Optimal. Leaf size=191 \[ \frac{1}{168} a^2 c^2 x^6-\frac{5 c^2 x^2}{504 a^2}-\frac{2 c^2 \log \left (a^2 x^2+1\right )}{63 a^4}+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac{1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac{c^2 x \tan ^{-1}(a x)}{12 a^3}-\frac{c^2 \tan ^{-1}(a x)^2}{24 a^4}-\frac{1}{12} a c^2 x^5 \tan ^{-1}(a x)+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^2-\frac{c^2 x^3 \tan ^{-1}(a x)}{36 a}+\frac{c^2 x^4}{84} \]
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Rubi [A] time = 0.788643, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 47, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4948, 4852, 4916, 266, 43, 4846, 260, 4884} \[ \frac{1}{168} a^2 c^2 x^6-\frac{5 c^2 x^2}{504 a^2}-\frac{2 c^2 \log \left (a^2 x^2+1\right )}{63 a^4}+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac{1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac{c^2 x \tan ^{-1}(a x)}{12 a^3}-\frac{c^2 \tan ^{-1}(a x)^2}{24 a^4}-\frac{1}{12} a c^2 x^5 \tan ^{-1}(a x)+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^2-\frac{c^2 x^3 \tan ^{-1}(a x)}{36 a}+\frac{c^2 x^4}{84} \]
Antiderivative was successfully verified.
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Rule 4948
Rule 4852
Rule 4916
Rule 266
Rule 43
Rule 4846
Rule 260
Rule 4884
Rubi steps
\begin{align*} \int x^3 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^2 \, dx &=\int \left (c^2 x^3 \tan ^{-1}(a x)^2+2 a^2 c^2 x^5 \tan ^{-1}(a x)^2+a^4 c^2 x^7 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^2 \int x^3 \tan ^{-1}(a x)^2 \, dx+\left (2 a^2 c^2\right ) \int x^5 \tan ^{-1}(a x)^2 \, dx+\left (a^4 c^2\right ) \int x^7 \tan ^{-1}(a x)^2 \, dx\\ &=\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac{1}{2} \left (a c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{3} \left (2 a^3 c^2\right ) \int \frac{x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{4} \left (a^5 c^2\right ) \int \frac{x^8 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac{c^2 \int x^2 \tan ^{-1}(a x) \, dx}{2 a}+\frac{c^2 \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a}-\frac{1}{3} \left (2 a c^2\right ) \int x^4 \tan ^{-1}(a x) \, dx+\frac{1}{3} \left (2 a c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{4} \left (a^3 c^2\right ) \int x^6 \tan ^{-1}(a x) \, dx+\frac{1}{4} \left (a^3 c^2\right ) \int \frac{x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{c^2 x^3 \tan ^{-1}(a x)}{6 a}-\frac{2}{15} a c^2 x^5 \tan ^{-1}(a x)-\frac{1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2+\frac{1}{6} c^2 \int \frac{x^3}{1+a^2 x^2} \, dx+\frac{c^2 \int \tan ^{-1}(a x) \, dx}{2 a^3}-\frac{c^2 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^3}+\frac{\left (2 c^2\right ) \int x^2 \tan ^{-1}(a x) \, dx}{3 a}-\frac{\left (2 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}+\frac{1}{4} \left (a c^2\right ) \int x^4 \tan ^{-1}(a x) \, dx-\frac{1}{4} \left (a c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{15} \left (2 a^2 c^2\right ) \int \frac{x^5}{1+a^2 x^2} \, dx+\frac{1}{28} \left (a^4 c^2\right ) \int \frac{x^7}{1+a^2 x^2} \, dx\\ &=\frac{c^2 x \tan ^{-1}(a x)}{2 a^3}+\frac{c^2 x^3 \tan ^{-1}(a x)}{18 a}-\frac{1}{12} a c^2 x^5 \tan ^{-1}(a x)-\frac{1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)-\frac{c^2 \tan ^{-1}(a x)^2}{4 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2+\frac{1}{12} c^2 \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{9} \left (2 c^2\right ) \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{\left (2 c^2\right ) \int \tan ^{-1}(a x) \, dx}{3 a^3}+\frac{\left (2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^3}-\frac{c^2 \int \frac{x}{1+a^2 x^2} \, dx}{2 a^2}-\frac{c^2 \int x^2 \tan ^{-1}(a x) \, dx}{4 a}+\frac{c^2 \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a}-\frac{1}{20} \left (a^2 c^2\right ) \int \frac{x^5}{1+a^2 x^2} \, dx+\frac{1}{15} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{56} \left (a^4 c^2\right ) \operatorname{Subst}\left (\int \frac{x^3}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{c^2 x \tan ^{-1}(a x)}{6 a^3}-\frac{c^2 x^3 \tan ^{-1}(a x)}{36 a}-\frac{1}{12} a c^2 x^5 \tan ^{-1}(a x)-\frac{1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)+\frac{c^2 \tan ^{-1}(a x)^2}{12 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac{c^2 \log \left (1+a^2 x^2\right )}{4 a^4}+\frac{1}{12} c^2 \int \frac{x^3}{1+a^2 x^2} \, dx+\frac{1}{12} c^2 \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{1}{9} c^2 \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )+\frac{c^2 \int \tan ^{-1}(a x) \, dx}{4 a^3}-\frac{c^2 \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a^3}+\frac{\left (2 c^2\right ) \int \frac{x}{1+a^2 x^2} \, dx}{3 a^2}-\frac{1}{40} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{15} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac{1}{56} \left (a^4 c^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^6}-\frac{x}{a^4}+\frac{x^2}{a^2}-\frac{1}{a^6 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{29 c^2 x^2}{840 a^2}+\frac{41 c^2 x^4}{1680}+\frac{1}{168} a^2 c^2 x^6+\frac{c^2 x \tan ^{-1}(a x)}{12 a^3}-\frac{c^2 x^3 \tan ^{-1}(a x)}{36 a}-\frac{1}{12} a c^2 x^5 \tan ^{-1}(a x)-\frac{1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)-\frac{c^2 \tan ^{-1}(a x)^2}{24 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2+\frac{41 c^2 \log \left (1+a^2 x^2\right )}{840 a^4}+\frac{1}{24} c^2 \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{9} c^2 \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{c^2 \int \frac{x}{1+a^2 x^2} \, dx}{4 a^2}-\frac{1}{40} \left (a^2 c^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^4}+\frac{x}{a^2}+\frac{1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{13 c^2 x^2}{252 a^2}+\frac{c^2 x^4}{84}+\frac{1}{168} a^2 c^2 x^6+\frac{c^2 x \tan ^{-1}(a x)}{12 a^3}-\frac{c^2 x^3 \tan ^{-1}(a x)}{36 a}-\frac{1}{12} a c^2 x^5 \tan ^{-1}(a x)-\frac{1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)-\frac{c^2 \tan ^{-1}(a x)^2}{24 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2+\frac{5 c^2 \log \left (1+a^2 x^2\right )}{504 a^4}+\frac{1}{24} c^2 \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{5 c^2 x^2}{504 a^2}+\frac{c^2 x^4}{84}+\frac{1}{168} a^2 c^2 x^6+\frac{c^2 x \tan ^{-1}(a x)}{12 a^3}-\frac{c^2 x^3 \tan ^{-1}(a x)}{36 a}-\frac{1}{12} a c^2 x^5 \tan ^{-1}(a x)-\frac{1}{28} a^3 c^2 x^7 \tan ^{-1}(a x)-\frac{c^2 \tan ^{-1}(a x)^2}{24 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^2+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^2+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^2-\frac{2 c^2 \log \left (1+a^2 x^2\right )}{63 a^4}\\ \end{align*}
Mathematica [A] time = 0.0805569, size = 110, normalized size = 0.58 \[ \frac{c^2 \left (3 a^6 x^6+6 a^4 x^4-5 a^2 x^2-16 \log \left (a^2 x^2+1\right )-2 a x \left (9 a^6 x^6+21 a^4 x^4+7 a^2 x^2-21\right ) \tan ^{-1}(a x)+21 \left (a^2 x^2+1\right )^3 \left (3 a^2 x^2-1\right ) \tan ^{-1}(a x)^2\right )}{504 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 168, normalized size = 0.9 \begin{align*} -{\frac{5\,{c}^{2}{x}^{2}}{504\,{a}^{2}}}+{\frac{{c}^{2}{x}^{4}}{84}}+{\frac{{a}^{2}{c}^{2}{x}^{6}}{168}}+{\frac{{c}^{2}x\arctan \left ( ax \right ) }{12\,{a}^{3}}}-{\frac{{c}^{2}{x}^{3}\arctan \left ( ax \right ) }{36\,a}}-{\frac{a{c}^{2}{x}^{5}\arctan \left ( ax \right ) }{12}}-{\frac{{a}^{3}{c}^{2}{x}^{7}\arctan \left ( ax \right ) }{28}}-{\frac{{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{24\,{a}^{4}}}+{\frac{{c}^{2}{x}^{4} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4}}+{\frac{{a}^{2}{c}^{2}{x}^{6} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3}}+{\frac{{a}^{4}{c}^{2}{x}^{8} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{8}}-{\frac{2\,{c}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{63\,{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56325, size = 228, normalized size = 1.19 \begin{align*} -\frac{1}{252} \, a{\left (\frac{21 \, c^{2} \arctan \left (a x\right )}{a^{5}} + \frac{9 \, a^{6} c^{2} x^{7} + 21 \, a^{4} c^{2} x^{5} + 7 \, a^{2} c^{2} x^{3} - 21 \, c^{2} x}{a^{4}}\right )} \arctan \left (a x\right ) + \frac{1}{24} \,{\left (3 \, a^{4} c^{2} x^{8} + 8 \, a^{2} c^{2} x^{6} + 6 \, c^{2} x^{4}\right )} \arctan \left (a x\right )^{2} + \frac{3 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - 5 \, a^{2} c^{2} x^{2} + 21 \, c^{2} \arctan \left (a x\right )^{2} - 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{504 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17902, size = 319, normalized size = 1.67 \begin{align*} \frac{3 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - 5 \, a^{2} c^{2} x^{2} + 21 \,{\left (3 \, a^{8} c^{2} x^{8} + 8 \, a^{6} c^{2} x^{6} + 6 \, a^{4} c^{2} x^{4} - c^{2}\right )} \arctan \left (a x\right )^{2} - 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right ) - 2 \,{\left (9 \, a^{7} c^{2} x^{7} + 21 \, a^{5} c^{2} x^{5} + 7 \, a^{3} c^{2} x^{3} - 21 \, a c^{2} x\right )} \arctan \left (a x\right )}{504 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.81681, size = 185, normalized size = 0.97 \begin{align*} \begin{cases} \frac{a^{4} c^{2} x^{8} \operatorname{atan}^{2}{\left (a x \right )}}{8} - \frac{a^{3} c^{2} x^{7} \operatorname{atan}{\left (a x \right )}}{28} + \frac{a^{2} c^{2} x^{6} \operatorname{atan}^{2}{\left (a x \right )}}{3} + \frac{a^{2} c^{2} x^{6}}{168} - \frac{a c^{2} x^{5} \operatorname{atan}{\left (a x \right )}}{12} + \frac{c^{2} x^{4} \operatorname{atan}^{2}{\left (a x \right )}}{4} + \frac{c^{2} x^{4}}{84} - \frac{c^{2} x^{3} \operatorname{atan}{\left (a x \right )}}{36 a} - \frac{5 c^{2} x^{2}}{504 a^{2}} + \frac{c^{2} x \operatorname{atan}{\left (a x \right )}}{12 a^{3}} - \frac{2 c^{2} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{63 a^{4}} - \frac{c^{2} \operatorname{atan}^{2}{\left (a x \right )}}{24 a^{4}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18099, size = 217, normalized size = 1.14 \begin{align*} \frac{1}{24} \,{\left (3 \, a^{4} c^{2} x^{8} + 8 \, a^{2} c^{2} x^{6} + 6 \, c^{2} x^{4}\right )} \arctan \left (a x\right )^{2} - \frac{18 \, a^{7} c^{2} x^{7} \arctan \left (a x\right ) - 3 \, a^{6} c^{2} x^{6} + 42 \, a^{5} c^{2} x^{5} \arctan \left (a x\right ) - 6 \, a^{4} c^{2} x^{4} + 14 \, a^{3} c^{2} x^{3} \arctan \left (a x\right ) + 5 \, a^{2} c^{2} x^{2} - 42 \, a c^{2} x \arctan \left (a x\right ) + 21 \, c^{2} \arctan \left (a x\right )^{2} + 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right )}{504 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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