (a x \right )}{105 a}+\frac {19 a^{2} c^{3} x^{3}}{315}+\frac {a^{4} c^{3} x^{5}}{105}+\frac {38 c^{3} x}{105}-\frac {c^{3} \ln \left (i a x +1\right )^{2} x}{4}-\frac {c^{3} \ln \left (-i a x +1\right )^{2} x}{4}+\frac {20469 i c^{3}}{42875 a}+\frac {16 i c^{3} \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{35 a}+\frac {4 i c^{3} \ln \left (i a x +1\right )^{2}}{35 a}-\frac {4 i c^{3} \ln \left (-i a x +1\right )^{2}}{35 a}+\frac {c^{3} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x}{2}-\frac {3 c^{3} a^{4} \ln \left (i a x +1\right )^{2} x^{5}}{20}-\frac {c^{3} a^{2} \ln \left (i a x +1\right )^{2} x^{3}}{4}-\frac {c^{3} a^{6} \ln \left (i a x +1\right )^{2} x^{7}}{28}-\frac {3 c^{3} a^{4} \ln \left (-i a x +1\right )^{2} x^{5}}{20}-\frac {c^{3} a^{6} \ln \left (-i a x +1\right )^{2} x^{7}}{28}-\frac {c^{3} a^{2} \ln \left (-i a x +1\right )^{2} x^{3}}{4}-\frac {16 i c^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{35 a}+\frac {16 i c^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{35 a}+\frac {8 i c^{3} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{35 a}+\frac {c^{3} a^{6} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{7}}{14}+\frac {3 c^{3} a^{4} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{5}}{10}+\frac {c^{3} a^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{2}-\frac {i c^{3} a^{5} \ln \left (-i a x +1\right ) x^{6}}{42}-\frac {4 i c^{3} a^{3} \ln \left (-i a x +1\right ) x^{4}}{35}-\frac {19 i c^{3} a \ln \left (-i a x +1\right ) x^{2}}{70}+\frac {i c^{3} a^{5} \ln \left (i a x +1\right ) x^{6}}{42}+\frac {4 i c^{3} a^{3} \ln \left (i a x +1\right ) x^{4}}{35}+\frac {19 i c^{3} a \ln \left (i a x +1\right ) x^{2}}{70}\) | \(558\) |
[In]
[Out]
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2} \,d x } \]
[In]
[Out]
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=c^{3} \left (\int 3 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{6} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]
[In]
[Out]
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2} \,d x } \]
[In]
[Out]
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int {\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]
[In]
[Out]