Integrand size = 21, antiderivative size = 207 \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=-\frac {b^2}{54 c (i-c x)^3}+\frac {5 i b^2}{144 c (i-c x)^2}+\frac {11 b^2}{144 c (i-c x)}-\frac {11 b^2 \arctan (c x)}{144 c}-\frac {i b (a+b \arctan (c x))}{9 c (i-c x)^3}-\frac {b (a+b \arctan (c x))}{12 c (i-c x)^2}+\frac {i b (a+b \arctan (c x))}{12 c (i-c x)}-\frac {i (a+b \arctan (c x))^2}{24 c}+\frac {i (a+b \arctan (c x))^2}{3 c (1+i c x)^3} \]
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Time = 0.17 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4974, 4972, 641, 46, 209, 5004} \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=\frac {i b (a+b \arctan (c x))}{12 c (-c x+i)}-\frac {b (a+b \arctan (c x))}{12 c (-c x+i)^2}-\frac {i b (a+b \arctan (c x))}{9 c (-c x+i)^3}-\frac {i (a+b \arctan (c x))^2}{24 c}+\frac {i (a+b \arctan (c x))^2}{3 c (1+i c x)^3}-\frac {11 b^2 \arctan (c x)}{144 c}+\frac {11 b^2}{144 c (-c x+i)}+\frac {5 i b^2}{144 c (-c x+i)^2}-\frac {b^2}{54 c (-c x+i)^3} \]
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Rule 46
Rule 209
Rule 641
Rule 4972
Rule 4974
Rule 5004
Rubi steps \begin{align*} \text {integral}& = \frac {i (a+b \arctan (c x))^2}{3 c (1+i c x)^3}-\frac {1}{3} (2 i b) \int \left (\frac {a+b \arctan (c x)}{2 (-i+c x)^4}+\frac {i (a+b \arctan (c x))}{4 (-i+c x)^3}-\frac {a+b \arctan (c x)}{8 (-i+c x)^2}+\frac {a+b \arctan (c x)}{8 \left (1+c^2 x^2\right )}\right ) \, dx \\ & = \frac {i (a+b \arctan (c x))^2}{3 c (1+i c x)^3}+\frac {1}{12} (i b) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx-\frac {1}{12} (i b) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx-\frac {1}{3} (i b) \int \frac {a+b \arctan (c x)}{(-i+c x)^4} \, dx+\frac {1}{6} b \int \frac {a+b \arctan (c x)}{(-i+c x)^3} \, dx \\ & = -\frac {i b (a+b \arctan (c x))}{9 c (i-c x)^3}-\frac {b (a+b \arctan (c x))}{12 c (i-c x)^2}+\frac {i b (a+b \arctan (c x))}{12 c (i-c x)}-\frac {i (a+b \arctan (c x))^2}{24 c}+\frac {i (a+b \arctan (c x))^2}{3 c (1+i c x)^3}+\frac {1}{12} \left (i b^2\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx-\frac {1}{9} \left (i b^2\right ) \int \frac {1}{(-i+c x)^3 \left (1+c^2 x^2\right )} \, dx+\frac {1}{12} b^2 \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {i b (a+b \arctan (c x))}{9 c (i-c x)^3}-\frac {b (a+b \arctan (c x))}{12 c (i-c x)^2}+\frac {i b (a+b \arctan (c x))}{12 c (i-c x)}-\frac {i (a+b \arctan (c x))^2}{24 c}+\frac {i (a+b \arctan (c x))^2}{3 c (1+i c x)^3}+\frac {1}{12} \left (i b^2\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx-\frac {1}{9} \left (i b^2\right ) \int \frac {1}{(-i+c x)^4 (i+c x)} \, dx+\frac {1}{12} b^2 \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx \\ & = -\frac {i b (a+b \arctan (c x))}{9 c (i-c x)^3}-\frac {b (a+b \arctan (c x))}{12 c (i-c x)^2}+\frac {i b (a+b \arctan (c x))}{12 c (i-c x)}-\frac {i (a+b \arctan (c x))^2}{24 c}+\frac {i (a+b \arctan (c x))^2}{3 c (1+i c x)^3}+\frac {1}{12} \left (i b^2\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx-\frac {1}{9} \left (i b^2\right ) \int \left (-\frac {i}{2 (-i+c x)^4}+\frac {1}{4 (-i+c x)^3}+\frac {i}{8 (-i+c x)^2}-\frac {i}{8 \left (1+c^2 x^2\right )}\right ) \, dx+\frac {1}{12} b^2 \int \left (-\frac {i}{2 (-i+c x)^3}+\frac {1}{4 (-i+c x)^2}-\frac {1}{4 \left (1+c^2 x^2\right )}\right ) \, dx \\ & = -\frac {b^2}{54 c (i-c x)^3}+\frac {5 i b^2}{144 c (i-c x)^2}+\frac {11 b^2}{144 c (i-c x)}-\frac {i b (a+b \arctan (c x))}{9 c (i-c x)^3}-\frac {b (a+b \arctan (c x))}{12 c (i-c x)^2}+\frac {i b (a+b \arctan (c x))}{12 c (i-c x)}-\frac {i (a+b \arctan (c x))^2}{24 c}+\frac {i (a+b \arctan (c x))^2}{3 c (1+i c x)^3}-\frac {1}{72} b^2 \int \frac {1}{1+c^2 x^2} \, dx-\frac {1}{48} b^2 \int \frac {1}{1+c^2 x^2} \, dx-\frac {1}{24} b^2 \int \frac {1}{1+c^2 x^2} \, dx \\ & = -\frac {b^2}{54 c (i-c x)^3}+\frac {5 i b^2}{144 c (i-c x)^2}+\frac {11 b^2}{144 c (i-c x)}-\frac {11 b^2 \arctan (c x)}{144 c}-\frac {i b (a+b \arctan (c x))}{9 c (i-c x)^3}-\frac {b (a+b \arctan (c x))}{12 c (i-c x)^2}+\frac {i b (a+b \arctan (c x))}{12 c (i-c x)}-\frac {i (a+b \arctan (c x))^2}{24 c}+\frac {i (a+b \arctan (c x))^2}{3 c (1+i c x)^3} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=-\frac {144 a^2+12 a b \left (-10 i+9 c x+3 i c^2 x^2\right )+b^2 \left (-56-81 i c x+33 c^2 x^2\right )+3 b (i+c x) \left (12 a \left (-7 i+4 c x+i c^2 x^2\right )+b \left (-29-32 i c x+11 c^2 x^2\right )\right ) \arctan (c x)+18 b^2 \left (7-3 i c x+3 c^2 x^2+i c^3 x^3\right ) \arctan (c x)^2}{432 c (-i+c x)^3} \]
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Time = 2.41 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.43
method | result | size |
derivativedivides | \(\frac {\frac {i a^{2}}{3 \left (i c x +1\right )^{3}}+b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{3 \left (i c x +1\right )^{3}}-\frac {2 i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{16}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{16}-\frac {i \arctan \left (c x \right )}{8 \left (c x -i\right )^{2}}-\frac {\arctan \left (c x \right )}{6 \left (c x -i\right )^{3}}+\frac {\arctan \left (c x \right )}{8 c x -8 i}-\frac {11 i \arctan \left (c x \right )}{96}+\frac {i}{36 \left (c x -i\right )^{3}}-\frac {11 i}{96 \left (c x -i\right )}-\frac {5}{96 \left (c x -i\right )^{2}}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{32}+\frac {\ln \left (c x -i\right )^{2}}{64}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{32}+\frac {\ln \left (c x +i\right )^{2}}{64}\right )}{3}\right )+\frac {2 i a b \arctan \left (c x \right )}{3 \left (i c x +1\right )^{3}}-\frac {i a b \arctan \left (c x \right )}{12}-\frac {a b}{12 \left (c x -i\right )^{2}}+\frac {i a b}{9 \left (c x -i\right )^{3}}-\frac {i a b}{12 \left (c x -i\right )}}{c}\) | \(297\) |
default | \(\frac {\frac {i a^{2}}{3 \left (i c x +1\right )^{3}}+b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{3 \left (i c x +1\right )^{3}}-\frac {2 i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{16}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{16}-\frac {i \arctan \left (c x \right )}{8 \left (c x -i\right )^{2}}-\frac {\arctan \left (c x \right )}{6 \left (c x -i\right )^{3}}+\frac {\arctan \left (c x \right )}{8 c x -8 i}-\frac {11 i \arctan \left (c x \right )}{96}+\frac {i}{36 \left (c x -i\right )^{3}}-\frac {11 i}{96 \left (c x -i\right )}-\frac {5}{96 \left (c x -i\right )^{2}}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{32}+\frac {\ln \left (c x -i\right )^{2}}{64}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{32}+\frac {\ln \left (c x +i\right )^{2}}{64}\right )}{3}\right )+\frac {2 i a b \arctan \left (c x \right )}{3 \left (i c x +1\right )^{3}}-\frac {i a b \arctan \left (c x \right )}{12}-\frac {a b}{12 \left (c x -i\right )^{2}}+\frac {i a b}{9 \left (c x -i\right )^{3}}-\frac {i a b}{12 \left (c x -i\right )}}{c}\) | \(297\) |
parts | \(\frac {i a^{2}}{3 \left (i c x +1\right )^{3} c}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{3 \left (i c x +1\right )^{3}}-\frac {2 i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{16}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{16}-\frac {i \arctan \left (c x \right )}{8 \left (c x -i\right )^{2}}-\frac {\arctan \left (c x \right )}{6 \left (c x -i\right )^{3}}+\frac {\arctan \left (c x \right )}{8 c x -8 i}-\frac {11 i \arctan \left (c x \right )}{96}+\frac {i}{36 \left (c x -i\right )^{3}}-\frac {11 i}{96 \left (c x -i\right )}-\frac {5}{96 \left (c x -i\right )^{2}}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{32}+\frac {\ln \left (c x -i\right )^{2}}{64}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{32}+\frac {\ln \left (c x +i\right )^{2}}{64}\right )}{3}\right )}{c}+\frac {2 i a b \arctan \left (c x \right )}{3 c \left (i c x +1\right )^{3}}-\frac {i a b \arctan \left (c x \right )}{12 c}-\frac {a b}{12 c \left (c x -i\right )^{2}}+\frac {i a b}{9 c \left (c x -i\right )^{3}}-\frac {i a b}{12 c \left (c x -i\right )}\) | \(314\) |
risch | \(\frac {i b^{2} \left (c^{3} x^{3}-3 i c^{2} x^{2}-3 c x -7 i\right ) \ln \left (i c x +1\right )^{2}}{96 \left (c x -i\right )^{3} c}+\frac {i b \left (-3 b \,c^{3} x^{3} \ln \left (-i c x +1\right )+9 i b \,x^{2} \ln \left (-i c x +1\right ) c^{2}+6 i b \,c^{2} x^{2}+9 b c x \ln \left (-i c x +1\right )+21 i b \ln \left (-i c x +1\right )+18 x b c -20 i b +48 a \right ) \ln \left (i c x +1\right )}{144 \left (c x -i\right )^{3} c}-\frac {288 a^{2}+66 b^{2} c^{2} x^{2}+288 i \ln \left (-i c x +1\right ) a b -112 b^{2}+216 a b c x +120 b^{2} \ln \left (-i c x +1\right )+27 i b^{2} c x \ln \left (-i c x +1\right )^{2}-240 i a b -63 b^{2} \ln \left (-i c x +1\right )^{2}-162 i b^{2} c x -36 b^{2} \ln \left (-i c x +1\right ) c^{2} x^{2}+108 i \ln \left (-i c x +1\right ) b^{2} c x -27 b^{2} c^{2} x^{2} \ln \left (-i c x +1\right )^{2}+99 \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) b^{2} c^{2} x^{2}-99 \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) b^{2} c^{2} x^{2}-36 i \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) a b +36 i \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) a b -108 i \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) a b \,c^{2} x^{2}+108 i \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) a b \,c^{2} x^{2}-33 \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) b^{2}+33 \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) b^{2}-9 i c^{3} b^{2} x^{3} \ln \left (-i c x +1\right )^{2}+33 i \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) b^{2} c^{3} x^{3}-33 i \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) b^{2} c^{3} x^{3}-99 i \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) b^{2} c x +99 i \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) b^{2} c x -36 \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) a b \,c^{3} x^{3}+36 \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) a b \,c^{3} x^{3}+108 \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) a b c x -108 \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) a b c x +72 i a b \,c^{2} x^{2}}{864 \left (c x -i\right )^{3} c}\) | \(832\) |
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Time = 0.25 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=-\frac {6 \, {\left (12 i \, a b + 11 \, b^{2}\right )} c^{2} x^{2} + 54 \, {\left (4 \, a b - 3 i \, b^{2}\right )} c x + 9 \, {\left (-i \, b^{2} c^{3} x^{3} - 3 \, b^{2} c^{2} x^{2} + 3 i \, b^{2} c x - 7 \, b^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + 288 \, a^{2} - 240 i \, a b - 112 \, b^{2} - 3 \, {\left ({\left (12 \, a b - 11 i \, b^{2}\right )} c^{3} x^{3} - 3 \, {\left (12 i \, a b + 7 \, b^{2}\right )} c^{2} x^{2} - 3 \, {\left (12 \, a b + i \, b^{2}\right )} c x - 84 i \, a b - 29 \, b^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{864 \, {\left (c^{4} x^{3} - 3 i \, c^{3} x^{2} - 3 \, c^{2} x + i \, c\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (158) = 316\).
Time = 21.42 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.67 \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=- \frac {b \left (12 a - 11 i b\right ) \log {\left (- \frac {i b \left (12 a - 11 i b\right )}{c} + x \left (12 a b - 11 i b^{2}\right ) \right )}}{288 c} + \frac {b \left (12 a - 11 i b\right ) \log {\left (\frac {i b \left (12 a - 11 i b\right )}{c} + x \left (12 a b - 11 i b^{2}\right ) \right )}}{288 c} + \frac {- 144 a^{2} + 120 i a b + 56 b^{2} + x^{2} \left (- 36 i a b c^{2} - 33 b^{2} c^{2}\right ) + x \left (- 108 a b c + 81 i b^{2} c\right )}{432 c^{4} x^{3} - 1296 i c^{3} x^{2} - 1296 c^{2} x + 432 i c} + \frac {\left (- 48 i a b - 3 i b^{2} c^{3} x^{3} \log {\left (i c x + 1 \right )} - 9 b^{2} c^{2} x^{2} \log {\left (i c x + 1 \right )} + 6 b^{2} c^{2} x^{2} + 9 i b^{2} c x \log {\left (i c x + 1 \right )} - 18 i b^{2} c x - 21 b^{2} \log {\left (i c x + 1 \right )} - 20 b^{2}\right ) \log {\left (- i c x + 1 \right )}}{144 c^{4} x^{3} - 432 i c^{3} x^{2} - 432 c^{2} x + 144 i c} + \frac {\left (i b^{2} c^{3} x^{3} + 3 b^{2} c^{2} x^{2} - 3 i b^{2} c x + 7 b^{2}\right ) \log {\left (- i c x + 1 \right )}^{2}}{96 c^{4} x^{3} - 288 i c^{3} x^{2} - 288 c^{2} x + 96 i c} + \frac {\left (i b^{2} c^{3} x^{3} + 3 b^{2} c^{2} x^{2} - 3 i b^{2} c x + 7 b^{2}\right ) \log {\left (i c x + 1 \right )}^{2}}{96 c^{4} x^{3} - 288 i c^{3} x^{2} - 288 c^{2} x + 96 i c} + \frac {\left (24 i a b - 3 b^{2} c^{2} x^{2} + 9 i b^{2} c x + 10 b^{2}\right ) \log {\left (i c x + 1 \right )}}{72 c^{4} x^{3} - 216 i c^{3} x^{2} - 216 c^{2} x + 72 i c} \]
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Time = 0.25 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=\frac {3 \, {\left (-12 i \, a b - 11 \, b^{2}\right )} c^{2} x^{2} - 27 \, {\left (4 \, a b - 3 i \, b^{2}\right )} c x + 18 \, {\left (-i \, b^{2} c^{3} x^{3} - 3 \, b^{2} c^{2} x^{2} + 3 i \, b^{2} c x - 7 \, b^{2}\right )} \arctan \left (c x\right )^{2} - 144 \, a^{2} + 120 i \, a b + 56 \, b^{2} + 3 \, {\left ({\left (-12 i \, a b - 11 \, b^{2}\right )} c^{3} x^{3} - 3 \, {\left (12 \, a b - 7 i \, b^{2}\right )} c^{2} x^{2} + 3 \, {\left (12 i \, a b - b^{2}\right )} c x - 84 \, a b + 29 i \, b^{2}\right )} \arctan \left (c x\right )}{432 \, {\left (c^{4} x^{3} - 3 i \, c^{3} x^{2} - 3 \, c^{2} x + i \, c\right )}} \]
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\[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c x + 1\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (1+c\,x\,1{}\mathrm {i}\right )}^4} \,d x \]
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