Integrand size = 25, antiderivative size = 410 \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=-\frac {3 b (a+b \arctan (c x))^2}{2 c^3 d}+\frac {3 i b x (a+b \arctan (c x))^2}{2 c^2 d}+\frac {i (a+b \arctan (c x))^3}{2 c^3 d}+\frac {x (a+b \arctan (c x))^3}{c^2 d}-\frac {i x^2 (a+b \arctan (c x))^3}{2 c d}+\frac {3 i b^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}-\frac {i (a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}-\frac {3 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d}+\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )}{4 c^3 d} \]
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Time = 0.62 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {4986, 4946, 5036, 4930, 5040, 4964, 2449, 2352, 5004, 5114, 6745, 5118} \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\frac {3 i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c^3 d}-\frac {3 i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{2 c^3 d}+\frac {3 i b^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c^3 d}+\frac {3 b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))^2}{2 c^3 d}-\frac {3 b (a+b \arctan (c x))^2}{2 c^3 d}+\frac {i (a+b \arctan (c x))^3}{2 c^3 d}+\frac {3 b \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c^3 d}-\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^3}{c^3 d}+\frac {3 i b x (a+b \arctan (c x))^2}{2 c^2 d}+\frac {x (a+b \arctan (c x))^3}{c^2 d}-\frac {i x^2 (a+b \arctan (c x))^3}{2 c d}-\frac {3 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c^3 d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c^3 d}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{i c x+1}\right )}{4 c^3 d} \]
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Rule 2352
Rule 2449
Rule 4930
Rule 4946
Rule 4964
Rule 4986
Rule 5004
Rule 5036
Rule 5040
Rule 5114
Rule 5118
Rule 6745
Rubi steps \begin{align*} \text {integral}& = \frac {i \int \frac {x (a+b \arctan (c x))^3}{d+i c d x} \, dx}{c}-\frac {i \int x (a+b \arctan (c x))^3 \, dx}{c d} \\ & = -\frac {i x^2 (a+b \arctan (c x))^3}{2 c d}-\frac {\int \frac {(a+b \arctan (c x))^3}{d+i c d x} \, dx}{c^2}+\frac {(3 i b) \int \frac {x^2 (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{2 d}+\frac {\int (a+b \arctan (c x))^3 \, dx}{c^2 d} \\ & = \frac {x (a+b \arctan (c x))^3}{c^2 d}-\frac {i x^2 (a+b \arctan (c x))^3}{2 c d}-\frac {i (a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}+\frac {(3 i b) \int (a+b \arctan (c x))^2 \, dx}{2 c^2 d}-\frac {(3 i b) \int \frac {(a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{2 c^2 d}+\frac {(3 i b) \int \frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d}-\frac {(3 b) \int \frac {x (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{c d} \\ & = \frac {3 i b x (a+b \arctan (c x))^2}{2 c^2 d}+\frac {i (a+b \arctan (c x))^3}{2 c^3 d}+\frac {x (a+b \arctan (c x))^3}{c^2 d}-\frac {i x^2 (a+b \arctan (c x))^3}{2 c d}-\frac {i (a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d}+\frac {(3 b) \int \frac {(a+b \arctan (c x))^2}{i-c x} \, dx}{c^2 d}-\frac {\left (3 b^2\right ) \int \frac {(a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d}-\frac {\left (3 i b^2\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{c d} \\ & = -\frac {3 b (a+b \arctan (c x))^2}{2 c^3 d}+\frac {3 i b x (a+b \arctan (c x))^2}{2 c^2 d}+\frac {i (a+b \arctan (c x))^3}{2 c^3 d}+\frac {x (a+b \arctan (c x))^3}{c^2 d}-\frac {i x^2 (a+b \arctan (c x))^3}{2 c d}+\frac {3 b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}-\frac {i (a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d}+\frac {\left (3 i b^2\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{c^2 d}-\frac {\left (6 b^2\right ) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d}+\frac {\left (3 i b^3\right ) \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 c^2 d} \\ & = -\frac {3 b (a+b \arctan (c x))^2}{2 c^3 d}+\frac {3 i b x (a+b \arctan (c x))^2}{2 c^2 d}+\frac {i (a+b \arctan (c x))^3}{2 c^3 d}+\frac {x (a+b \arctan (c x))^3}{c^2 d}-\frac {i x^2 (a+b \arctan (c x))^3}{2 c d}+\frac {3 i b^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}-\frac {i (a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )}{4 c^3 d}-\frac {\left (3 i b^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d}-\frac {\left (3 i b^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d} \\ & = -\frac {3 b (a+b \arctan (c x))^2}{2 c^3 d}+\frac {3 i b x (a+b \arctan (c x))^2}{2 c^2 d}+\frac {i (a+b \arctan (c x))^3}{2 c^3 d}+\frac {x (a+b \arctan (c x))^3}{c^2 d}-\frac {i x^2 (a+b \arctan (c x))^3}{2 c d}+\frac {3 i b^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}-\frac {i (a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )}{4 c^3 d}-\frac {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^3 d} \\ & = -\frac {3 b (a+b \arctan (c x))^2}{2 c^3 d}+\frac {3 i b x (a+b \arctan (c x))^2}{2 c^2 d}+\frac {i (a+b \arctan (c x))^3}{2 c^3 d}+\frac {x (a+b \arctan (c x))^3}{c^2 d}-\frac {i x^2 (a+b \arctan (c x))^3}{2 c d}+\frac {3 i b^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}-\frac {i (a+b \arctan (c x))^3 \log \left (\frac {2}{1+i c x}\right )}{c^3 d}-\frac {3 b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d}+\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3 d}+\frac {3 b (a+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^3 d}+\frac {3 b^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 i b^2 (a+b \arctan (c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )}{4 c^3 d} \\ \end{align*}
Time = 1.03 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.32 \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=-\frac {i \left (4 i a^3 c x-6 a^2 b c x+2 a^3 c^2 x^2-4 i a^3 \arctan (c x)+6 a^2 b \arctan (c x)+12 i a^2 b c x \arctan (c x)-12 a b^2 c x \arctan (c x)+6 a^2 b c^2 x^2 \arctan (c x)-12 i a^2 b \arctan (c x)^2+18 a b^2 \arctan (c x)^2+6 i b^3 \arctan (c x)^2+12 i a b^2 c x \arctan (c x)^2-6 b^3 c x \arctan (c x)^2+6 a b^2 c^2 x^2 \arctan (c x)^2-8 i a b^2 \arctan (c x)^3+6 b^3 \arctan (c x)^3+4 i b^3 c x \arctan (c x)^3+2 b^3 c^2 x^2 \arctan (c x)^3-2 i b^3 \arctan (c x)^4+12 a^2 b \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+24 i a b^2 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-12 b^3 \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )+12 a b^2 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+12 i b^3 \arctan (c x)^2 \log \left (1+e^{2 i \arctan (c x)}\right )+4 b^3 \arctan (c x)^3 \log \left (1+e^{2 i \arctan (c x)}\right )-2 a^3 \log \left (1+c^2 x^2\right )-6 i a^2 b \log \left (1+c^2 x^2\right )+6 a b^2 \log \left (1+c^2 x^2\right )-6 i b (a+i b+b \arctan (c x))^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+6 b^2 (a+i b+b \arctan (c x)) \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c x)}\right )+3 i b^3 \operatorname {PolyLog}\left (4,-e^{2 i \arctan (c x)}\right )\right )}{4 c^3 d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 14.23 (sec) , antiderivative size = 1359, normalized size of antiderivative = 3.31
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1359\) |
default | \(\text {Expression too large to display}\) | \(1359\) |
parts | \(\text {Expression too large to display}\) | \(1406\) |
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\[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{2}}{i \, c d x + d} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{2}}{i \, c d x + d} \,d x } \]
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\[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{2}}{i \, c d x + d} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \arctan (c x))^3}{d+i c d x} \, dx=\int \frac {x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{d+c\,d\,x\,1{}\mathrm {i}} \,d x \]
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