Integrand size = 22, antiderivative size = 98 \[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\frac {i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c d}-\frac {b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c d}+\frac {i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c d} \]
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Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4964, 5004, 5114, 6745} \[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) (a+b \arctan (c x))}{c d}+\frac {i \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))^2}{c d}+\frac {i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i c x+1}\right )}{2 c d} \]
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Rule 4964
Rule 5004
Rule 5114
Rule 6745
Rubi steps \begin{align*} \text {integral}& = \frac {i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c d}-\frac {(2 i b) \int \frac {(a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d} \\ & = \frac {i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c d}-\frac {b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c d}+\frac {b^2 \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d} \\ & = \frac {i (a+b \arctan (c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c d}-\frac {b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c d}+\frac {i b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\frac {i \left (2 (a+b \arctan (c x))^2 \log \left (\frac {2 d}{d+i c d x}\right )+2 i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,\frac {i+c x}{-i+c x}\right )+b^2 \operatorname {PolyLog}\left (3,\frac {i+c x}{-i+c x}\right )\right )}{2 c d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 24.67 (sec) , antiderivative size = 851, normalized size of antiderivative = 8.68
method | result | size |
derivativedivides | \(\frac {-\frac {i a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {a^{2} \arctan \left (c x \right )}{d}+\frac {b^{2} \left (-i \ln \left (i c x +1\right ) \arctan \left (c x \right )^{2}+2 i \left (\frac {\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}-\frac {i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \left (\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )+\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3}-\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3}+\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )+\operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-1\right ) \arctan \left (c x \right )^{2}}{4}-\frac {i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{4}\right )\right )}{d}+\frac {2 a b \left (-i \ln \left (i c x +1\right ) \arctan \left (c x \right )-\frac {\left (\ln \left (i c x +1\right )-\ln \left (\frac {1}{2}+\frac {i c x}{2}\right )\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {1}{2}+\frac {i c x}{2}\right )}{2}+\frac {\ln \left (i c x +1\right )^{2}}{4}\right )}{d}}{c}\) | \(851\) |
default | \(\frac {-\frac {i a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {a^{2} \arctan \left (c x \right )}{d}+\frac {b^{2} \left (-i \ln \left (i c x +1\right ) \arctan \left (c x \right )^{2}+2 i \left (\frac {\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}-\frac {i \arctan \left (c x \right )^{3}}{3}+\frac {i \pi \left (\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )+\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3}-\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3}+\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )+\operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-1\right ) \arctan \left (c x \right )^{2}}{4}-\frac {i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{4}\right )\right )}{d}+\frac {2 a b \left (-i \ln \left (i c x +1\right ) \arctan \left (c x \right )-\frac {\left (\ln \left (i c x +1\right )-\ln \left (\frac {1}{2}+\frac {i c x}{2}\right )\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {1}{2}+\frac {i c x}{2}\right )}{2}+\frac {\ln \left (i c x +1\right )^{2}}{4}\right )}{d}}{c}\) | \(851\) |
parts | \(-\frac {i a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d c}+\frac {a^{2} \arctan \left (c x \right )}{d c}+\frac {b^{2} \left (-i \ln \left (i c x +1\right ) \arctan \left (c x \right )^{2}+2 i \left (\frac {\arctan \left (c x \right )^{2} \ln \left (\frac {2 i \left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}-\frac {i \pi \left (-\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )-\operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}+\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )-\operatorname {csgn}\left (\frac {i}{1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}+\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}-\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3}+\operatorname {csgn}\left (\frac {\left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right ) \operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{2}+\operatorname {csgn}\left (\frac {i \left (i c x +1\right )^{2}}{\left (c^{2} x^{2}+1\right ) \left (1+\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}\right )^{3}+1\right ) \arctan \left (c x \right )^{2}}{4}-\frac {i \arctan \left (c x \right ) \operatorname {polylog}\left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{2}+\frac {\operatorname {polylog}\left (3, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right )}{4}-\frac {i \arctan \left (c x \right )^{3}}{3}\right )\right )}{d c}+\frac {2 a b \left (-i \ln \left (i c x +1\right ) \arctan \left (c x \right )-\frac {\left (\ln \left (i c x +1\right )-\ln \left (\frac {1}{2}+\frac {i c x}{2}\right )\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {1}{2}+\frac {i c x}{2}\right )}{2}+\frac {\ln \left (i c x +1\right )^{2}}{4}\right )}{d c}\) | \(860\) |
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\[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{i \, c d x + d} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{i \, c d x + d} \,d x } \]
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\[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{i \, c d x + d} \,d x } \]
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Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{d+i c d x} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{d+c\,d\,x\,1{}\mathrm {i}} \,d x \]
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