Optimal. Leaf size=136 \[ -\frac{1}{72} a^5 c^3 x^8-\frac{10}{189} a^3 c^3 x^6+\frac{8 c^3 \log \left (a^2 x^2+1\right )}{315 a^3}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)-\frac{89 a c^3 x^4}{1260}-\frac{8 c^3 x^2}{315 a}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x) \]
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Rubi [A] time = 0.233357, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4948, 4852, 266, 43} \[ -\frac{1}{72} a^5 c^3 x^8-\frac{10}{189} a^3 c^3 x^6+\frac{8 c^3 \log \left (a^2 x^2+1\right )}{315 a^3}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)-\frac{89 a c^3 x^4}{1260}-\frac{8 c^3 x^2}{315 a}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4948
Rule 4852
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x) \, dx &=\int \left (c^3 x^2 \tan ^{-1}(a x)+3 a^2 c^3 x^4 \tan ^{-1}(a x)+3 a^4 c^3 x^6 \tan ^{-1}(a x)+a^6 c^3 x^8 \tan ^{-1}(a x)\right ) \, dx\\ &=c^3 \int x^2 \tan ^{-1}(a x) \, dx+\left (3 a^2 c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx+\left (3 a^4 c^3\right ) \int x^6 \tan ^{-1}(a x) \, dx+\left (a^6 c^3\right ) \int x^8 \tan ^{-1}(a x) \, dx\\ &=\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac{1}{3} \left (a c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{1}{5} \left (3 a^3 c^3\right ) \int \frac{x^5}{1+a^2 x^2} \, dx-\frac{1}{7} \left (3 a^5 c^3\right ) \int \frac{x^7}{1+a^2 x^2} \, dx-\frac{1}{9} \left (a^7 c^3\right ) \int \frac{x^9}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac{1}{6} \left (a c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{10} \left (3 a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{14} \left (3 a^5 c^3\right ) \operatorname{Subst}\left (\int \frac{x^3}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{18} \left (a^7 c^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)-\frac{1}{6} \left (a c^3\right ) \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}{10} \left (3 a^3 c^3\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}{14} \left (3 a^5 c^3\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{1}{18} \left (a^7 c^3\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{a^8}+\frac{x}{a^6}-\frac{x^2}{a^4}+\frac{x^3}{a^2}+\frac{1}{a^8 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{8 c^3 x^2}{315 a}-\frac{89 a c^3 x^4}{1260}-\frac{10}{189} a^3 c^3 x^6-\frac{1}{72} a^5 c^3 x^8+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)+\frac{8 c^3 \log \left (1+a^2 x^2\right )}{315 a^3}\\ \end{align*}
Mathematica [A] time = 0.084463, size = 136, normalized size = 1. \[ -\frac{1}{72} a^5 c^3 x^8-\frac{10}{189} a^3 c^3 x^6+\frac{8 c^3 \log \left (a^2 x^2+1\right )}{315 a^3}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)-\frac{89 a c^3 x^4}{1260}-\frac{8 c^3 x^2}{315 a}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 119, normalized size = 0.9 \begin{align*} -{\frac{8\,{c}^{3}{x}^{2}}{315\,a}}-{\frac{89\,a{c}^{3}{x}^{4}}{1260}}-{\frac{10\,{a}^{3}{c}^{3}{x}^{6}}{189}}-{\frac{{a}^{5}{c}^{3}{x}^{8}}{72}}+{\frac{{c}^{3}{x}^{3}\arctan \left ( ax \right ) }{3}}+{\frac{3\,{a}^{2}{c}^{3}{x}^{5}\arctan \left ( ax \right ) }{5}}+{\frac{3\,{a}^{4}{c}^{3}{x}^{7}\arctan \left ( ax \right ) }{7}}+{\frac{{a}^{6}{c}^{3}{x}^{9}\arctan \left ( ax \right ) }{9}}+{\frac{8\,{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{315\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979308, size = 159, normalized size = 1.17 \begin{align*} \frac{1}{7560} \, a{\left (\frac{192 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{a^{4}} - \frac{105 \, a^{6} c^{3} x^{8} + 400 \, a^{4} c^{3} x^{6} + 534 \, a^{2} c^{3} x^{4} + 192 \, c^{3} x^{2}}{a^{2}}\right )} + \frac{1}{315} \,{\left (35 \, a^{6} c^{3} x^{9} + 135 \, a^{4} c^{3} x^{7} + 189 \, a^{2} c^{3} x^{5} + 105 \, c^{3} x^{3}\right )} \arctan \left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72335, size = 270, normalized size = 1.99 \begin{align*} -\frac{105 \, a^{8} c^{3} x^{8} + 400 \, a^{6} c^{3} x^{6} + 534 \, a^{4} c^{3} x^{4} + 192 \, a^{2} c^{3} x^{2} - 192 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) - 24 \,{\left (35 \, a^{9} c^{3} x^{9} + 135 \, a^{7} c^{3} x^{7} + 189 \, a^{5} c^{3} x^{5} + 105 \, a^{3} c^{3} x^{3}\right )} \arctan \left (a x\right )}{7560 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.32547, size = 138, normalized size = 1.01 \begin{align*} \begin{cases} \frac{a^{6} c^{3} x^{9} \operatorname{atan}{\left (a x \right )}}{9} - \frac{a^{5} c^{3} x^{8}}{72} + \frac{3 a^{4} c^{3} x^{7} \operatorname{atan}{\left (a x \right )}}{7} - \frac{10 a^{3} c^{3} x^{6}}{189} + \frac{3 a^{2} c^{3} x^{5} \operatorname{atan}{\left (a x \right )}}{5} - \frac{89 a c^{3} x^{4}}{1260} + \frac{c^{3} x^{3} \operatorname{atan}{\left (a x \right )}}{3} - \frac{8 c^{3} x^{2}}{315 a} + \frac{8 c^{3} \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{315 a^{3}} & \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 = 1.17765, size = 158, normalized size = 1.16 \begin{align*} \frac{1}{315} \,{\left (35 \, a^{6} c^{3} x^{9} + 135 \, a^{4} c^{3} x^{7} + 189 \, a^{2} c^{3} x^{5} + 105 \, c^{3} x^{3}\right )} \arctan \left (a x\right ) + \frac{8 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{315 \, a^{3}} - \frac{105 \, a^{13} c^{3} x^{8} + 400 \, a^{11} c^{3} x^{6} + 534 \, a^{9} c^{3} x^{4} + 192 \, a^{7} c^{3} x^{2}}{7560 \, a^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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