Integrand size = 23, antiderivative size = 580 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {b^2 d^2 x}{3 c^2}-\frac {3 b^2 d e x}{5 c^4}+\frac {11 b^2 e^2 x}{42 c^6}+\frac {b^2 d e x^3}{15 c^2}-\frac {5 b^2 e^2 x^3}{126 c^4}+\frac {b^2 e^2 x^5}{105 c^2}-\frac {b^2 d^2 \arctan (c x)}{3 c^3}+\frac {3 b^2 d e \arctan (c x)}{5 c^5}-\frac {11 b^2 e^2 \arctan (c x)}{42 c^7}-\frac {b d^2 x^2 (a+b \arctan (c x))}{3 c}+\frac {2 b d e x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b e^2 x^2 (a+b \arctan (c x))}{7 c^5}-\frac {b d e x^4 (a+b \arctan (c x))}{5 c}+\frac {b e^2 x^4 (a+b \arctan (c x))}{14 c^3}-\frac {b e^2 x^6 (a+b \arctan (c x))}{21 c}-\frac {i d^2 (a+b \arctan (c x))^2}{3 c^3}+\frac {2 i d e (a+b \arctan (c x))^2}{5 c^5}-\frac {i e^2 (a+b \arctan (c x))^2}{7 c^7}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {2}{5} d e x^5 (a+b \arctan (c x))^2+\frac {1}{7} e^2 x^7 (a+b \arctan (c x))^2-\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {4 b d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {2 b e^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{7 c^7}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {2 i b^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^5}-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{7 c^7} \]
[Out]
Time = 0.75 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.00, number of steps used = 44, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5100, 4946, 5036, 327, 209, 5040, 4964, 2449, 2352, 308} \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=-\frac {i e^2 (a+b \arctan (c x))^2}{7 c^7}-\frac {2 b e^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{7 c^7}+\frac {2 i d e (a+b \arctan (c x))^2}{5 c^5}+\frac {4 b d e \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{5 c^5}-\frac {b e^2 x^2 (a+b \arctan (c x))}{7 c^5}-\frac {i d^2 (a+b \arctan (c x))^2}{3 c^3}-\frac {2 b d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^3}+\frac {2 b d e x^2 (a+b \arctan (c x))}{5 c^3}+\frac {b e^2 x^4 (a+b \arctan (c x))}{14 c^3}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2-\frac {b d^2 x^2 (a+b \arctan (c x))}{3 c}+\frac {2}{5} d e x^5 (a+b \arctan (c x))^2-\frac {b d e x^4 (a+b \arctan (c x))}{5 c}+\frac {1}{7} e^2 x^7 (a+b \arctan (c x))^2-\frac {b e^2 x^6 (a+b \arctan (c x))}{21 c}-\frac {11 b^2 e^2 \arctan (c x)}{42 c^7}+\frac {3 b^2 d e \arctan (c x)}{5 c^5}-\frac {b^2 d^2 \arctan (c x)}{3 c^3}-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{7 c^7}+\frac {11 b^2 e^2 x}{42 c^6}+\frac {2 i b^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{5 c^5}-\frac {3 b^2 d e x}{5 c^4}-\frac {5 b^2 e^2 x^3}{126 c^4}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^3}+\frac {b^2 d^2 x}{3 c^2}+\frac {b^2 d e x^3}{15 c^2}+\frac {b^2 e^2 x^5}{105 c^2} \]
[In]
[Out]
Rule 209
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 4946
Rule 4964
Rule 5036
Rule 5040
Rule 5100
Rubi steps \begin{align*} \text {integral}& = \int \left (d^2 x^2 (a+b \arctan (c x))^2+2 d e x^4 (a+b \arctan (c x))^2+e^2 x^6 (a+b \arctan (c x))^2\right ) \, dx \\ & = d^2 \int x^2 (a+b \arctan (c x))^2 \, dx+(2 d e) \int x^4 (a+b \arctan (c x))^2 \, dx+e^2 \int x^6 (a+b \arctan (c x))^2 \, dx \\ & = \frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {2}{5} d e x^5 (a+b \arctan (c x))^2+\frac {1}{7} e^2 x^7 (a+b \arctan (c x))^2-\frac {1}{3} \left (2 b c d^2\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {1}{5} (4 b c d e) \int \frac {x^5 (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {1}{7} \left (2 b c e^2\right ) \int \frac {x^7 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {2}{5} d e x^5 (a+b \arctan (c x))^2+\frac {1}{7} e^2 x^7 (a+b \arctan (c x))^2-\frac {\left (2 b d^2\right ) \int x (a+b \arctan (c x)) \, dx}{3 c}+\frac {\left (2 b d^2\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 c}-\frac {(4 b d e) \int x^3 (a+b \arctan (c x)) \, dx}{5 c}+\frac {(4 b d e) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{5 c}-\frac {\left (2 b e^2\right ) \int x^5 (a+b \arctan (c x)) \, dx}{7 c}+\frac {\left (2 b e^2\right ) \int \frac {x^5 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{7 c} \\ & = -\frac {b d^2 x^2 (a+b \arctan (c x))}{3 c}-\frac {b d e x^4 (a+b \arctan (c x))}{5 c}-\frac {b e^2 x^6 (a+b \arctan (c x))}{21 c}-\frac {i d^2 (a+b \arctan (c x))^2}{3 c^3}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {2}{5} d e x^5 (a+b \arctan (c x))^2+\frac {1}{7} e^2 x^7 (a+b \arctan (c x))^2+\frac {1}{3} \left (b^2 d^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {\left (2 b d^2\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{3 c^2}+\frac {1}{5} \left (b^2 d e\right ) \int \frac {x^4}{1+c^2 x^2} \, dx+\frac {(4 b d e) \int x (a+b \arctan (c x)) \, dx}{5 c^3}-\frac {(4 b d e) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{5 c^3}+\frac {1}{21} \left (b^2 e^2\right ) \int \frac {x^6}{1+c^2 x^2} \, dx+\frac {\left (2 b e^2\right ) \int x^3 (a+b \arctan (c x)) \, dx}{7 c^3}-\frac {\left (2 b e^2\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{7 c^3} \\ & = \frac {b^2 d^2 x}{3 c^2}-\frac {b d^2 x^2 (a+b \arctan (c x))}{3 c}+\frac {2 b d e x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b d e x^4 (a+b \arctan (c x))}{5 c}+\frac {b e^2 x^4 (a+b \arctan (c x))}{14 c^3}-\frac {b e^2 x^6 (a+b \arctan (c x))}{21 c}-\frac {i d^2 (a+b \arctan (c x))^2}{3 c^3}+\frac {2 i d e (a+b \arctan (c x))^2}{5 c^5}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {2}{5} d e x^5 (a+b \arctan (c x))^2+\frac {1}{7} e^2 x^7 (a+b \arctan (c x))^2-\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}-\frac {\left (b^2 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}+\frac {1}{5} \left (b^2 d e\right ) \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx+\frac {(4 b d e) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{5 c^4}-\frac {\left (2 b^2 d e\right ) \int \frac {x^2}{1+c^2 x^2} \, dx}{5 c^2}+\frac {1}{21} \left (b^2 e^2\right ) \int \left (\frac {1}{c^6}-\frac {x^2}{c^4}+\frac {x^4}{c^2}-\frac {1}{c^6 \left (1+c^2 x^2\right )}\right ) \, dx-\frac {\left (2 b e^2\right ) \int x (a+b \arctan (c x)) \, dx}{7 c^5}+\frac {\left (2 b e^2\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{7 c^5}-\frac {\left (b^2 e^2\right ) \int \frac {x^4}{1+c^2 x^2} \, dx}{14 c^2} \\ & = \frac {b^2 d^2 x}{3 c^2}-\frac {3 b^2 d e x}{5 c^4}+\frac {b^2 e^2 x}{21 c^6}+\frac {b^2 d e x^3}{15 c^2}-\frac {b^2 e^2 x^3}{63 c^4}+\frac {b^2 e^2 x^5}{105 c^2}-\frac {b^2 d^2 \arctan (c x)}{3 c^3}-\frac {b d^2 x^2 (a+b \arctan (c x))}{3 c}+\frac {2 b d e x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b e^2 x^2 (a+b \arctan (c x))}{7 c^5}-\frac {b d e x^4 (a+b \arctan (c x))}{5 c}+\frac {b e^2 x^4 (a+b \arctan (c x))}{14 c^3}-\frac {b e^2 x^6 (a+b \arctan (c x))}{21 c}-\frac {i d^2 (a+b \arctan (c x))^2}{3 c^3}+\frac {2 i d e (a+b \arctan (c x))^2}{5 c^5}-\frac {i e^2 (a+b \arctan (c x))^2}{7 c^7}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {2}{5} d e x^5 (a+b \arctan (c x))^2+\frac {1}{7} e^2 x^7 (a+b \arctan (c x))^2-\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {4 b d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {\left (2 i b^2 d^2\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 (b^2 d e\right ) \int \frac {1}{1+c^2 x^2} \, dx}{5 c^4}+\frac {\left (2 b^2 d e\right ) \int \frac {1}{1+c^2 x^2} \, dx}{5 c^4}-\frac {\left (4 b^2 d e\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^4}-\frac {\left (2 b e^2\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{7 c^6}-\frac {\left (b^2 e^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{21 c^6}+\frac {\left (b^2 e^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx}{7 c^4}-\frac {\left (b^2 e^2\right ) \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{14 c^2} \\ & = \frac {b^2 d^2 x}{3 c^2}-\frac {3 b^2 d e x}{5 c^4}+\frac {11 b^2 e^2 x}{42 c^6}+\frac {b^2 d e x^3}{15 c^2}-\frac {5 b^2 e^2 x^3}{126 c^4}+\frac {b^2 e^2 x^5}{105 c^2}-\frac {b^2 d^2 \arctan (c x)}{3 c^3}+\frac {3 b^2 d e \arctan (c x)}{5 c^5}-\frac {b^2 e^2 \arctan (c x)}{21 c^7}-\frac {b d^2 x^2 (a+b \arctan (c x))}{3 c}+\frac {2 b d e x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b e^2 x^2 (a+b \arctan (c x))}{7 c^5}-\frac {b d e x^4 (a+b \arctan (c x))}{5 c}+\frac {b e^2 x^4 (a+b \arctan (c x))}{14 c^3}-\frac {b e^2 x^6 (a+b \arctan (c x))}{21 c}-\frac {i d^2 (a+b \arctan (c x))^2}{3 c^3}+\frac {2 i d e (a+b \arctan (c x))^2}{5 c^5}-\frac {i e^2 (a+b \arctan (c x))^2}{7 c^7}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {2}{5} d e x^5 (a+b \arctan (c x))^2+\frac {1}{7} e^2 x^7 (a+b \arctan (c x))^2-\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {4 b d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {2 b e^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{7 c^7}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {\left (4 i b^2 d e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{5 c^5}-\frac {\left (b^2 e^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{14 c^6}-\frac {\left (b^2 e^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{7 c^6}+\frac {\left (2 b^2 e^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{7 c^6} \\ & = \frac {b^2 d^2 x}{3 c^2}-\frac {3 b^2 d e x}{5 c^4}+\frac {11 b^2 e^2 x}{42 c^6}+\frac {b^2 d e x^3}{15 c^2}-\frac {5 b^2 e^2 x^3}{126 c^4}+\frac {b^2 e^2 x^5}{105 c^2}-\frac {b^2 d^2 \arctan (c x)}{3 c^3}+\frac {3 b^2 d e \arctan (c x)}{5 c^5}-\frac {11 b^2 e^2 \arctan (c x)}{42 c^7}-\frac {b d^2 x^2 (a+b \arctan (c x))}{3 c}+\frac {2 b d e x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b e^2 x^2 (a+b \arctan (c x))}{7 c^5}-\frac {b d e x^4 (a+b \arctan (c x))}{5 c}+\frac {b e^2 x^4 (a+b \arctan (c x))}{14 c^3}-\frac {b e^2 x^6 (a+b \arctan (c x))}{21 c}-\frac {i d^2 (a+b \arctan (c x))^2}{3 c^3}+\frac {2 i d e (a+b \arctan (c x))^2}{5 c^5}-\frac {i e^2 (a+b \arctan (c x))^2}{7 c^7}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {2}{5} d e x^5 (a+b \arctan (c x))^2+\frac {1}{7} e^2 x^7 (a+b \arctan (c x))^2-\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {4 b d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {2 b e^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{7 c^7}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {2 i b^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^5}-\frac {\left (2 i b^2 e^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{7 c^7} \\ & = \frac {b^2 d^2 x}{3 c^2}-\frac {3 b^2 d e x}{5 c^4}+\frac {11 b^2 e^2 x}{42 c^6}+\frac {b^2 d e x^3}{15 c^2}-\frac {5 b^2 e^2 x^3}{126 c^4}+\frac {b^2 e^2 x^5}{105 c^2}-\frac {b^2 d^2 \arctan (c x)}{3 c^3}+\frac {3 b^2 d e \arctan (c x)}{5 c^5}-\frac {11 b^2 e^2 \arctan (c x)}{42 c^7}-\frac {b d^2 x^2 (a+b \arctan (c x))}{3 c}+\frac {2 b d e x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b e^2 x^2 (a+b \arctan (c x))}{7 c^5}-\frac {b d e x^4 (a+b \arctan (c x))}{5 c}+\frac {b e^2 x^4 (a+b \arctan (c x))}{14 c^3}-\frac {b e^2 x^6 (a+b \arctan (c x))}{21 c}-\frac {i d^2 (a+b \arctan (c x))^2}{3 c^3}+\frac {2 i d e (a+b \arctan (c x))^2}{5 c^5}-\frac {i e^2 (a+b \arctan (c x))^2}{7 c^7}+\frac {1}{3} d^2 x^3 (a+b \arctan (c x))^2+\frac {2}{5} d e x^5 (a+b \arctan (c x))^2+\frac {1}{7} e^2 x^7 (a+b \arctan (c x))^2-\frac {2 b d^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {4 b d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {2 b e^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{7 c^7}-\frac {i b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {2 i b^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^5}-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{7 c^7} \\ \end{align*}
Time = 1.18 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.88 \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\frac {378 a b c^2 d e-165 a b e^2+210 b^2 c^5 d^2 x-378 b^2 c^3 d e x+165 b^2 c e^2 x-210 a b c^6 d^2 x^2+252 a b c^4 d e x^2-90 a b c^2 e^2 x^2+210 a^2 c^7 d^2 x^3+42 b^2 c^5 d e x^3-25 b^2 c^3 e^2 x^3-126 a b c^6 d e x^4+45 a b c^4 e^2 x^4+252 a^2 c^7 d e x^5+6 b^2 c^5 e^2 x^5-30 a b c^6 e^2 x^6+90 a^2 c^7 e^2 x^7+6 b^2 \left (35 i c^4 d^2-42 i c^2 d e+15 i e^2+c^7 \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right )\right ) \arctan (c x)^2-3 b \arctan (c x) \left (-4 a c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )+b \left (1+c^2 x^2\right ) \left (55 e^2-c^2 e \left (126 d+25 e x^2\right )+2 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )\right )+4 b \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \log \left (1+e^{2 i \arctan (c x)}\right )\right )+210 a b c^4 d^2 \log \left (1+c^2 x^2\right )-252 a b c^2 d e \log \left (1+c^2 x^2\right )+90 a b e^2 \log \left (1+c^2 x^2\right )+6 i b^2 \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{630 c^7} \]
[In]
[Out]
Time = 0.87 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.09
method | result | size |
parts | \(a^{2} \left (\frac {1}{7} e^{2} x^{7}+\frac {2}{5} e d \,x^{5}+\frac {1}{3} d^{2} x^{3}\right )+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c^{3} e^{2} x^{7}}{7}+\frac {2 \arctan \left (c x \right )^{2} c^{3} d e \,x^{5}}{5}+\frac {\arctan \left (c x \right )^{2} d^{2} c^{3} x^{3}}{3}-\frac {2 \left (\frac {35 \arctan \left (c x \right ) c^{6} d^{2} x^{2}}{2}+\frac {21 \arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{2}+\frac {5 \arctan \left (c x \right ) e^{2} c^{6} x^{6}}{2}-21 \arctan \left (c x \right ) d \,c^{4} e \,x^{2}-\frac {15 \arctan \left (c x \right ) e^{2} c^{4} x^{4}}{4}+\frac {15 \arctan \left (c x \right ) e^{2} c^{2} x^{2}}{2}-\frac {35 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{4} d^{2}}{2}+21 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d e -\frac {15 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{2}}{2}-\frac {e^{2} c^{5} x^{5}}{2}-\frac {7 d \,c^{5} e \,x^{3}}{2}-\frac {35 c^{5} x \,d^{2}}{2}+\frac {25 e^{2} c^{3} x^{3}}{12}+\frac {63 c^{3} d e x}{2}-\frac {55 c x \,e^{2}}{4}-\frac {\left (-70 c^{4} d^{2}+126 c^{2} d e -55 e^{2}\right ) \arctan \left (c x \right )}{4}-\frac {\left (-70 c^{4} d^{2}+84 c^{2} d e -30 e^{2}\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 )}{4}\right )}{105 c^{4}}\right )}{c^{3}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) c^{3} e^{2} x^{7}}{7}+\frac {2 \arctan \left (c x \right ) c^{3} d e \,x^{5}}{5}+\frac {\arctan \left (c x \right ) d^{2} c^{3} x^{3}}{3}-\frac {\frac {35 d^{2} c^{6} x^{2}}{2}+\frac {21 d \,c^{6} e \,x^{4}}{2}+\frac {5 e^{2} c^{6} x^{6}}{2}-21 d \,c^{4} e \,x^{2}-\frac {15 e^{2} c^{4} x^{4}}{4}+\frac {15 e^{2} c^{2} x^{2}}{2}+\frac {\left (-35 c^{4} d^{2}+42 c^{2} d e -15 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{105 c^{4}}\right )}{c^{3}}\) | \(634\) |
derivativedivides | \(\frac {\frac {a^{2} \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} d^{2} c^{7} x^{3}}{3}+\frac {2 \arctan \left (c x \right )^{2} d \,c^{7} e \,x^{5}}{5}+\frac {\arctan \left (c x \right )^{2} e^{2} c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} d^{2} x^{2}}{3}-\frac {\arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{5}+\frac {2 \arctan \left (c x \right ) d \,c^{4} e \,x^{2}}{5}-\frac {\arctan \left (c x \right ) e^{2} c^{6} x^{6}}{21}+\frac {\arctan \left (c x \right ) e^{2} c^{4} x^{4}}{14}-\frac {\arctan \left (c x \right ) e^{2} c^{2} x^{2}}{7}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{4} d^{2}}{3}-\frac {2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d e}{5}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{2}}{7}+\frac {c^{5} x \,d^{2}}{3}+\frac {d \,c^{5} e \,x^{3}}{15}+\frac {e^{2} c^{5} x^{5}}{105}-\frac {3 c^{3} d e x}{5}-\frac {5 e^{2} c^{3} x^{3}}{126}+\frac {11 c x \,e^{2}}{42}+\frac {\left (-70 c^{4} d^{2}+126 c^{2} d e -55 e^{2}\right ) \arctan \left (c x \right )}{210}+\frac {\left (-70 c^{4} d^{2}+84 c^{2} d e -30 e^{2}\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 )}{210}\right )}{c^{4}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \arctan \left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\arctan \left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {d^{2} c^{6} x^{2}}{6}-\frac {d \,c^{6} e \,x^{4}}{10}+\frac {d \,c^{4} e \,x^{2}}{5}-\frac {e^{2} c^{6} x^{6}}{42}+\frac {e^{2} c^{4} x^{4}}{28}-\frac {e^{2} c^{2} x^{2}}{14}-\frac {\left (-35 c^{4} d^{2}+42 c^{2} d e -15 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{210}\right )}{c^{4}}}{c^{3}}\) | \(638\) |
default | \(\frac {\frac {a^{2} \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} d^{2} c^{7} x^{3}}{3}+\frac {2 \arctan \left (c x \right )^{2} d \,c^{7} e \,x^{5}}{5}+\frac {\arctan \left (c x \right )^{2} e^{2} c^{7} x^{7}}{7}-\frac {\arctan \left (c x \right ) c^{6} d^{2} x^{2}}{3}-\frac {\arctan \left (c x \right ) e \,c^{6} d \,x^{4}}{5}+\frac {2 \arctan \left (c x \right ) d \,c^{4} e \,x^{2}}{5}-\frac {\arctan \left (c x \right ) e^{2} c^{6} x^{6}}{21}+\frac {\arctan \left (c x \right ) e^{2} c^{4} x^{4}}{14}-\frac {\arctan \left (c x \right ) e^{2} c^{2} x^{2}}{7}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{4} d^{2}}{3}-\frac {2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d e}{5}+\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{2}}{7}+\frac {c^{5} x \,d^{2}}{3}+\frac {d \,c^{5} e \,x^{3}}{15}+\frac {e^{2} c^{5} x^{5}}{105}-\frac {3 c^{3} d e x}{5}-\frac {5 e^{2} c^{3} x^{3}}{126}+\frac {11 c x \,e^{2}}{42}+\frac {\left (-70 c^{4} d^{2}+126 c^{2} d e -55 e^{2}\right ) \arctan \left (c x \right )}{210}+\frac {\left (-70 c^{4} d^{2}+84 c^{2} d e -30 e^{2}\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 )}{210}\right )}{c^{4}}+\frac {2 a b \left (\frac {\arctan \left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \arctan \left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\arctan \left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {d^{2} c^{6} x^{2}}{6}-\frac {d \,c^{6} e \,x^{4}}{10}+\frac {d \,c^{4} e \,x^{2}}{5}-\frac {e^{2} c^{6} x^{6}}{42}+\frac {e^{2} c^{4} x^{4}}{28}-\frac {e^{2} c^{2} x^{2}}{14}-\frac {\left (-35 c^{4} d^{2}+42 c^{2} d e -15 e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{210}\right )}{c^{4}}}{c^{3}}\) | \(638\) |
risch | \(\text {Expression too large to display}\) | \(1597\) |
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\[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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\[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{2}\, dx \]
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\[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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\[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int { {\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (d+e x^2\right )^2 (a+b \arctan (c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2 \,d x \]
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