Integrand size = 22, antiderivative size = 271 \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^3} \, dx=\frac {3 b^3}{64 c d^3 (i-c x)^2}-\frac {21 i b^3}{64 c d^3 (i-c x)}+\frac {21 i b^3 \arctan (c x)}{64 c d^3}+\frac {3 i b^2 (a+b \arctan (c x))}{16 c d^3 (i-c x)^2}+\frac {9 b^2 (a+b \arctan (c x))}{16 c d^3 (i-c x)}-\frac {9 b (a+b \arctan (c x))^2}{32 c d^3}-\frac {3 b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)^2}+\frac {3 i b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)}-\frac {i (a+b \arctan (c x))^3}{8 c d^3}+\frac {i (a+b \arctan (c x))^3}{2 c d^3 (1+i c x)^2} \]
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Time = 0.29 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4974, 4972, 641, 46, 209, 5004} \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^3} \, dx=\frac {9 b^2 (a+b \arctan (c x))}{16 c d^3 (-c x+i)}+\frac {3 i b^2 (a+b \arctan (c x))}{16 c d^3 (-c x+i)^2}+\frac {3 i b (a+b \arctan (c x))^2}{8 c d^3 (-c x+i)}-\frac {3 b (a+b \arctan (c x))^2}{8 c d^3 (-c x+i)^2}-\frac {9 b (a+b \arctan (c x))^2}{32 c d^3}+\frac {i (a+b \arctan (c x))^3}{2 c d^3 (1+i c x)^2}-\frac {i (a+b \arctan (c x))^3}{8 c d^3}+\frac {21 i b^3 \arctan (c x)}{64 c d^3}-\frac {21 i b^3}{64 c d^3 (-c x+i)}+\frac {3 b^3}{64 c d^3 (-c x+i)^2} \]
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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))^3}{2 c d^3 (1+i c x)^2}-\frac {(3 i b) \int \left (\frac {i (a+b \arctan (c x))^2}{2 d^2 (-i+c x)^3}-\frac {(a+b \arctan (c x))^2}{4 d^2 (-i+c x)^2}+\frac {(a+b \arctan (c x))^2}{4 d^2 \left (1+c^2 x^2\right )}\right ) \, dx}{2 d} \\ & = \frac {i (a+b \arctan (c x))^3}{2 c d^3 (1+i c x)^2}+\frac {(3 i b) \int \frac {(a+b \arctan (c x))^2}{(-i+c x)^2} \, dx}{8 d^3}-\frac {(3 i b) \int \frac {(a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{8 d^3}+\frac {(3 b) \int \frac {(a+b \arctan (c x))^2}{(-i+c x)^3} \, dx}{4 d^3} \\ & = -\frac {3 b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)^2}+\frac {3 i b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)}-\frac {i (a+b \arctan (c x))^3}{8 c d^3}+\frac {i (a+b \arctan (c x))^3}{2 c d^3 (1+i c x)^2}+\frac {\left (3 i b^2\right ) \int \left (-\frac {i (a+b \arctan (c x))}{2 (-i+c x)^2}+\frac {i (a+b \arctan (c x))}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{4 d^3}+\frac {\left (3 b^2\right ) \int \left (-\frac {i (a+b \arctan (c x))}{2 (-i+c x)^3}+\frac {a+b \arctan (c x)}{4 (-i+c x)^2}-\frac {a+b \arctan (c x)}{4 \left (1+c^2 x^2\right )}\right ) \, dx}{4 d^3} \\ & = -\frac {3 b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)^2}+\frac {3 i b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)}-\frac {i (a+b \arctan (c x))^3}{8 c d^3}+\frac {i (a+b \arctan (c x))^3}{2 c d^3 (1+i c x)^2}-\frac {\left (3 i b^2\right ) \int \frac {a+b \arctan (c x)}{(-i+c x)^3} \, dx}{8 d^3}+\frac {\left (3 b^2\right ) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{16 d^3}-\frac {\left (3 b^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{16 d^3}+\frac {\left (3 b^2\right ) \int \frac {a+b \arctan (c x)}{(-i+c x)^2} \, dx}{8 d^3}-\frac {\left (3 b^2\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{8 d^3} \\ & = \frac {3 i b^2 (a+b \arctan (c x))}{16 c d^3 (i-c x)^2}+\frac {9 b^2 (a+b \arctan (c x))}{16 c d^3 (i-c x)}-\frac {9 b (a+b \arctan (c x))^2}{32 c d^3}-\frac {3 b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)^2}+\frac {3 i b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)}-\frac {i (a+b \arctan (c x))^3}{8 c d^3}+\frac {i (a+b \arctan (c x))^3}{2 c d^3 (1+i c x)^2}-\frac {\left (3 i b^3\right ) \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{16 d^3}+\frac {\left (3 b^3\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{16 d^3}+\frac {\left (3 b^3\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{8 d^3} \\ & = \frac {3 i b^2 (a+b \arctan (c x))}{16 c d^3 (i-c x)^2}+\frac {9 b^2 (a+b \arctan (c x))}{16 c d^3 (i-c x)}-\frac {9 b (a+b \arctan (c x))^2}{32 c d^3}-\frac {3 b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)^2}+\frac {3 i b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)}-\frac {i (a+b \arctan (c x))^3}{8 c d^3}+\frac {i (a+b \arctan (c x))^3}{2 c d^3 (1+i c x)^2}-\frac {\left (3 i b^3\right ) \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{16 d^3}+\frac {\left (3 b^3\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{16 d^3}+\frac {\left (3 b^3\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{8 d^3} \\ & = \frac {3 i b^2 (a+b \arctan (c x))}{16 c d^3 (i-c x)^2}+\frac {9 b^2 (a+b \arctan (c x))}{16 c d^3 (i-c x)}-\frac {9 b (a+b \arctan (c x))^2}{32 c d^3}-\frac {3 b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)^2}+\frac {3 i b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)}-\frac {i (a+b \arctan (c x))^3}{8 c d^3}+\frac {i (a+b \arctan (c x))^3}{2 c d^3 (1+i c x)^2}-\frac {\left (3 i b^3\right ) \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}{16 d^3}+\frac {\left (3 b^3\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{16 d^3}+\frac {\left (3 b^3\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{8 d^3} \\ & = \frac {3 b^3}{64 c d^3 (i-c x)^2}-\frac {21 i b^3}{64 c d^3 (i-c x)}+\frac {3 i b^2 (a+b \arctan (c x))}{16 c d^3 (i-c x)^2}+\frac {9 b^2 (a+b \arctan (c x))}{16 c d^3 (i-c x)}-\frac {9 b (a+b \arctan (c x))^2}{32 c d^3}-\frac {3 b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)^2}+\frac {3 i b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)}-\frac {i (a+b \arctan (c x))^3}{8 c d^3}+\frac {i (a+b \arctan (c x))^3}{2 c d^3 (1+i c x)^2}+\frac {\left (3 i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{64 d^3}+\frac {\left (3 i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{32 d^3}+\frac {\left (3 i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{16 d^3} \\ & = \frac {3 b^3}{64 c d^3 (i-c x)^2}-\frac {21 i b^3}{64 c d^3 (i-c x)}+\frac {21 i b^3 \arctan (c x)}{64 c d^3}+\frac {3 i b^2 (a+b \arctan (c x))}{16 c d^3 (i-c x)^2}+\frac {9 b^2 (a+b \arctan (c x))}{16 c d^3 (i-c x)}-\frac {9 b (a+b \arctan (c x))^2}{32 c d^3}-\frac {3 b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)^2}+\frac {3 i b (a+b \arctan (c x))^2}{8 c d^3 (i-c x)}-\frac {i (a+b \arctan (c x))^3}{8 c d^3}+\frac {i (a+b \arctan (c x))^3}{2 c d^3 (1+i c x)^2} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.68 \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^3} \, dx=-\frac {i \left (32 a^3+3 b^3 (8 i-7 c x)+12 a b^2 (-4-3 i c x)+24 a^2 b (-2 i+c x)+3 b (i+c x) \left (b^2 (9 i-7 c x)+4 a b (-5-3 i c x)+8 a^2 (-3 i+c x)\right ) \arctan (c x)+6 b^2 (i+c x) (b (-5-3 i c x)+4 a (-3 i+c x)) \arctan (c x)^2+8 b^3 \left (3-2 i c x+c^2 x^2\right ) \arctan (c x)^3\right )}{64 c d^3 (-i+c x)^2} \]
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Time = 1.08 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.57
method | result | size |
derivativedivides | \(\frac {\frac {i a^{3}}{2 d^{3} \left (i c x +1\right )^{2}}+\frac {b^{3} \left (\frac {i \arctan \left (c x \right )^{3}}{2 \left (i c x +1\right )^{2}}-\frac {i \left (-21 c x +24 i-16 i \arctan \left (c x \right )^{3} c x +8 \arctan \left (c x \right )^{3} c^{2} x^{2}-8 \arctan \left (c x \right )^{3}-18 i \arctan \left (c x \right )^{2} c^{2} x^{2}-30 i \arctan \left (c x \right )^{2}-12 \arctan \left (c x \right )^{2} c x +6 i \arctan \left (c x \right ) c x -21 c^{2} x^{2} \arctan \left (c x \right )-27 \arctan \left (c x \right )\right )}{64 \left (c x -i\right )^{2}}\right )}{d^{3}}+\frac {3 a \,b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{2 \left (i c x +1\right )^{2}}-i \left (-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{8}-\frac {i \arctan \left (c x \right )}{4 \left (c x -i\right )^{2}}+\frac {\arctan \left (c x \right )}{4 c x -4 i}+\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{8}+\frac {\ln \left (c x +i\right )^{2}}{32}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{16}-\frac {3 i}{16 \left (c x -i\right )}-\frac {1}{16 \left (c x -i\right )^{2}}-\frac {3 i \arctan \left (c x \right )}{16}+\frac {\ln \left (c x -i\right )^{2}}{32}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{16}\right )\right )}{d^{3}}+\frac {3 i a^{2} b \arctan \left (c x \right )}{2 d^{3} \left (i c x +1\right )^{2}}-\frac {3 i a^{2} b \arctan \left (c x \right )}{8 d^{3}}-\frac {3 a^{2} b}{8 d^{3} \left (c x -i\right )^{2}}-\frac {3 i a^{2} b}{8 d^{3} \left (c x -i\right )}}{c}\) | \(425\) |
default | \(\frac {\frac {i a^{3}}{2 d^{3} \left (i c x +1\right )^{2}}+\frac {b^{3} \left (\frac {i \arctan \left (c x \right )^{3}}{2 \left (i c x +1\right )^{2}}-\frac {i \left (-21 c x +24 i-16 i \arctan \left (c x \right )^{3} c x +8 \arctan \left (c x \right )^{3} c^{2} x^{2}-8 \arctan \left (c x \right )^{3}-18 i \arctan \left (c x \right )^{2} c^{2} x^{2}-30 i \arctan \left (c x \right )^{2}-12 \arctan \left (c x \right )^{2} c x +6 i \arctan \left (c x \right ) c x -21 c^{2} x^{2} \arctan \left (c x \right )-27 \arctan \left (c x \right )\right )}{64 \left (c x -i\right )^{2}}\right )}{d^{3}}+\frac {3 a \,b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{2 \left (i c x +1\right )^{2}}-i \left (-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{8}-\frac {i \arctan \left (c x \right )}{4 \left (c x -i\right )^{2}}+\frac {\arctan \left (c x \right )}{4 c x -4 i}+\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{8}+\frac {\ln \left (c x +i\right )^{2}}{32}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{16}-\frac {3 i}{16 \left (c x -i\right )}-\frac {1}{16 \left (c x -i\right )^{2}}-\frac {3 i \arctan \left (c x \right )}{16}+\frac {\ln \left (c x -i\right )^{2}}{32}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{16}\right )\right )}{d^{3}}+\frac {3 i a^{2} b \arctan \left (c x \right )}{2 d^{3} \left (i c x +1\right )^{2}}-\frac {3 i a^{2} b \arctan \left (c x \right )}{8 d^{3}}-\frac {3 a^{2} b}{8 d^{3} \left (c x -i\right )^{2}}-\frac {3 i a^{2} b}{8 d^{3} \left (c x -i\right )}}{c}\) | \(425\) |
parts | \(\frac {i a^{3}}{2 d^{3} \left (i c x +1\right )^{2} c}+\frac {b^{3} \left (\frac {i \arctan \left (c x \right )^{3}}{2 \left (i c x +1\right )^{2}}-\frac {i \left (-21 c x +24 i-16 i \arctan \left (c x \right )^{3} c x +8 \arctan \left (c x \right )^{3} c^{2} x^{2}-8 \arctan \left (c x \right )^{3}-18 i \arctan \left (c x \right )^{2} c^{2} x^{2}-30 i \arctan \left (c x \right )^{2}-12 \arctan \left (c x \right )^{2} c x +6 i \arctan \left (c x \right ) c x -21 c^{2} x^{2} \arctan \left (c x \right )-27 \arctan \left (c x \right )\right )}{64 \left (c x -i\right )^{2}}\right )}{d^{3} c}+\frac {3 a \,b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{2 \left (i c x +1\right )^{2}}-i \left (-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{8}-\frac {i \arctan \left (c x \right )}{4 \left (c x -i\right )^{2}}+\frac {\arctan \left (c x \right )}{4 c x -4 i}+\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{8}+\frac {\ln \left (c x +i\right )^{2}}{32}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{16}-\frac {3 i}{16 \left (c x -i\right )}-\frac {1}{16 \left (c x -i\right )^{2}}-\frac {3 i \arctan \left (c x \right )}{16}+\frac {\ln \left (c x -i\right )^{2}}{32}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{16}\right )\right )}{d^{3} c}+\frac {3 i a^{2} b \arctan \left (c x \right )}{2 d^{3} c \left (i c x +1\right )^{2}}-\frac {3 i a^{2} b \arctan \left (c x \right )}{8 d^{3} c}-\frac {3 a^{2} b}{8 d^{3} c \left (c x -i\right )^{2}}-\frac {3 i a^{2} b}{8 d^{3} c \left (c x -i\right )}\) | \(442\) |
risch | \(\text {Expression too large to display}\) | \(1562\) |
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Time = 0.25 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^3} \, dx=-\frac {2 \, {\left (b^{3} c^{2} x^{2} - 2 i \, b^{3} c x + 3 \, b^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{3} + 64 i \, a^{3} + 96 \, a^{2} b - 96 i \, a b^{2} - 48 \, b^{3} + 6 \, {\left (8 i \, a^{2} b + 12 \, a b^{2} - 7 i \, b^{3}\right )} c x + 3 \, {\left ({\left (-4 i \, a b^{2} - 3 \, b^{3}\right )} c^{2} x^{2} - 12 i \, a b^{2} - 5 \, b^{3} - 2 \, {\left (4 \, a b^{2} - i \, b^{3}\right )} c x\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} - 3 \, {\left ({\left (8 \, a^{2} b - 12 i \, a b^{2} - 7 \, b^{3}\right )} c^{2} x^{2} + 24 \, a^{2} b - 20 i \, a b^{2} - 9 \, b^{3} - 2 \, {\left (8 i \, a^{2} b + 4 \, a b^{2} - i \, b^{3}\right )} c x\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{128 \, {\left (c^{3} d^{3} x^{2} - 2 i \, c^{2} d^{3} x - c d^{3}\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 954 vs. \(2 (230) = 460\).
Time = 119.50 (sec) , antiderivative size = 954, normalized size of antiderivative = 3.52 \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^3} \, dx=- \frac {3 b \left (8 a^{2} - 12 i a b - 7 b^{2}\right ) \log {\left (- \frac {3 i b \left (8 a^{2} - 12 i a b - 7 b^{2}\right )}{c} + x \left (24 a^{2} b - 36 i a b^{2} - 21 b^{3}\right ) \right )}}{128 c d^{3}} + \frac {3 b \left (8 a^{2} - 12 i a b - 7 b^{2}\right ) \log {\left (\frac {3 i b \left (8 a^{2} - 12 i a b - 7 b^{2}\right )}{c} + x \left (24 a^{2} b - 36 i a b^{2} - 21 b^{3}\right ) \right )}}{128 c d^{3}} + \frac {\left (- b^{3} c^{2} x^{2} + 2 i b^{3} c x - 3 b^{3}\right ) \log {\left (- i c x + 1 \right )}^{3}}{64 c^{3} d^{3} x^{2} - 128 i c^{2} d^{3} x - 64 c d^{3}} + \frac {\left (b^{3} c^{2} x^{2} - 2 i b^{3} c x + 3 b^{3}\right ) \log {\left (i c x + 1 \right )}^{3}}{64 c^{3} d^{3} x^{2} - 128 i c^{2} d^{3} x - 64 c d^{3}} + \frac {\left (12 i a b^{2} c^{2} x^{2} + 24 a b^{2} c x + 36 i a b^{2} + 9 b^{3} c^{2} x^{2} - 6 i b^{3} c x + 15 b^{3}\right ) \log {\left (i c x + 1 \right )}^{2}}{128 c^{3} d^{3} x^{2} - 256 i c^{2} d^{3} x - 128 c d^{3}} + \frac {\left (12 i a b^{2} c^{2} x^{2} + 24 a b^{2} c x + 36 i a b^{2} + 6 b^{3} c^{2} x^{2} \log {\left (i c x + 1 \right )} + 9 b^{3} c^{2} x^{2} - 12 i b^{3} c x \log {\left (i c x + 1 \right )} - 6 i b^{3} c x + 18 b^{3} \log {\left (i c x + 1 \right )} + 15 b^{3}\right ) \log {\left (- i c x + 1 \right )}^{2}}{128 c^{3} d^{3} x^{2} - 256 i c^{2} d^{3} x - 128 c d^{3}} + \frac {- 32 i a^{3} - 48 a^{2} b + 48 i a b^{2} + 24 b^{3} + x \left (- 24 i a^{2} b c - 36 a b^{2} c + 21 i b^{3} c\right )}{64 c^{3} d^{3} x^{2} - 128 i c^{2} d^{3} x - 64 c d^{3}} + \frac {\left (48 a^{2} b - 12 i a b^{2} c^{2} x^{2} \log {\left (i c x + 1 \right )} - 24 a b^{2} c x \log {\left (i c x + 1 \right )} + 24 a b^{2} c x - 36 i a b^{2} \log {\left (i c x + 1 \right )} - 48 i a b^{2} - 3 b^{3} c^{2} x^{2} \log {\left (i c x + 1 \right )}^{2} - 9 b^{3} c^{2} x^{2} \log {\left (i c x + 1 \right )} + 6 i b^{3} c x \log {\left (i c x + 1 \right )}^{2} + 6 i b^{3} c x \log {\left (i c x + 1 \right )} - 18 i b^{3} c x - 9 b^{3} \log {\left (i c x + 1 \right )}^{2} - 15 b^{3} \log {\left (i c x + 1 \right )} - 24 b^{3}\right ) \log {\left (- i c x + 1 \right )}}{64 c^{3} d^{3} x^{2} - 128 i c^{2} d^{3} x - 64 c d^{3}} + \frac {\left (- 24 a^{2} b - 12 a b^{2} c x + 24 i a b^{2} + 9 i b^{3} c x + 12 b^{3}\right ) \log {\left (i c x + 1 \right )}}{32 c^{3} d^{3} x^{2} - 64 i c^{2} d^{3} x - 32 c d^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^3} \, dx=\frac {8 \, {\left (-i \, b^{3} c^{2} x^{2} - 2 \, b^{3} c x - 3 i \, b^{3}\right )} \arctan \left (c x\right )^{3} - 32 i \, a^{3} - 48 \, a^{2} b + 48 i \, a b^{2} + 24 \, b^{3} + 3 \, {\left (-8 i \, a^{2} b - 12 \, a b^{2} + 7 i \, b^{3}\right )} c x + 6 \, {\left ({\left (-4 i \, a b^{2} - 3 \, b^{3}\right )} c^{2} x^{2} - 12 i \, a b^{2} - 5 \, b^{3} - 2 \, {\left (4 \, a b^{2} - i \, b^{3}\right )} c x\right )} \arctan \left (c x\right )^{2} + 3 \, {\left ({\left (-8 i \, a^{2} b - 12 \, a b^{2} + 7 i \, b^{3}\right )} c^{2} x^{2} - 24 i \, a^{2} b - 20 \, a b^{2} + 9 i \, b^{3} - 2 \, {\left (8 \, a^{2} b - 4 i \, a b^{2} - b^{3}\right )} c x\right )} \arctan \left (c x\right )}{64 \, {\left (c^{3} d^{3} x^{2} - 2 i \, c^{2} d^{3} x - c d^{3}\right )}} \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^3}{(d+i c d x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
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