Integrand size = 23, antiderivative size = 255 \[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\frac {i a b d x}{2 c^2}+\frac {b^2 d x}{3 c^2}+\frac {i b^2 d x^2}{12 c}-\frac {b^2 d \arctan (c x)}{3 c^3}+\frac {i b^2 d x \arctan (c x)}{2 c^2}-\frac {b d x^2 (a+b \arctan (c x))}{3 c}-\frac {1}{6} i b d x^3 (a+b \arctan (c x))-\frac {7 i d (a+b \arctan (c x))^2}{12 c^3}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{4} i c d x^4 (a+b \arctan (c x))^2-\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}-\frac {i b^2 d \log \left (1+c^2 x^2\right )}{3 c^3}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3} \]
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Time = 0.35 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {4996, 4946, 5036, 327, 209, 5040, 4964, 2449, 2352, 272, 45, 4930, 266, 5004} \[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=-\frac {7 i d (a+b \arctan (c x))^2}{12 c^3}-\frac {2 b d \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^3}+\frac {1}{4} i c d x^4 (a+b \arctan (c x))^2+\frac {1}{3} d x^3 (a+b \arctan (c x))^2-\frac {1}{6} i b d x^3 (a+b \arctan (c x))-\frac {b d x^2 (a+b \arctan (c x))}{3 c}+\frac {i a b d x}{2 c^2}-\frac {b^2 d \arctan (c x)}{3 c^3}+\frac {i b^2 d x \arctan (c x)}{2 c^2}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^3}+\frac {b^2 d x}{3 c^2}-\frac {i b^2 d \log \left (c^2 x^2+1\right )}{3 c^3}+\frac {i b^2 d x^2}{12 c} \]
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Rule 45
Rule 209
Rule 266
Rule 272
Rule 327
Rule 2352
Rule 2449
Rule 4930
Rule 4946
Rule 4964
Rule 4996
Rule 5004
Rule 5036
Rule 5040
Rubi steps \begin{align*} \text {integral}& = \int \left (d x^2 (a+b \arctan (c x))^2+i c d x^3 (a+b \arctan (c x))^2\right ) \, dx \\ & = d \int x^2 (a+b \arctan (c x))^2 \, dx+(i c d) \int x^3 (a+b \arctan (c x))^2 \, dx \\ & = \frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{4} i c d x^4 (a+b \arctan (c x))^2-\frac {1}{3} (2 b c d) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {1}{2} \left (i b c^2 d\right ) \int \frac {x^4 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{4} i c d x^4 (a+b \arctan (c x))^2-\frac {1}{2} (i b d) \int x^2 (a+b \arctan (c x)) \, dx+\frac {1}{2} (i b d) \int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {(2 b d) \int x (a+b \arctan (c x)) \, dx}{3 c}+\frac {(2 b d) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 c} \\ & = -\frac {b d x^2 (a+b \arctan (c x))}{3 c}-\frac {1}{6} i b d x^3 (a+b \arctan (c x))-\frac {i d (a+b \arctan (c x))^2}{3 c^3}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{4} i c d x^4 (a+b \arctan (c x))^2+\frac {1}{3} \left (b^2 d\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {(i b d) \int (a+b \arctan (c x)) \, dx}{2 c^2}-\frac {(i b d) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{2 c^2}-\frac {(2 b d) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{3 c^2}+\frac {1}{6} \left (i b^2 c d\right ) \int \frac {x^3}{1+c^2 x^2} \, dx \\ & = \frac {i a b d x}{2 c^2}+\frac {b^2 d x}{3 c^2}-\frac {b d x^2 (a+b \arctan (c x))}{3 c}-\frac {1}{6} i b d x^3 (a+b \arctan (c x))-\frac {7 i d (a+b \arctan (c x))^2}{12 c^3}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{4} i c d x^4 (a+b \arctan (c x))^2-\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {\left (i b^2 d\right ) \int \arctan (c x) \, dx}{2 c^2}-\frac {\left (b^2 d\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}+\frac {1}{12} \left (i b^2 c d\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right ) \\ & = \frac {i a b d x}{2 c^2}+\frac {b^2 d x}{3 c^2}-\frac {b^2 d \arctan (c x)}{3 c^3}+\frac {i b^2 d x \arctan (c x)}{2 c^2}-\frac {b d x^2 (a+b \arctan (c x))}{3 c}-\frac {1}{6} i b d x^3 (a+b \arctan (c x))-\frac {7 i d (a+b \arctan (c x))^2}{12 c^3}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{4} i c d x^4 (a+b \arctan (c x))^2-\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}-\frac {\left (2 i b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^3}-\frac {\left (i b^2 d\right ) \int \frac {x}{1+c^2 x^2} \, dx}{2 c}+\frac {1}{12} \left (i b^2 c d\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right ) \\ & = \frac {i a b d x}{2 c^2}+\frac {b^2 d x}{3 c^2}+\frac {i b^2 d x^2}{12 c}-\frac {b^2 d \arctan (c x)}{3 c^3}+\frac {i b^2 d x \arctan (c x)}{2 c^2}-\frac {b d x^2 (a+b \arctan (c x))}{3 c}-\frac {1}{6} i b d x^3 (a+b \arctan (c x))-\frac {7 i d (a+b \arctan (c x))^2}{12 c^3}+\frac {1}{3} d x^3 (a+b \arctan (c x))^2+\frac {1}{4} i c d x^4 (a+b \arctan (c x))^2-\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}-\frac {i b^2 d \log \left (1+c^2 x^2\right )}{3 c^3}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.95 \[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\frac {i d \left (b^2+6 a b c x-4 i b^2 c x+4 i a b c^2 x^2+b^2 c^2 x^2-4 i a^2 c^3 x^3-2 a b c^3 x^3+3 a^2 c^4 x^4+b^2 \left (1-4 i c^3 x^3+3 c^4 x^4\right ) \arctan (c x)^2+2 b \arctan (c x) \left (b \left (2 i+3 c x+2 i c^2 x^2-c^3 x^3\right )+a \left (-3-4 i c^3 x^3+3 c^4 x^4\right )+4 i b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-4 i a b \log \left (1+c^2 x^2\right )-4 b^2 \log \left (1+c^2 x^2\right )+4 b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{12 c^3} \]
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Time = 1.42 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.38
method | result | size |
parts | \(a^{2} d \left (\frac {1}{4} i c \,x^{4}+\frac {1}{3} x^{3}\right )+\frac {d \,b^{2} \left (\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{6}-\frac {i \arctan \left (c x \right )^{2}}{4}+\frac {i \arctan \left (c x \right ) c x}{2}-\frac {c^{2} x^{2} \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c x}{3}+\frac {i c^{2} x^{2}}{12}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}\right )}{c^{3}}+\frac {2 a b d \left (\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {i c x}{4}-\frac {i c^{3} x^{3}}{12}-\frac {c^{2} x^{2}}{6}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i \arctan \left (c x \right )}{4}\right )}{c^{3}}\) | \(352\) |
derivativedivides | \(\frac {a^{2} d \left (\frac {1}{4} i c^{4} x^{4}+\frac {1}{3} c^{3} x^{3}\right )+d \,b^{2} \left (\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{6}-\frac {i \arctan \left (c x \right )^{2}}{4}+\frac {i \arctan \left (c x \right ) c x}{2}-\frac {c^{2} x^{2} \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c x}{3}+\frac {i c^{2} x^{2}}{12}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}\right )+2 a b d \left (\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {i c x}{4}-\frac {i c^{3} x^{3}}{12}-\frac {c^{2} x^{2}}{6}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i \arctan \left (c x \right )}{4}\right )}{c^{3}}\) | \(355\) |
default | \(\frac {a^{2} d \left (\frac {1}{4} i c^{4} x^{4}+\frac {1}{3} c^{3} x^{3}\right )+d \,b^{2} \left (\frac {i \arctan \left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{3}-\frac {i \arctan \left (c x \right ) c^{3} x^{3}}{6}-\frac {i \arctan \left (c x \right )^{2}}{4}+\frac {i \arctan \left (c x \right ) c x}{2}-\frac {c^{2} x^{2} \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {c x}{3}+\frac {i c^{2} x^{2}}{12}-\frac {i \ln \left (c^{2} x^{2}+1\right )}{3}-\frac {\arctan \left (c x \right )}{3}+\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{6}\right )+2 a b d \left (\frac {i \arctan \left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{3}+\frac {i c x}{4}-\frac {i c^{3} x^{3}}{12}-\frac {c^{2} x^{2}}{6}+\frac {\ln \left (c^{2} x^{2}+1\right )}{6}-\frac {i \arctan \left (c x \right )}{4}\right )}{c^{3}}\) | \(355\) |
risch | \(\frac {b^{2} d x}{3 c^{2}}-\frac {115 b^{2} d \arctan \left (c x \right )}{288 c^{3}}-\frac {a b d}{c^{3}}+\frac {a^{2} d \,x^{3}}{3}+\frac {b d a \ln \left (c^{2} x^{2}+1\right )}{3 c^{3}}-\frac {d a b \,x^{2}}{3 c}+\frac {i d a b \ln \left (-i c x +1\right ) x^{3}}{3}-\frac {i d c \,b^{2} \ln \left (-i c x +1\right )^{2} x^{4}}{16}-\frac {i b^{2} d \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{3}}+\frac {i b^{2} d \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{3 c^{3}}-\frac {i b d a \arctan \left (c x \right )}{2 c^{3}}-\frac {i d \,b^{2} \ln \left (-i c x +1\right ) x^{2}}{6 c}-\frac {d c a b \ln \left (-i c x +1\right ) x^{4}}{4}+\frac {i a b d x}{2 c^{2}}-\frac {i d \,b^{2} \left (3 c^{4} x^{4}-4 i c^{3} x^{3}+1\right ) \ln \left (i c x +1\right )^{2}}{48 c^{3}}+\left (\frac {i d \,b^{2} \left (3 c \,x^{4}-4 i x^{3}\right ) \ln \left (-i c x +1\right )}{24}+\frac {b d \left (6 a \,c^{4} x^{4}-8 i a \,c^{3} x^{3}-2 b \,c^{3} x^{3}+4 i b \,c^{2} x^{2}-7 i b \ln \left (-i c x +1\right )+6 x b c \right )}{24 c^{3}}\right ) \ln \left (i c x +1\right )-\frac {211 i b^{2} d \ln \left (c^{2} x^{2}+1\right )}{576 c^{3}}-\frac {i d a b \,x^{3}}{6}+\frac {19 i b^{2} d \ln \left (-i c x +1\right )}{288 c^{3}}-\frac {i b^{2} d \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{3}}-\frac {d \,b^{2} \ln \left (-i c x +1\right ) x}{4 c^{2}}+\frac {7 i d \,b^{2} \ln \left (-i c x +1\right )^{2}}{48 c^{3}}+\frac {i a^{2} c d \,x^{4}}{4}+\frac {i b^{2} d \,x^{2}}{12 c}+\frac {5 i b^{2} d}{12 c^{3}}-\frac {7 i d \,a^{2}}{12 c^{3}}-\frac {d \,b^{2} \ln \left (-i c x +1\right )^{2} x^{3}}{12}+\frac {d \,b^{2} \ln \left (-i c x +1\right ) x^{3}}{12}\) | \(545\) |
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\[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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Timed out. \[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\text {Timed out} \]
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\[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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\[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int { {\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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Timed out. \[ \int x^2 (d+i c d x) (a+b \arctan (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\,1{}\mathrm {i}\right ) \,d x \]
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