Optimal. Leaf size=225 \[ -\frac{8 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{105 a^3}+\frac{1}{105} a^2 c^2 x^5+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2-\frac{c^2 x}{210 a^2}-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{c^2 \tan ^{-1}(a x)}{210 a^3}-\frac{16 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{105 a^3}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}+\frac{17 c^2 x^3}{630} \]
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Rubi [A] time = 0.752474, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 44, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {4948, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302} \[ -\frac{8 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{105 a^3}+\frac{1}{105} a^2 c^2 x^5+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2-\frac{c^2 x}{210 a^2}-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{c^2 \tan ^{-1}(a x)}{210 a^3}-\frac{16 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{105 a^3}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}+\frac{17 c^2 x^3}{630} \]
Antiderivative was successfully verified.
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Rule 4948
Rule 4852
Rule 4916
Rule 321
Rule 203
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 302
Rubi steps
\begin{align*} \int x^2 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx &=\int \left (c^2 x^2 \tan ^{-1}(a x)^2+2 a^2 c^2 x^4 \tan ^{-1}(a x)^2+a^4 c^2 x^6 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^2 \int x^2 \tan ^{-1}(a x)^2 \, dx+\left (2 a^2 c^2\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx+\left (a^4 c^2\right ) \int x^6 \tan ^{-1}(a x)^2 \, dx\\ &=\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{1}{3} \left (2 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (4 a^3 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{7} \left (2 a^5 c^2\right ) \int \frac{x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{\left (2 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{3 a}+\frac{\left (2 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}-\frac{1}{5} \left (4 a c^2\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac{1}{5} \left (4 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{7} \left (2 a^3 c^2\right ) \int x^5 \tan ^{-1}(a x) \, dx+\frac{1}{7} \left (2 a^3 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{c^2 x^2 \tan ^{-1}(a x)}{3 a}-\frac{1}{5} a c^2 x^4 \tan ^{-1}(a x)-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac{i c^2 \tan ^{-1}(a x)^2}{3 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2+\frac{1}{3} c^2 \int \frac{x^2}{1+a^2 x^2} \, dx-\frac{\left (2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{3 a^2}+\frac{\left (4 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{5 a}-\frac{\left (4 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a}+\frac{1}{7} \left (2 a c^2\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac{1}{7} \left (2 a c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{5} \left (a^2 c^2\right ) \int \frac{x^4}{1+a^2 x^2} \, dx+\frac{1}{21} \left (a^4 c^2\right ) \int \frac{x^6}{1+a^2 x^2} \, dx\\ &=\frac{c^2 x}{3 a^2}+\frac{c^2 x^2 \tan ^{-1}(a x)}{15 a}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)+\frac{i c^2 \tan ^{-1}(a x)^2}{15 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{2 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{3 a^3}-\frac{1}{5} \left (2 c^2\right ) \int \frac{x^2}{1+a^2 x^2} \, dx-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{3 a^2}+\frac{\left (2 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}+\frac{\left (4 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^2}-\frac{\left (2 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{7 a}+\frac{\left (2 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{7 a}-\frac{1}{14} \left (a^2 c^2\right ) \int \frac{x^4}{1+a^2 x^2} \, dx+\frac{1}{5} \left (a^2 c^2\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx+\frac{1}{21} \left (a^4 c^2\right ) \int \left (\frac{1}{a^6}-\frac{x^2}{a^4}+\frac{x^4}{a^2}-\frac{1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=-\frac{23 c^2 x}{105 a^2}+\frac{16 c^2 x^3}{315}+\frac{1}{105} a^2 c^2 x^5-\frac{c^2 \tan ^{-1}(a x)}{3 a^3}-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2+\frac{2 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^3}+\frac{1}{7} c^2 \int \frac{x^2}{1+a^2 x^2} \, dx-\frac{\left (2 i c^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{3 a^3}-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{21 a^2}+\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{5 a^2}-\frac{\left (2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{7 a^2}+\frac{\left (2 c^2\right ) \int \frac{1}{1+a^2 x^2} \, dx}{5 a^2}-\frac{\left (4 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}-\frac{1}{14} \left (a^2 c^2\right ) \int \left (-\frac{1}{a^4}+\frac{x^2}{a^2}+\frac{1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=-\frac{c^2 x}{210 a^2}+\frac{17 c^2 x^3}{630}+\frac{1}{105} a^2 c^2 x^5+\frac{23 c^2 \tan ^{-1}(a x)}{105 a^3}-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{16 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{105 a^3}-\frac{i c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{3 a^3}+\frac{\left (4 i c^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{5 a^3}-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{14 a^2}-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{7 a^2}+\frac{\left (2 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}\\ &=-\frac{c^2 x}{210 a^2}+\frac{17 c^2 x^3}{630}+\frac{1}{105} a^2 c^2 x^5+\frac{c^2 \tan ^{-1}(a x)}{210 a^3}-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{16 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{105 a^3}+\frac{i c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{15 a^3}-\frac{\left (2 i c^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{7 a^3}\\ &=-\frac{c^2 x}{210 a^2}+\frac{17 c^2 x^3}{630}+\frac{1}{105} a^2 c^2 x^5+\frac{c^2 \tan ^{-1}(a x)}{210 a^3}-\frac{8 c^2 x^2 \tan ^{-1}(a x)}{105 a}-\frac{9}{70} a c^2 x^4 \tan ^{-1}(a x)-\frac{1}{21} a^3 c^2 x^6 \tan ^{-1}(a x)-\frac{8 i c^2 \tan ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} c^2 x^3 \tan ^{-1}(a x)^2+\frac{2}{5} a^2 c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{7} a^4 c^2 x^7 \tan ^{-1}(a x)^2-\frac{16 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{105 a^3}-\frac{8 i c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{105 a^3}\\ \end{align*}
Mathematica [A] time = 1.24216, size = 133, normalized size = 0.59 \[ \frac{c^2 \left (48 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+a x \left (6 a^4 x^4+17 a^2 x^2-3\right )+6 \left (15 a^7 x^7+42 a^5 x^5+35 a^3 x^3+8 i\right ) \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \left (10 a^6 x^6+27 a^4 x^4+16 a^2 x^2+32 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-1\right )\right )}{630 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.092, size = 333, normalized size = 1.5 \begin{align*}{\frac{{a}^{4}{c}^{2}{x}^{7} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{7}}+{\frac{2\,{a}^{2}{c}^{2}{x}^{5} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{5}}+{\frac{{c}^{2}{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3}}-{\frac{{a}^{3}{c}^{2}{x}^{6}\arctan \left ( ax \right ) }{21}}-{\frac{9\,a{c}^{2}{x}^{4}\arctan \left ( ax \right ) }{70}}-{\frac{8\,{c}^{2}{x}^{2}\arctan \left ( ax \right ) }{105\,a}}+{\frac{8\,{c}^{2}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{105\,{a}^{3}}}+{\frac{{a}^{2}{c}^{2}{x}^{5}}{105}}+{\frac{17\,{c}^{2}{x}^{3}}{630}}-{\frac{{c}^{2}x}{210\,{a}^{2}}}+{\frac{{c}^{2}\arctan \left ( ax \right ) }{210\,{a}^{3}}}+{\frac{{\frac{4\,i}{105}}{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax-i \right ) }{{a}^{3}}}+{\frac{{\frac{4\,i}{105}}{c}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}}}+{\frac{{\frac{4\,i}{105}}{c}^{2}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}}}-{\frac{{\frac{2\,i}{105}}{c}^{2} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{3}}}+{\frac{{\frac{2\,i}{105}}{c}^{2} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{3}}}-{\frac{{\frac{4\,i}{105}}{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) }{{a}^{3}}}-{\frac{{\frac{4\,i}{105}}{c}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}}}-{\frac{{\frac{4\,i}{105}}{c}^{2}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{420} \,{\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} x^{3}\right )} \arctan \left (a x\right )^{2} - \frac{1}{1680} \,{\left (15 \, a^{4} c^{2} x^{7} + 42 \, a^{2} c^{2} x^{5} + 35 \, c^{2} x^{3}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{1260 \,{\left (a^{6} c^{2} x^{8} + 3 \, a^{4} c^{2} x^{6} + 3 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )^{2} + 105 \,{\left (a^{6} c^{2} x^{8} + 3 \, a^{4} c^{2} x^{6} + 3 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} - 8 \,{\left (15 \, a^{5} c^{2} x^{7} + 42 \, a^{3} c^{2} x^{5} + 35 \, a c^{2} x^{3}\right )} \arctan \left (a x\right ) + 4 \,{\left (15 \, a^{6} c^{2} x^{8} + 42 \, a^{4} c^{2} x^{6} + 35 \, a^{2} c^{2} x^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{1680 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{6} + 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \arctan \left (a x\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \left (\int x^{2} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int 2 a^{2} x^{4} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int a^{4} x^{6} \operatorname{atan}^{2}{\left (a x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{2} x^{2} \arctan \left (a x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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