Integrand size = 25, antiderivative size = 513 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=-\frac {b^2 c^2 d^3}{60 x^4}-\frac {i b^2 c^3 d^3}{10 x^3}+\frac {61 b^2 c^4 d^3}{180 x^2}+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {37}{30} i b^2 c^6 d^3 \arctan (c x)-\frac {b c d^3 (a+b \arctan (c x))}{15 x^5}-\frac {3 i b c^2 d^3 (a+b \arctan (c x))}{10 x^4}+\frac {11 b c^3 d^3 (a+b \arctan (c x))}{18 x^3}+\frac {14 i b c^4 d^3 (a+b \arctan (c x))}{15 x^2}-\frac {11 b c^5 d^3 (a+b \arctan (c x))}{6 x}-\frac {d^3 (a+b \arctan (c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{5 x^5}+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{4 x^4}+\frac {i c^3 d^3 (a+b \arctan (c x))^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {113}{90} b^2 c^6 d^3 \log \left (1+c^2 x^2\right )-\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,-i c x)+\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,i c x)+\frac {37}{40} b^2 c^6 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \]
[Out]
Time = 0.37 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {45, 4994, 4946, 272, 46, 331, 209, 36, 29, 31, 4940, 2438, 4964, 2449, 2352} \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\frac {37}{20} i b c^6 d^3 \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))+\frac {1}{60} i b c^6 d^3 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))-\frac {11 b c^5 d^3 (a+b \arctan (c x))}{6 x}+\frac {14 i b c^4 d^3 (a+b \arctan (c x))}{15 x^2}+\frac {i c^3 d^3 (a+b \arctan (c x))^2}{3 x^3}+\frac {11 b c^3 d^3 (a+b \arctan (c x))}{18 x^3}+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{4 x^4}-\frac {3 i b c^2 d^3 (a+b \arctan (c x))}{10 x^4}-\frac {d^3 (a+b \arctan (c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{5 x^5}-\frac {b c d^3 (a+b \arctan (c x))}{15 x^5}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {37}{30} i b^2 c^6 d^3 \arctan (c x)-\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,-i c x)+\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,i c x)+\frac {37}{40} b^2 c^6 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {61 b^2 c^4 d^3}{180 x^2}-\frac {i b^2 c^3 d^3}{10 x^3}-\frac {b^2 c^2 d^3}{60 x^4}-\frac {113}{90} b^2 c^6 d^3 \log \left (c^2 x^2+1\right ) \]
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 45
Rule 46
Rule 209
Rule 272
Rule 331
Rule 2352
Rule 2438
Rule 2449
Rule 4940
Rule 4946
Rule 4964
Rule 4994
Rubi steps \begin{align*} \text {integral}& = -\frac {d^3 (a+b \arctan (c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{5 x^5}+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{4 x^4}+\frac {i c^3 d^3 (a+b \arctan (c x))^2}{3 x^3}-(2 b c) \int \left (-\frac {d^3 (a+b \arctan (c x))}{6 x^6}-\frac {3 i c d^3 (a+b \arctan (c x))}{5 x^5}+\frac {11 c^2 d^3 (a+b \arctan (c x))}{12 x^4}+\frac {14 i c^3 d^3 (a+b \arctan (c x))}{15 x^3}-\frac {11 c^4 d^3 (a+b \arctan (c x))}{12 x^2}-\frac {14 i c^5 d^3 (a+b \arctan (c x))}{15 x}+\frac {i c^6 d^3 (a+b \arctan (c x))}{120 (-i+c x)}+\frac {37 i c^6 d^3 (a+b \arctan (c x))}{40 (i+c x)}\right ) \, dx \\ & = -\frac {d^3 (a+b \arctan (c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{5 x^5}+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{4 x^4}+\frac {i c^3 d^3 (a+b \arctan (c x))^2}{3 x^3}+\frac {1}{3} \left (b c d^3\right ) \int \frac {a+b \arctan (c x)}{x^6} \, dx+\frac {1}{5} \left (6 i b c^2 d^3\right ) \int \frac {a+b \arctan (c x)}{x^5} \, dx-\frac {1}{6} \left (11 b c^3 d^3\right ) \int \frac {a+b \arctan (c x)}{x^4} \, dx-\frac {1}{15} \left (28 i b c^4 d^3\right ) \int \frac {a+b \arctan (c x)}{x^3} \, dx+\frac {1}{6} \left (11 b c^5 d^3\right ) \int \frac {a+b \arctan (c x)}{x^2} \, dx+\frac {1}{15} \left (28 i b c^6 d^3\right ) \int \frac {a+b \arctan (c x)}{x} \, dx-\frac {1}{60} \left (i b c^7 d^3\right ) \int \frac {a+b \arctan (c x)}{-i+c x} \, dx-\frac {1}{20} \left (37 i b c^7 d^3\right ) \int \frac {a+b \arctan (c x)}{i+c x} \, dx \\ & = -\frac {b c d^3 (a+b \arctan (c x))}{15 x^5}-\frac {3 i b c^2 d^3 (a+b \arctan (c x))}{10 x^4}+\frac {11 b c^3 d^3 (a+b \arctan (c x))}{18 x^3}+\frac {14 i b c^4 d^3 (a+b \arctan (c x))}{15 x^2}-\frac {11 b c^5 d^3 (a+b \arctan (c x))}{6 x}-\frac {d^3 (a+b \arctan (c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{5 x^5}+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{4 x^4}+\frac {i c^3 d^3 (a+b \arctan (c x))^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )+\frac {1}{15} \left (b^2 c^2 d^3\right ) \int \frac {1}{x^5 \left (1+c^2 x^2\right )} \, dx+\frac {1}{10} \left (3 i b^2 c^3 d^3\right ) \int \frac {1}{x^4 \left (1+c^2 x^2\right )} \, dx-\frac {1}{18} \left (11 b^2 c^4 d^3\right ) \int \frac {1}{x^3 \left (1+c^2 x^2\right )} \, dx-\frac {1}{15} \left (14 i b^2 c^5 d^3\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac {1}{15} \left (14 b^2 c^6 d^3\right ) \int \frac {\log (1-i c x)}{x} \, dx+\frac {1}{15} \left (14 b^2 c^6 d^3\right ) \int \frac {\log (1+i c x)}{x} \, dx+\frac {1}{6} \left (11 b^2 c^6 d^3\right ) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx-\frac {1}{60} \left (i b^2 c^7 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\frac {1}{20} \left (37 i b^2 c^7 d^3\right ) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx \\ & = -\frac {i b^2 c^3 d^3}{10 x^3}+\frac {14 i b^2 c^5 d^3}{15 x}-\frac {b c d^3 (a+b \arctan (c x))}{15 x^5}-\frac {3 i b c^2 d^3 (a+b \arctan (c x))}{10 x^4}+\frac {11 b c^3 d^3 (a+b \arctan (c x))}{18 x^3}+\frac {14 i b c^4 d^3 (a+b \arctan (c x))}{15 x^2}-\frac {11 b c^5 d^3 (a+b \arctan (c x))}{6 x}-\frac {d^3 (a+b \arctan (c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{5 x^5}+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{4 x^4}+\frac {i c^3 d^3 (a+b \arctan (c x))^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,-i c x)+\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,i c x)+\frac {1}{30} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \frac {1}{x^3 \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{36} \left (11 b^2 c^4 d^3\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{10} \left (3 i b^2 c^5 d^3\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac {1}{60} \left (b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )+\frac {1}{12} \left (11 b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{20} \left (37 b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )+\frac {1}{15} \left (14 i b^2 c^7 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx \\ & = -\frac {i b^2 c^3 d^3}{10 x^3}+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {14}{15} i b^2 c^6 d^3 \arctan (c x)-\frac {b c d^3 (a+b \arctan (c x))}{15 x^5}-\frac {3 i b c^2 d^3 (a+b \arctan (c x))}{10 x^4}+\frac {11 b c^3 d^3 (a+b \arctan (c x))}{18 x^3}+\frac {14 i b c^4 d^3 (a+b \arctan (c x))}{15 x^2}-\frac {11 b c^5 d^3 (a+b \arctan (c x))}{6 x}-\frac {d^3 (a+b \arctan (c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{5 x^5}+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{4 x^4}+\frac {i c^3 d^3 (a+b \arctan (c x))^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,-i c x)+\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,i c x)+\frac {37}{40} b^2 c^6 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )+\frac {1}{30} \left (b^2 c^2 d^3\right ) \text {Subst}\left (\int \left (\frac {1}{x^3}-\frac {c^2}{x^2}+\frac {c^4}{x}-\frac {c^6}{1+c^2 x}\right ) \, dx,x,x^2\right )-\frac {1}{36} \left (11 b^2 c^4 d^3\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {c^2}{x}+\frac {c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )+\frac {1}{12} \left (11 b^2 c^6 d^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{10} \left (3 i b^2 c^7 d^3\right ) \int \frac {1}{1+c^2 x^2} \, dx-\frac {1}{12} \left (11 b^2 c^8 d^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right ) \\ & = -\frac {b^2 c^2 d^3}{60 x^4}-\frac {i b^2 c^3 d^3}{10 x^3}+\frac {61 b^2 c^4 d^3}{180 x^2}+\frac {37 i b^2 c^5 d^3}{30 x}+\frac {37}{30} i b^2 c^6 d^3 \arctan (c x)-\frac {b c d^3 (a+b \arctan (c x))}{15 x^5}-\frac {3 i b c^2 d^3 (a+b \arctan (c x))}{10 x^4}+\frac {11 b c^3 d^3 (a+b \arctan (c x))}{18 x^3}+\frac {14 i b c^4 d^3 (a+b \arctan (c x))}{15 x^2}-\frac {11 b c^5 d^3 (a+b \arctan (c x))}{6 x}-\frac {d^3 (a+b \arctan (c x))^2}{6 x^6}-\frac {3 i c d^3 (a+b \arctan (c x))^2}{5 x^5}+\frac {3 c^2 d^3 (a+b \arctan (c x))^2}{4 x^4}+\frac {i c^3 d^3 (a+b \arctan (c x))^2}{3 x^3}+\frac {28}{15} i a b c^6 d^3 \log (x)+\frac {113}{45} b^2 c^6 d^3 \log (x)+\frac {37}{20} i b c^6 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )+\frac {1}{60} i b c^6 d^3 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )-\frac {113}{90} b^2 c^6 d^3 \log \left (1+c^2 x^2\right )-\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,-i c x)+\frac {14}{15} b^2 c^6 d^3 \operatorname {PolyLog}(2,i c x)+\frac {37}{40} b^2 c^6 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )-\frac {1}{120} b^2 c^6 d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \\ \end{align*}
Time = 1.37 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.78 \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\frac {d^3 \left (-30 a^2-108 i a^2 c x-12 a b c x+135 a^2 c^2 x^2-54 i a b c^2 x^2-3 b^2 c^2 x^2+60 i a^2 c^3 x^3+110 a b c^3 x^3-18 i b^2 c^3 x^3+168 i a b c^4 x^4+61 b^2 c^4 x^4-330 a b c^5 x^5+222 i b^2 c^5 x^5+64 b^2 c^6 x^6+3 b^2 (-i+c x)^4 \left (-10+4 i c x+c^2 x^2\right ) \arctan (c x)^2+2 b \arctan (c x) \left (b c x \left (-6-27 i c x+55 c^2 x^2+84 i c^3 x^3-165 c^4 x^4+111 i c^5 x^5\right )-3 a \left (10+36 i c x-45 c^2 x^2-20 i c^3 x^3+55 c^6 x^6\right )+168 i b c^6 x^6 \log \left (1-e^{2 i \arctan (c x)}\right )\right )+336 i a b c^6 x^6 \log (c x)+452 b^2 c^6 x^6 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )-168 i a b c^6 x^6 \log \left (1+c^2 x^2\right )+168 b^2 c^6 x^6 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )}{180 x^6} \]
[In]
[Out]
Time = 4.67 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.10
method | result | size |
parts | \(d^{3} a^{2} \left (\frac {3 c^{2}}{4 x^{4}}-\frac {3 i c}{5 x^{5}}-\frac {1}{6 x^{6}}+\frac {i c^{3}}{3 x^{3}}\right )+b^{2} d^{3} c^{6} \left (-\frac {\arctan \left (c x \right )}{15 c^{5} x^{5}}-\frac {113 \ln \left (c^{2} x^{2}+1\right )}{90}-\frac {1}{60 c^{4} x^{4}}-\frac {11 \arctan \left (c x \right )^{2}}{12}+\frac {113 \ln \left (c x \right )}{45}+\frac {61}{180 c^{2} x^{2}}+\frac {3 \arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {7 \ln \left (c x -i\right )^{2}}{30}-\frac {7 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{15}+\frac {7 \ln \left (c x +i\right )^{2}}{30}+\frac {7 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{15}-\frac {11 \arctan \left (c x \right )}{6 c x}+\frac {14 \operatorname {dilog}\left (-i c x +1\right )}{15}-\frac {14 \operatorname {dilog}\left (i c x +1\right )}{15}+\frac {7 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {7 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{15}-\frac {7 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{15}+\frac {37 i \arctan \left (c x \right )}{30}-\frac {14 \ln \left (c x \right ) \ln \left (i c x +1\right )}{15}+\frac {14 \ln \left (c x \right ) \ln \left (-i c x +1\right )}{15}+\frac {28 i \arctan \left (c x \right ) \ln \left (c x \right )}{15}+\frac {11 \arctan \left (c x \right )}{18 c^{3} x^{3}}+\frac {37 i}{30 c x}+\frac {14 i \arctan \left (c x \right )}{15 c^{2} x^{2}}-\frac {i}{10 c^{3} x^{3}}-\frac {3 i \arctan \left (c x \right )}{10 c^{4} x^{4}}-\frac {14 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {i \arctan \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\arctan \left (c x \right )^{2}}{6 c^{6} x^{6}}-\frac {3 i \arctan \left (c x \right )^{2}}{5 c^{5} x^{5}}\right )+2 a \,d^{3} b \,c^{6} \left (-\frac {3 i \arctan \left (c x \right )}{5 c^{5} x^{5}}+\frac {3 \arctan \left (c x \right )}{4 c^{4} x^{4}}+\frac {i \arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{6 c^{6} x^{6}}+\frac {14 i \ln \left (c x \right )}{15}-\frac {3 i}{20 c^{4} x^{4}}+\frac {7 i}{15 c^{2} x^{2}}-\frac {1}{30 c^{5} x^{5}}+\frac {11}{36 c^{3} x^{3}}-\frac {11}{12 c x}-\frac {7 i \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {11 \arctan \left (c x \right )}{12}\right )\) | \(564\) |
derivativedivides | \(c^{6} \left (d^{3} a^{2} \left (-\frac {3 i}{5 c^{5} x^{5}}+\frac {3}{4 c^{4} x^{4}}+\frac {i}{3 c^{3} x^{3}}-\frac {1}{6 c^{6} x^{6}}\right )+b^{2} d^{3} \left (-\frac {\arctan \left (c x \right )}{15 c^{5} x^{5}}-\frac {113 \ln \left (c^{2} x^{2}+1\right )}{90}-\frac {1}{60 c^{4} x^{4}}-\frac {11 \arctan \left (c x \right )^{2}}{12}+\frac {113 \ln \left (c x \right )}{45}+\frac {61}{180 c^{2} x^{2}}+\frac {3 \arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {7 \ln \left (c x -i\right )^{2}}{30}-\frac {7 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{15}+\frac {7 \ln \left (c x +i\right )^{2}}{30}+\frac {7 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{15}-\frac {11 \arctan \left (c x \right )}{6 c x}+\frac {14 \operatorname {dilog}\left (-i c x +1\right )}{15}-\frac {14 \operatorname {dilog}\left (i c x +1\right )}{15}+\frac {7 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {7 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{15}-\frac {7 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{15}+\frac {37 i \arctan \left (c x \right )}{30}-\frac {14 \ln \left (c x \right ) \ln \left (i c x +1\right )}{15}+\frac {14 \ln \left (c x \right ) \ln \left (-i c x +1\right )}{15}+\frac {28 i \arctan \left (c x \right ) \ln \left (c x \right )}{15}+\frac {11 \arctan \left (c x \right )}{18 c^{3} x^{3}}+\frac {37 i}{30 c x}+\frac {14 i \arctan \left (c x \right )}{15 c^{2} x^{2}}-\frac {i}{10 c^{3} x^{3}}-\frac {3 i \arctan \left (c x \right )}{10 c^{4} x^{4}}-\frac {14 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {i \arctan \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\arctan \left (c x \right )^{2}}{6 c^{6} x^{6}}-\frac {3 i \arctan \left (c x \right )^{2}}{5 c^{5} x^{5}}\right )+2 a \,d^{3} b \left (-\frac {3 i \arctan \left (c x \right )}{5 c^{5} x^{5}}+\frac {3 \arctan \left (c x \right )}{4 c^{4} x^{4}}+\frac {i \arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{6 c^{6} x^{6}}+\frac {14 i \ln \left (c x \right )}{15}-\frac {3 i}{20 c^{4} x^{4}}+\frac {7 i}{15 c^{2} x^{2}}-\frac {1}{30 c^{5} x^{5}}+\frac {11}{36 c^{3} x^{3}}-\frac {11}{12 c x}-\frac {7 i \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {11 \arctan \left (c x \right )}{12}\right )\right )\) | \(567\) |
default | \(c^{6} \left (d^{3} a^{2} \left (-\frac {3 i}{5 c^{5} x^{5}}+\frac {3}{4 c^{4} x^{4}}+\frac {i}{3 c^{3} x^{3}}-\frac {1}{6 c^{6} x^{6}}\right )+b^{2} d^{3} \left (-\frac {\arctan \left (c x \right )}{15 c^{5} x^{5}}-\frac {113 \ln \left (c^{2} x^{2}+1\right )}{90}-\frac {1}{60 c^{4} x^{4}}-\frac {11 \arctan \left (c x \right )^{2}}{12}+\frac {113 \ln \left (c x \right )}{45}+\frac {61}{180 c^{2} x^{2}}+\frac {3 \arctan \left (c x \right )^{2}}{4 c^{4} x^{4}}-\frac {7 \ln \left (c x -i\right )^{2}}{30}-\frac {7 \operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )}{15}+\frac {7 \ln \left (c x +i\right )^{2}}{30}+\frac {7 \operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )}{15}-\frac {11 \arctan \left (c x \right )}{6 c x}+\frac {14 \operatorname {dilog}\left (-i c x +1\right )}{15}-\frac {14 \operatorname {dilog}\left (i c x +1\right )}{15}+\frac {7 \ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {7 \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{15}-\frac {7 \ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {7 \ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )}{15}+\frac {37 i \arctan \left (c x \right )}{30}-\frac {14 \ln \left (c x \right ) \ln \left (i c x +1\right )}{15}+\frac {14 \ln \left (c x \right ) \ln \left (-i c x +1\right )}{15}+\frac {28 i \arctan \left (c x \right ) \ln \left (c x \right )}{15}+\frac {11 \arctan \left (c x \right )}{18 c^{3} x^{3}}+\frac {37 i}{30 c x}+\frac {14 i \arctan \left (c x \right )}{15 c^{2} x^{2}}-\frac {i}{10 c^{3} x^{3}}-\frac {3 i \arctan \left (c x \right )}{10 c^{4} x^{4}}-\frac {14 i \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {i \arctan \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\arctan \left (c x \right )^{2}}{6 c^{6} x^{6}}-\frac {3 i \arctan \left (c x \right )^{2}}{5 c^{5} x^{5}}\right )+2 a \,d^{3} b \left (-\frac {3 i \arctan \left (c x \right )}{5 c^{5} x^{5}}+\frac {3 \arctan \left (c x \right )}{4 c^{4} x^{4}}+\frac {i \arctan \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arctan \left (c x \right )}{6 c^{6} x^{6}}+\frac {14 i \ln \left (c x \right )}{15}-\frac {3 i}{20 c^{4} x^{4}}+\frac {7 i}{15 c^{2} x^{2}}-\frac {1}{30 c^{5} x^{5}}+\frac {11}{36 c^{3} x^{3}}-\frac {11}{12 c x}-\frac {7 i \ln \left (c^{2} x^{2}+1\right )}{15}-\frac {11 \arctan \left (c x \right )}{12}\right )\right )\) | \(567\) |
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\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{7}} \,d x } \]
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Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{3} {\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{7}} \,d x } \]
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Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(d+i c d x)^3 (a+b \arctan (c x))^2}{x^7} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x^7} \,d x \]
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