Optimal. Leaf size=380 \[ \frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2}+\frac{a b d e x}{c^3}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4}+\frac{b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{a b e^2 x}{3 c^5}+\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{a b d^2 x}{c}+\frac{1}{2} d e x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{1}{6} e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{b^2 d^2 \log \left (c^2 x^2+1\right )}{2 c^2}+\frac{b^2 d e x^2}{6 c^2}-\frac{2 b^2 d e \log \left (c^2 x^2+1\right )}{3 c^4}+\frac{b^2 d e x \tan ^{-1}(c x)}{c^3}+\frac{b^2 e^2 x^4}{60 c^2}-\frac{4 b^2 e^2 x^2}{45 c^4}+\frac{23 b^2 e^2 \log \left (c^2 x^2+1\right )}{90 c^6}-\frac{b^2 e^2 x \tan ^{-1}(c x)}{3 c^5}-\frac{b^2 d^2 x \tan ^{-1}(c x)}{c} \]
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Rubi [A] time = 0.75363, antiderivative size = 380, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {4980, 4852, 4916, 4846, 260, 4884, 266, 43} \[ \frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2}+\frac{a b d e x}{c^3}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4}+\frac{b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{a b e^2 x}{3 c^5}+\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{a b d^2 x}{c}+\frac{1}{2} d e x^4 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{1}{6} e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{b^2 d^2 \log \left (c^2 x^2+1\right )}{2 c^2}+\frac{b^2 d e x^2}{6 c^2}-\frac{2 b^2 d e \log \left (c^2 x^2+1\right )}{3 c^4}+\frac{b^2 d e x \tan ^{-1}(c x)}{c^3}+\frac{b^2 e^2 x^4}{60 c^2}-\frac{4 b^2 e^2 x^2}{45 c^4}+\frac{23 b^2 e^2 \log \left (c^2 x^2+1\right )}{90 c^6}-\frac{b^2 e^2 x \tan ^{-1}(c x)}{3 c^5}-\frac{b^2 d^2 x \tan ^{-1}(c x)}{c} \]
Antiderivative was successfully verified.
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Rule 4980
Rule 4852
Rule 4916
Rule 4846
Rule 260
Rule 4884
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x \left (d+e x^2\right )^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x \left (a+b \tan ^{-1}(c x)\right )^2+2 d e x^3 \left (a+b \tan ^{-1}(c x)\right )^2+e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+(2 d e) \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+e^2 \int x^5 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{2} d e x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\left (b c d^2\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-(b c d e) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{1}{3} \left (b c e^2\right ) \int \frac{x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{2} d e x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{\left (b d^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c}+\frac{\left (b d^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c}-\frac{(b d e) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c}+\frac{(b d e) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c}-\frac{\left (b e^2\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac{\left (b e^2\right ) \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}\\ &=-\frac{a b d^2 x}{c}-\frac{b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2}+\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{2} d e x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{\left (b^2 d^2\right ) \int \tan ^{-1}(c x) \, dx}{c}+\frac{1}{3} \left (b^2 d e\right ) \int \frac{x^3}{1+c^2 x^2} \, dx+\frac{(b d e) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^3}-\frac{(b d e) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^3}+\frac{1}{15} \left (b^2 e^2\right ) \int \frac{x^5}{1+c^2 x^2} \, dx+\frac{\left (b e^2\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^3}-\frac{\left (b e^2\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^3}\\ &=-\frac{a b d^2 x}{c}+\frac{a b d e x}{c^3}-\frac{b^2 d^2 x \tan ^{-1}(c x)}{c}-\frac{b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4}+\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{2} d e x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\left (b^2 d^2\right ) \int \frac{x}{1+c^2 x^2} \, dx+\frac{1}{6} \left (b^2 d e\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )+\frac{\left (b^2 d e\right ) \int \tan ^{-1}(c x) \, dx}{c^3}+\frac{1}{30} \left (b^2 e^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+c^2 x} \, dx,x,x^2\right )-\frac{\left (b e^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^5}+\frac{\left (b e^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^5}-\frac{\left (b^2 e^2\right ) \int \frac{x^3}{1+c^2 x^2} \, dx}{9 c^2}\\ &=-\frac{a b d^2 x}{c}+\frac{a b d e x}{c^3}-\frac{a b e^2 x}{3 c^5}-\frac{b^2 d^2 x \tan ^{-1}(c x)}{c}+\frac{b^2 d e x \tan ^{-1}(c x)}{c^3}-\frac{b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4}+\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{2} d e x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{b^2 d^2 \log \left (1+c^2 x^2\right )}{2 c^2}+\frac{1}{6} \left (b^2 d e\right ) \operatorname{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{\left (b^2 d e\right ) \int \frac{x}{1+c^2 x^2} \, dx}{c^2}+\frac{1}{30} \left (b^2 e^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}+\frac{x}{c^2}+\frac{1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{\left (b^2 e^2\right ) \int \tan ^{-1}(c x) \, dx}{3 c^5}-\frac{\left (b^2 e^2\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )}{18 c^2}\\ &=-\frac{a b d^2 x}{c}+\frac{a b d e x}{c^3}-\frac{a b e^2 x}{3 c^5}+\frac{b^2 d e x^2}{6 c^2}-\frac{b^2 e^2 x^2}{30 c^4}+\frac{b^2 e^2 x^4}{60 c^2}-\frac{b^2 d^2 x \tan ^{-1}(c x)}{c}+\frac{b^2 d e x \tan ^{-1}(c x)}{c^3}-\frac{b^2 e^2 x \tan ^{-1}(c x)}{3 c^5}-\frac{b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4}+\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{2} d e x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{b^2 d^2 \log \left (1+c^2 x^2\right )}{2 c^2}-\frac{2 b^2 d e \log \left (1+c^2 x^2\right )}{3 c^4}+\frac{b^2 e^2 \log \left (1+c^2 x^2\right )}{30 c^6}+\frac{\left (b^2 e^2\right ) \int \frac{x}{1+c^2 x^2} \, dx}{3 c^4}-\frac{\left (b^2 e^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{18 c^2}\\ &=-\frac{a b d^2 x}{c}+\frac{a b d e x}{c^3}-\frac{a b e^2 x}{3 c^5}+\frac{b^2 d e x^2}{6 c^2}-\frac{4 b^2 e^2 x^2}{45 c^4}+\frac{b^2 e^2 x^4}{60 c^2}-\frac{b^2 d^2 x \tan ^{-1}(c x)}{c}+\frac{b^2 d e x \tan ^{-1}(c x)}{c^3}-\frac{b^2 e^2 x \tan ^{-1}(c x)}{3 c^5}-\frac{b d e x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{b e^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac{b e^2 x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2}-\frac{d e \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4}+\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{2} d e x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{6} e^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{b^2 d^2 \log \left (1+c^2 x^2\right )}{2 c^2}-\frac{2 b^2 d e \log \left (1+c^2 x^2\right )}{3 c^4}+\frac{23 b^2 e^2 \log \left (1+c^2 x^2\right )}{90 c^6}\\ \end{align*}
Mathematica [A] time = 0.315931, size = 317, normalized size = 0.83 \[ \frac{c x \left (30 a^2 c^5 x \left (3 d^2+3 d e x^2+e^2 x^4\right )-4 a b \left (3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )-5 c^2 e \left (9 d+e x^2\right )+15 e^2\right )+b^2 c e x \left (3 c^2 \left (10 d+e x^2\right )-16 e\right )\right )+4 b \tan ^{-1}(c x) \left (15 a \left (c^6 \left (3 d^2 x^2+3 d e x^4+e^2 x^6\right )+3 c^4 d^2-3 c^2 d e+e^2\right )-b c x \left (3 c^4 \left (15 d^2+5 d e x^2+e^2 x^4\right )-5 c^2 e \left (9 d+e x^2\right )+15 e^2\right )\right )+2 b^2 \left (45 c^4 d^2-60 c^2 d e+23 e^2\right ) \log \left (c^2 x^2+1\right )+30 b^2 \tan ^{-1}(c x)^2 \left (c^6 \left (3 d^2 x^2+3 d e x^4+e^2 x^6\right )+3 c^4 d^2-3 c^2 d e+e^2\right )}{180 c^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 484, normalized size = 1.3 \begin{align*} -{\frac{{b}^{2}\arctan \left ( cx \right ){x}^{5}{e}^{2}}{15\,c}}+ab\arctan \left ( cx \right ){x}^{2}{d}^{2}+{\frac{{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}{x}^{4}de}{2}}+{\frac{ab\arctan \left ( cx \right ){e}^{2}{x}^{6}}{3}}+{\frac{ab{x}^{3}{e}^{2}}{9\,{c}^{3}}}+{\frac{ab\arctan \left ( cx \right ){e}^{2}}{3\,{c}^{6}}}+{\frac{{b}^{2}\arctan \left ( cx \right ){x}^{3}{e}^{2}}{9\,{c}^{3}}}-{\frac{ab{x}^{5}{e}^{2}}{15\,c}}-{\frac{{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}de}{2\,{c}^{4}}}+{\frac{ab\arctan \left ( cx \right ){d}^{2}}{{c}^{2}}}+{\frac{23\,{b}^{2}{e}^{2}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{90\,{c}^{6}}}-{\frac{4\,{b}^{2}{e}^{2}{x}^{2}}{45\,{c}^{4}}}+{\frac{{b}^{2}{e}^{2}{x}^{4}}{60\,{c}^{2}}}+{\frac{{a}^{2}{x}^{2}{d}^{2}}{2}}+{\frac{{a}^{2}{e}^{2}{x}^{6}}{6}}-{\frac{{b}^{2}\arctan \left ( cx \right ){x}^{3}de}{3\,c}}-{\frac{ab{x}^{3}de}{3\,c}}+ab\arctan \left ( cx \right ){x}^{4}de-{\frac{ab\arctan \left ( cx \right ) de}{{c}^{4}}}+{\frac{{a}^{2}{x}^{4}de}{2}}+{\frac{{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}{e}^{2}{x}^{6}}{6}}+{\frac{{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}{x}^{2}{d}^{2}}{2}}+{\frac{{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}{d}^{2}}{2\,{c}^{2}}}+{\frac{{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}{e}^{2}}{6\,{c}^{6}}}-{\frac{ab{d}^{2}x}{c}}-{\frac{{b}^{2}{d}^{2}x\arctan \left ( cx \right ) }{c}}+{\frac{{b}^{2}{d}^{2}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,{c}^{2}}}-{\frac{ab{e}^{2}x}{3\,{c}^{5}}}+{\frac{{b}^{2}de{x}^{2}}{6\,{c}^{2}}}-{\frac{{b}^{2}{e}^{2}x\arctan \left ( cx \right ) }{3\,{c}^{5}}}-{\frac{2\,{b}^{2}de\ln \left ({c}^{2}{x}^{2}+1 \right ) }{3\,{c}^{4}}}+{\frac{abdex}{{c}^{3}}}+{\frac{{b}^{2}dex\arctan \left ( cx \right ) }{{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.62129, size = 585, normalized size = 1.54 \begin{align*} \frac{1}{6} \, b^{2} e^{2} x^{6} \arctan \left (c x\right )^{2} + \frac{1}{6} \, a^{2} e^{2} x^{6} + \frac{1}{2} \, b^{2} d e x^{4} \arctan \left (c x\right )^{2} + \frac{1}{2} \, a^{2} d e x^{4} + \frac{1}{2} \, b^{2} d^{2} x^{2} \arctan \left (c x\right )^{2} + \frac{1}{2} \, a^{2} d^{2} x^{2} +{\left (x^{2} \arctan \left (c x\right ) - c{\left (\frac{x}{c^{2}} - \frac{\arctan \left (c x\right )}{c^{3}}\right )}\right )} a b d^{2} - \frac{1}{2} \,{\left (2 \, c{\left (\frac{x}{c^{2}} - \frac{\arctan \left (c x\right )}{c^{3}}\right )} \arctan \left (c x\right ) + \frac{\arctan \left (c x\right )^{2} - \log \left (c^{2} x^{2} + 1\right )}{c^{2}}\right )} b^{2} d^{2} + \frac{1}{3} \,{\left (3 \, x^{4} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} a b d e - \frac{1}{6} \,{\left (2 \, c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\right )}{c^{5}}\right )} \arctan \left (c x\right ) - \frac{c^{2} x^{2} + 3 \, \arctan \left (c x\right )^{2} - 4 \, \log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )} b^{2} d e + \frac{1}{45} \,{\left (15 \, x^{6} \arctan \left (c x\right ) - c{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} a b e^{2} - \frac{1}{180} \,{\left (4 \, c{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )} \arctan \left (c x\right ) - \frac{3 \, c^{4} x^{4} - 16 \, c^{2} x^{2} - 30 \, \arctan \left (c x\right )^{2} + 46 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )} b^{2} e^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30814, size = 894, normalized size = 2.35 \begin{align*} \frac{30 \, a^{2} c^{6} e^{2} x^{6} - 12 \, a b c^{5} e^{2} x^{5} + 3 \,{\left (30 \, a^{2} c^{6} d e + b^{2} c^{4} e^{2}\right )} x^{4} - 20 \,{\left (3 \, a b c^{5} d e - a b c^{3} e^{2}\right )} x^{3} + 2 \,{\left (45 \, a^{2} c^{6} d^{2} + 15 \, b^{2} c^{4} d e - 8 \, b^{2} c^{2} e^{2}\right )} x^{2} + 30 \,{\left (b^{2} c^{6} e^{2} x^{6} + 3 \, b^{2} c^{6} d e x^{4} + 3 \, b^{2} c^{6} d^{2} x^{2} + 3 \, b^{2} c^{4} d^{2} - 3 \, b^{2} c^{2} d e + b^{2} e^{2}\right )} \arctan \left (c x\right )^{2} - 60 \,{\left (3 \, a b c^{5} d^{2} - 3 \, a b c^{3} d e + a b c e^{2}\right )} x + 4 \,{\left (15 \, a b c^{6} e^{2} x^{6} + 45 \, a b c^{6} d e x^{4} - 3 \, b^{2} c^{5} e^{2} x^{5} + 45 \, a b c^{6} d^{2} x^{2} + 45 \, a b c^{4} d^{2} - 45 \, a b c^{2} d e + 15 \, a b e^{2} - 5 \,{\left (3 \, b^{2} c^{5} d e - b^{2} c^{3} e^{2}\right )} x^{3} - 15 \,{\left (3 \, b^{2} c^{5} d^{2} - 3 \, b^{2} c^{3} d e + b^{2} c e^{2}\right )} x\right )} \arctan \left (c x\right ) + 2 \,{\left (45 \, b^{2} c^{4} d^{2} - 60 \, b^{2} c^{2} d e + 23 \, b^{2} e^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{180 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.65463, size = 575, normalized size = 1.51 \begin{align*} \begin{cases} \frac{a^{2} d^{2} x^{2}}{2} + \frac{a^{2} d e x^{4}}{2} + \frac{a^{2} e^{2} x^{6}}{6} + a b d^{2} x^{2} \operatorname{atan}{\left (c x \right )} + a b d e x^{4} \operatorname{atan}{\left (c x \right )} + \frac{a b e^{2} x^{6} \operatorname{atan}{\left (c x \right )}}{3} - \frac{a b d^{2} x}{c} - \frac{a b d e x^{3}}{3 c} - \frac{a b e^{2} x^{5}}{15 c} + \frac{a b d^{2} \operatorname{atan}{\left (c x \right )}}{c^{2}} + \frac{a b d e x}{c^{3}} + \frac{a b e^{2} x^{3}}{9 c^{3}} - \frac{a b d e \operatorname{atan}{\left (c x \right )}}{c^{4}} - \frac{a b e^{2} x}{3 c^{5}} + \frac{a b e^{2} \operatorname{atan}{\left (c x \right )}}{3 c^{6}} + \frac{b^{2} d^{2} x^{2} \operatorname{atan}^{2}{\left (c x \right )}}{2} + \frac{b^{2} d e x^{4} \operatorname{atan}^{2}{\left (c x \right )}}{2} + \frac{b^{2} e^{2} x^{6} \operatorname{atan}^{2}{\left (c x \right )}}{6} - \frac{b^{2} d^{2} x \operatorname{atan}{\left (c x \right )}}{c} - \frac{b^{2} d e x^{3} \operatorname{atan}{\left (c x \right )}}{3 c} - \frac{b^{2} e^{2} x^{5} \operatorname{atan}{\left (c x \right )}}{15 c} + \frac{b^{2} d^{2} \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{2 c^{2}} + \frac{b^{2} d^{2} \operatorname{atan}^{2}{\left (c x \right )}}{2 c^{2}} + \frac{b^{2} d e x^{2}}{6 c^{2}} + \frac{b^{2} e^{2} x^{4}}{60 c^{2}} + \frac{b^{2} d e x \operatorname{atan}{\left (c x \right )}}{c^{3}} + \frac{b^{2} e^{2} x^{3} \operatorname{atan}{\left (c x \right )}}{9 c^{3}} - \frac{2 b^{2} d e \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{3 c^{4}} - \frac{b^{2} d e \operatorname{atan}^{2}{\left (c x \right )}}{2 c^{4}} - \frac{4 b^{2} e^{2} x^{2}}{45 c^{4}} - \frac{b^{2} e^{2} x \operatorname{atan}{\left (c x \right )}}{3 c^{5}} + \frac{23 b^{2} e^{2} \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{90 c^{6}} + \frac{b^{2} e^{2} \operatorname{atan}^{2}{\left (c x \right )}}{6 c^{6}} & \text{for}\: c \neq 0 \\a^{2} \left (\frac{d^{2} x^{2}}{2} + \frac{d e x^{4}}{2} + \frac{e^{2} x^{6}}{6}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.62472, size = 716, normalized size = 1.88 \begin{align*} \frac{30 \, b^{2} c^{6} x^{6} \arctan \left (c x\right )^{2} e^{2} + 60 \, a b c^{6} x^{6} \arctan \left (c x\right ) e^{2} + 90 \, b^{2} c^{6} d x^{4} \arctan \left (c x\right )^{2} e + 30 \, a^{2} c^{6} x^{6} e^{2} + 180 \, a b c^{6} d x^{4} \arctan \left (c x\right ) e + 90 \, b^{2} c^{6} d^{2} x^{2} \arctan \left (c x\right )^{2} - 12 \, b^{2} c^{5} x^{5} \arctan \left (c x\right ) e^{2} + 90 \, a^{2} c^{6} d x^{4} e + 180 \, a b c^{6} d^{2} x^{2} \arctan \left (c x\right ) - 12 \, a b c^{5} x^{5} e^{2} - 60 \, b^{2} c^{5} d x^{3} \arctan \left (c x\right ) e + 90 \, a^{2} c^{6} d^{2} x^{2} - 60 \, a b c^{5} d x^{3} e - 180 \, b^{2} c^{5} d^{2} x \arctan \left (c x\right ) + 3 \, b^{2} c^{4} x^{4} e^{2} - 180 \, \pi a b c^{4} d^{2} \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 180 \, a b c^{5} d^{2} x + 90 \, b^{2} c^{4} d^{2} \arctan \left (c x\right )^{2} + 20 \, b^{2} c^{3} x^{3} \arctan \left (c x\right ) e^{2} + 30 \, b^{2} c^{4} d x^{2} e + 180 \, a b c^{4} d^{2} \arctan \left (c x\right ) + 20 \, a b c^{3} x^{3} e^{2} + 180 \, b^{2} c^{3} d x \arctan \left (c x\right ) e + 90 \, b^{2} c^{4} d^{2} \log \left (c^{2} x^{2} + 1\right ) + 180 \, a b c^{3} d x e - 90 \, b^{2} c^{2} d \arctan \left (c x\right )^{2} e - 16 \, b^{2} c^{2} x^{2} e^{2} - 180 \, a b c^{2} d \arctan \left (c x\right ) e - 120 \, b^{2} c^{2} d e \log \left (c^{2} x^{2} + 1\right ) - 60 \, b^{2} c x \arctan \left (c x\right ) e^{2} - 60 \, \pi a b e^{2} \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 60 \, a b c x e^{2} + 30 \, b^{2} \arctan \left (c x\right )^{2} e^{2} + 60 \, a b \arctan \left (c x\right ) e^{2} + 46 \, b^{2} e^{2} \log \left (c^{2} x^{2} + 1\right )}{180 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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