Integrand size = 18, antiderivative size = 231 \[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {b^2 e x}{3 c^2}-\frac {b^2 e \arctan (c x)}{3 c^3}-\frac {b e x^2 (a+b \arctan (c x))}{3 c}+\frac {i d (a+b \arctan (c x))^2}{c}-\frac {i e (a+b \arctan (c x))^2}{3 c^3}+d x (a+b \arctan (c x))^2+\frac {1}{3} e x^3 (a+b \arctan (c x))^2+\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}-\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3} \]
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Time = 0.25 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5034, 4930, 5040, 4964, 2449, 2352, 4946, 5036, 327, 209} \[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=-\frac {i e (a+b \arctan (c x))^2}{3 c^3}-\frac {2 b e \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^3}+d x (a+b \arctan (c x))^2+\frac {i d (a+b \arctan (c x))^2}{c}+\frac {2 b d \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {1}{3} e x^3 (a+b \arctan (c x))^2-\frac {b e x^2 (a+b \arctan (c x))}{3 c}-\frac {b^2 e \arctan (c x)}{3 c^3}-\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^3}+\frac {b^2 e x}{3 c^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c} \]
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Rule 209
Rule 327
Rule 2352
Rule 2449
Rule 4930
Rule 4946
Rule 4964
Rule 5034
Rule 5036
Rule 5040
Rubi steps \begin{align*} \text {integral}& = \int \left (d (a+b \arctan (c x))^2+e x^2 (a+b \arctan (c x))^2\right ) \, dx \\ & = d \int (a+b \arctan (c x))^2 \, dx+e \int x^2 (a+b \arctan (c x))^2 \, dx \\ & = d x (a+b \arctan (c x))^2+\frac {1}{3} e x^3 (a+b \arctan (c x))^2-(2 b c d) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {1}{3} (2 b c e) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {i d (a+b \arctan (c x))^2}{c}+d x (a+b \arctan (c x))^2+\frac {1}{3} e x^3 (a+b \arctan (c x))^2+(2 b d) \int \frac {a+b \arctan (c x)}{i-c x} \, dx-\frac {(2 b e) \int x (a+b \arctan (c x)) \, dx}{3 c}+\frac {(2 b e) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 c} \\ & = -\frac {b e x^2 (a+b \arctan (c x))}{3 c}+\frac {i d (a+b \arctan (c x))^2}{c}-\frac {i e (a+b \arctan (c x))^2}{3 c^3}+d x (a+b \arctan (c x))^2+\frac {1}{3} e x^3 (a+b \arctan (c x))^2+\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\frac {1}{3} \left (b^2 e\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {(2 b e) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{3 c^2} \\ & = \frac {b^2 e x}{3 c^2}-\frac {b e x^2 (a+b \arctan (c x))}{3 c}+\frac {i d (a+b \arctan (c x))^2}{c}-\frac {i e (a+b \arctan (c x))^2}{3 c^3}+d x (a+b \arctan (c x))^2+\frac {1}{3} e x^3 (a+b \arctan (c x))^2+\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e (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 )}{c}-\frac {\left (b^2 e\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 e\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2} \\ & = \frac {b^2 e x}{3 c^2}-\frac {b^2 e \arctan (c x)}{3 c^3}-\frac {b e x^2 (a+b \arctan (c x))}{3 c}+\frac {i d (a+b \arctan (c x))^2}{c}-\frac {i e (a+b \arctan (c x))^2}{3 c^3}+d x (a+b \arctan (c x))^2+\frac {1}{3} e x^3 (a+b \arctan (c x))^2+\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}-\frac {\left (2 i b^2 e\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 {b^2 e x}{3 c^2}-\frac {b^2 e \arctan (c x)}{3 c^3}-\frac {b e x^2 (a+b \arctan (c x))}{3 c}+\frac {i d (a+b \arctan (c x))^2}{c}-\frac {i e (a+b \arctan (c x))^2}{3 c^3}+d x (a+b \arctan (c x))^2+\frac {1}{3} e x^3 (a+b \arctan (c x))^2+\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}-\frac {i b^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.90 \[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\frac {3 a^2 c^3 d x+b^2 c e x-a b c^2 e x^2+a^2 c^3 e x^3+b^2 \left (-3 i c^2 d+i e+c^3 \left (3 d x+e x^3\right )\right ) \arctan (c x)^2-b \arctan (c x) \left (-2 a c^3 x \left (3 d+e x^2\right )+b \left (e+c^2 e x^2\right )+2 b \left (-3 c^2 d+e\right ) \log \left (1+e^{2 i \arctan (c x)}\right )\right )-3 a b c^2 d \log \left (1+c^2 x^2\right )+a b e \log \left (1+c^2 x^2\right )-i b^2 \left (3 c^2 d-e\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{3 c^3} \]
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Time = 0.54 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.39
method | result | size |
parts | \(a^{2} \left (\frac {1}{3} e \,x^{3}+x d \right )+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c \,x^{3} e}{3}+\arctan \left (c x \right )^{2} c x d -\frac {2 \left (\frac {\arctan \left (c x \right ) e \,c^{2} x^{2}}{2}+\frac {3 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d}{2}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e}{2}-\frac {e \left (c x -\arctan \left (c x \right )\right )}{2}-\frac {\left (3 c^{2} d -e \right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{2}\right )}{3 c^{2}}\right )}{c}+\frac {2 a b \left (\frac {c \arctan \left (c x \right ) x^{3} e}{3}+\arctan \left (c x \right ) c x d -\frac {\frac {e \,c^{2} x^{2}}{2}+\frac {\left (3 c^{2} d -e \right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{3 c^{2}}\right )}{c}\) | \(321\) |
derivativedivides | \(\frac {\frac {a^{2} \left (c^{3} x d +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (\arctan \left (c x \right )^{2} c^{3} x d +\frac {\arctan \left (c x \right )^{2} e \,c^{3} x^{3}}{3}-\frac {\arctan \left (c x \right ) e \,c^{2} x^{2}}{3}-\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d +\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e}{3}+\frac {e \left (c x -\arctan \left (c x \right )\right )}{3}+\frac {\left (3 c^{2} d -e \right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{3}\right )}{c^{2}}+\frac {2 a b \left (\arctan \left (c x \right ) c^{3} x d +\frac {\arctan \left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {e \,c^{2} x^{2}}{6}-\frac {\left (3 c^{2} d -e \right ) \ln \left (c^{2} x^{2}+1\right )}{6}\right )}{c^{2}}}{c}\) | \(330\) |
default | \(\frac {\frac {a^{2} \left (c^{3} x d +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (\arctan \left (c x \right )^{2} c^{3} x d +\frac {\arctan \left (c x \right )^{2} e \,c^{3} x^{3}}{3}-\frac {\arctan \left (c x \right ) e \,c^{2} x^{2}}{3}-\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d +\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e}{3}+\frac {e \left (c x -\arctan \left (c x \right )\right )}{3}+\frac {\left (3 c^{2} d -e \right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{3}\right )}{c^{2}}+\frac {2 a b \left (\arctan \left (c x \right ) c^{3} x d +\frac {\arctan \left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {e \,c^{2} x^{2}}{6}-\frac {\left (3 c^{2} d -e \right ) \ln \left (c^{2} x^{2}+1\right )}{6}\right )}{c^{2}}}{c}\) | \(330\) |
risch | \(-\frac {a b e \,x^{2}}{3 c}+\frac {2 a b d}{c}-\frac {11 a b e}{9 c^{3}}+\frac {i b^{2} d \ln \left (c^{2} x^{2}+1\right )}{2 c}+\frac {i b^{2} d \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{c}+\frac {b^{2} d \ln \left (i c x +1\right ) \ln \left (-i c x +1\right ) x}{2}+\frac {b^{2} e \ln \left (i c x +1\right ) \ln \left (-i c x +1\right ) x^{3}}{6}-\frac {b a d \ln \left (i c x +1\right )}{c}-\frac {i b^{2} d \ln \left (-i c x +1\right )}{2 c}-\frac {2 i b^{2} e \ln \left (c^{2} x^{2}+1\right )}{9 c^{3}}+\frac {5 i b^{2} e \ln \left (-i c x +1\right )}{36 c^{3}}-\frac {i b^{2} e \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{3 c^{3}}+\frac {i b^{2} \ln \left (i c x +1\right )^{2} d}{4 c}-\frac {i b^{2} \ln \left (i c x +1\right ) d}{2 c}+\frac {b^{2} e x}{3 c^{2}}-\frac {b^{2} e \arctan \left (c x \right )}{6 c^{3}}+\frac {b e a \ln \left (c^{2} x^{2}+1\right )}{3 c^{3}}+\frac {i a b d \arctan \left (c x \right )}{c}+\frac {i b^{2} e \ln \left (i c x +1\right ) x^{2}}{6 c}-\frac {i b^{2} e \ln \left (-i c x +1\right ) x^{2}}{6 c}+\frac {e \,a^{2} x^{3}}{3}+d x \,a^{2}-\frac {i b e a \ln \left (i c x +1\right ) x^{3}}{3}-i b a d \ln \left (i c x +1\right ) x +i \ln \left (-i c x +1\right ) x a b d +\frac {i e b a \ln \left (-i c x +1\right ) x^{3}}{3}+\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 )}{c}-\frac {i b^{2} d \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{c}-\frac {i b^{2} e \ln \left (i c x +1\right ) \ln \left (-i c x +1\right )}{6 c^{3}}+\frac {i b^{2} e \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{3 c^{3}}-\frac {i b^{2} e \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 (i c x +1\right ) \ln \left (-i c x +1\right )}{2 c}+\frac {i b^{2} d}{c}-\frac {17 i b^{2} e}{54 c^{3}}+\frac {i d \,a^{2}}{c}-\frac {i e \,a^{2}}{3 c^{3}}-\frac {i b^{2} e \ln \left (i c x +1\right )^{2}}{12 c^{3}}+\frac {11 i b^{2} e \ln \left (i c x +1\right )}{36 c^{3}}+\frac {i b^{2} e \ln \left (-i c x +1\right )^{2}}{12 c^{3}}-\frac {i \ln \left (-i c x +1\right )^{2} b^{2} d}{4 c}-\frac {a b d \ln \left (c^{2} x^{2}+1\right )}{2 c}-\frac {b^{2} e \ln \left (i c x +1\right )^{2} x^{3}}{12}-\frac {b^{2} \ln \left (i c x +1\right )^{2} x d}{4}-\frac {\ln \left (-i c x +1\right )^{2} x \,b^{2} d}{4}-\frac {b^{2} e \ln \left (-i c x +1\right )^{2} x^{3}}{12}\) | \(782\) |
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\[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]
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\[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )\, dx \]
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\[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]
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\[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int \left (d+e x^2\right ) (a+b \arctan (c x))^2 \, dx=\int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right ) \,d x \]
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