Optimal. Leaf size=580 \[ -\frac {i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{7 c^7}-\frac {2 b e^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{7 c^7}+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}+\frac {4 b d e \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{5 c^5}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{7 c^5}-\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {2 b d^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac {2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}+\frac {b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac {1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac {i b^2 e^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{7 c^7}-\frac {11 b^2 e^2 \tan ^{-1}(c x)}{42 c^7}+\frac {11 b^2 e^2 x}{42 c^6}+\frac {2 i b^2 d e \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{5 c^5}+\frac {3 b^2 d e \tan ^{-1}(c x)}{5 c^5}-\frac {3 b^2 d e x}{5 c^4}-\frac {5 b^2 e^2 x^3}{126 c^4}-\frac {i b^2 d^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{3 c^3}-\frac {b^2 d^2 \tan ^{-1}(c x)}{3 c^3}+\frac {b^2 d^2 x}{3 c^2}+\frac {b^2 d e x^3}{15 c^2}+\frac {b^2 e^2 x^5}{105 c^2} \]
[Out]
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Rubi [A] time = 1.07, antiderivative size = 580, normalized size of antiderivative = 1.00, number of steps used = 44, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4980, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302} \[ -\frac {i b^2 d^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {2 i b^2 d e \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^5}-\frac {i b^2 e^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{7 c^7}-\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac {2 b d^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac {2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}+\frac {4 b d e \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{5 c^5}+\frac {b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{7 c^5}-\frac {i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{7 c^7}-\frac {2 b e^2 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{7 c^7}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac {1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}+\frac {b^2 d^2 x}{3 c^2}-\frac {b^2 d^2 \tan ^{-1}(c x)}{3 c^3}+\frac {b^2 d e x^3}{15 c^2}-\frac {3 b^2 d e x}{5 c^4}+\frac {3 b^2 d e \tan ^{-1}(c x)}{5 c^5}+\frac {b^2 e^2 x^5}{105 c^2}-\frac {5 b^2 e^2 x^3}{126 c^4}+\frac {11 b^2 e^2 x}{42 c^6}-\frac {11 b^2 e^2 \tan ^{-1}(c x)}{42 c^7} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 4852
Rule 4854
Rule 4916
Rule 4920
Rule 4980
Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+2 d e x^4 \left (a+b \tan ^{-1}(c x)\right )^2+e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+(2 d e) \int x^4 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+e^2 \int x^6 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{3} \left (2 b c d^2\right ) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {1}{5} (4 b c d e) \int \frac {x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {1}{7} \left (2 b c e^2\right ) \int \frac {x^7 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {\left (2 b d^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac {\left (2 b d^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}-\frac {(4 b d e) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c}+\frac {(4 b d e) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c}-\frac {\left (2 b e^2\right ) \int x^5 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{7 c}+\frac {\left (2 b e^2\right ) \int \frac {x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{7 c}\\ &=-\frac {b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac {b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}-\frac {b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^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 \tan ^{-1}(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 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c^3}-\frac {(4 b d e) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{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 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{7 c^3}-\frac {\left (2 b e^2\right ) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{7 c^3}\\ &=\frac {b^2 d^2 x}{3 c^2}-\frac {b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac {b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac {b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac {b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \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 \tan ^{-1}(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 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{7 c^5}+\frac {\left (2 b e^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{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 \tan ^{-1}(c x)}{3 c^3}-\frac {b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{7 c^5}-\frac {b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac {b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac {b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}-\frac {i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{7 c^7}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {4 b d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {\left (2 i b^2 d^2\right ) \operatorname {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 \tan ^{-1}(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 \tan ^{-1}(c x)}{3 c^3}+\frac {3 b^2 d e \tan ^{-1}(c x)}{5 c^5}-\frac {b^2 e^2 \tan ^{-1}(c x)}{21 c^7}-\frac {b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{7 c^5}-\frac {b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac {b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac {b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}-\frac {i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{7 c^7}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {4 b d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {2 b e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{7 c^7}-\frac {i b^2 d^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {\left (4 i b^2 d e\right ) \operatorname {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 \tan ^{-1}(c x)}{3 c^3}+\frac {3 b^2 d e \tan ^{-1}(c x)}{5 c^5}-\frac {11 b^2 e^2 \tan ^{-1}(c x)}{42 c^7}-\frac {b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{7 c^5}-\frac {b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac {b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac {b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}-\frac {i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{7 c^7}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {4 b d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {2 b e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{7 c^7}-\frac {i b^2 d^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {2 i b^2 d e \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{5 c^5}-\frac {\left (2 i b^2 e^2\right ) \operatorname {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 \tan ^{-1}(c x)}{3 c^3}+\frac {3 b^2 d e \tan ^{-1}(c x)}{5 c^5}-\frac {11 b^2 e^2 \tan ^{-1}(c x)}{42 c^7}-\frac {b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac {2 b d e x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac {b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{7 c^5}-\frac {b d e x^4 \left (a+b \tan ^{-1}(c x)\right )}{5 c}+\frac {b e^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{14 c^3}-\frac {b e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )}{21 c}-\frac {i d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac {2 i d e \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^5}-\frac {i e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{7 c^7}+\frac {1}{3} d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} d e x^5 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{7} e^2 x^7 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {2 b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+\frac {4 b d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}-\frac {2 b e^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{7 c^7}-\frac {i b^2 d^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{3 c^3}+\frac {2 i b^2 d e \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{5 c^5}-\frac {i b^2 e^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{7 c^7}\\ \end {align*}
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Mathematica [A] time = 1.78, size = 513, normalized size = 0.88 \[ \frac {210 a^2 c^7 d^2 x^3+252 a^2 c^7 d e x^5+90 a^2 c^7 e^2 x^7-210 a b c^6 d^2 x^2-126 a b c^6 d e x^4-30 a b c^6 e^2 x^6+252 a b c^4 d e x^2+45 a b c^4 e^2 x^4-252 a b c^2 d e \log \left (c^2 x^2+1\right )+378 a b c^2 d e-90 a b c^2 e^2 x^2+90 a b e^2 \log \left (c^2 x^2+1\right )+210 a b c^4 d^2 \log \left (c^2 x^2+1\right )-3 b \tan ^{-1}(c x) \left (-4 a c^7 x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )+b \left (c^2 x^2+1\right ) \left (2 c^4 \left (35 d^2+21 d e x^2+5 e^2 x^4\right )-c^2 e \left (126 d+25 e x^2\right )+55 e^2\right )+4 b \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-165 a b e^2+210 b^2 c^5 d^2 x+42 b^2 c^5 d e x^3+6 b^2 c^5 e^2 x^5-378 b^2 c^3 d e x-25 b^2 c^3 e^2 x^3+6 i b^2 \left (35 c^4 d^2-42 c^2 d e+15 e^2\right ) \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+6 b^2 \tan ^{-1}(c x)^2 \left (c^7 \left (35 d^2 x^3+42 d e x^5+15 e^2 x^7\right )+35 i c^4 d^2-42 i c^2 d e+15 i e^2\right )+165 b^2 c e^2 x}{630 c^7} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} e^{2} x^{6} + 2 \, a^{2} d e x^{4} + a^{2} d^{2} x^{2} + {\left (b^{2} e^{2} x^{6} + 2 \, b^{2} d e x^{4} + b^{2} d^{2} x^{2}\right )} \arctan \left (c x\right )^{2} + 2 \, {\left (a b e^{2} x^{6} + 2 \, a b d e x^{4} + a b d^{2} x^{2}\right )} \arctan \left (c x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 1158, normalized size = 2.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{7} \, a^{2} e^{2} x^{7} + \frac {2}{5} \, a^{2} d e x^{5} + \frac {1}{3} \, a^{2} d^{2} x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} a b d^{2} + \frac {1}{5} \, {\left (4 \, x^{5} \arctan \left (c x\right ) - c {\left (\frac {c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} a b d e + \frac {1}{42} \, {\left (12 \, x^{7} \arctan \left (c x\right ) - c {\left (\frac {2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac {6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} a b e^{2} + \frac {1}{420} \, {\left (15 \, b^{2} e^{2} x^{7} + 42 \, b^{2} d e x^{5} + 35 \, b^{2} d^{2} x^{3}\right )} \arctan \left (c x\right )^{2} - \frac {1}{1680} \, {\left (15 \, b^{2} e^{2} x^{7} + 42 \, b^{2} d e x^{5} + 35 \, b^{2} d^{2} x^{3}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} + \int \frac {1260 \, {\left (b^{2} c^{2} e^{2} x^{8} + {\left (2 \, b^{2} c^{2} d e + b^{2} e^{2}\right )} x^{6} + b^{2} d^{2} x^{2} + {\left (b^{2} c^{2} d^{2} + 2 \, b^{2} d e\right )} x^{4}\right )} \arctan \left (c x\right )^{2} + 105 \, {\left (b^{2} c^{2} e^{2} x^{8} + {\left (2 \, b^{2} c^{2} d e + b^{2} e^{2}\right )} x^{6} + b^{2} d^{2} x^{2} + {\left (b^{2} c^{2} d^{2} + 2 \, b^{2} d e\right )} x^{4}\right )} \log \left (c^{2} x^{2} + 1\right )^{2} - 8 \, {\left (15 \, b^{2} c e^{2} x^{7} + 42 \, b^{2} c d e x^{5} + 35 \, b^{2} c d^{2} x^{3}\right )} \arctan \left (c x\right ) + 4 \, {\left (15 \, b^{2} c^{2} e^{2} x^{8} + 42 \, b^{2} c^{2} d e x^{6} + 35 \, b^{2} c^{2} d^{2} x^{4}\right )} \log \left (c^{2} x^{2} + 1\right )}{1680 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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