Optimal. Leaf size=124 \[ -\frac{c x^2}{180 a^2}-\frac{7 c \log \left (a^2 x^2+1\right )}{90 a^4}+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^2+\frac{c x \tan ^{-1}(a x)}{6 a^3}-\frac{c \tan ^{-1}(a x)^2}{12 a^4}-\frac{1}{15} a c x^5 \tan ^{-1}(a x)+\frac{1}{4} c x^4 \tan ^{-1}(a x)^2-\frac{c x^3 \tan ^{-1}(a x)}{18 a}+\frac{c x^4}{60} \]
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Rubi [A] time = 0.425179, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4950, 4852, 4916, 266, 43, 4846, 260, 4884} \[ -\frac{c x^2}{180 a^2}-\frac{7 c \log \left (a^2 x^2+1\right )}{90 a^4}+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^2+\frac{c x \tan ^{-1}(a x)}{6 a^3}-\frac{c \tan ^{-1}(a x)^2}{12 a^4}-\frac{1}{15} a c x^5 \tan ^{-1}(a x)+\frac{1}{4} c x^4 \tan ^{-1}(a x)^2-\frac{c x^3 \tan ^{-1}(a x)}{18 a}+\frac{c x^4}{60} \]
Antiderivative was successfully verified.
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Rule 4950
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 ) \tan ^{-1}(a x)^2 \, dx &=c \int x^3 \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^5 \tan ^{-1}(a x)^2 \, dx\\ &=\frac{1}{4} c x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^2-\frac{1}{2} (a c) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{3} \left (a^3 c\right ) \int \frac{x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} c x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^2-\frac{c \int x^2 \tan ^{-1}(a x) \, dx}{2 a}+\frac{c \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a}-\frac{1}{3} (a c) \int x^4 \tan ^{-1}(a x) \, dx+\frac{1}{3} (a c) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{c x^3 \tan ^{-1}(a x)}{6 a}-\frac{1}{15} a c x^5 \tan ^{-1}(a x)+\frac{1}{4} c x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^2+\frac{1}{6} c \int \frac{x^3}{1+a^2 x^2} \, dx+\frac{c \int \tan ^{-1}(a x) \, dx}{2 a^3}-\frac{c \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^3}+\frac{c \int x^2 \tan ^{-1}(a x) \, dx}{3 a}-\frac{c \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}+\frac{1}{15} \left (a^2 c\right ) \int \frac{x^5}{1+a^2 x^2} \, dx\\ &=\frac{c x \tan ^{-1}(a x)}{2 a^3}-\frac{c x^3 \tan ^{-1}(a x)}{18 a}-\frac{1}{15} a c x^5 \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)^2}{4 a^4}+\frac{1}{4} c x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^2+\frac{1}{12} c \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )-\frac{1}{9} c \int \frac{x^3}{1+a^2 x^2} \, dx-\frac{c \int \tan ^{-1}(a x) \, dx}{3 a^3}+\frac{c \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^3}-\frac{c \int \frac{x}{1+a^2 x^2} \, dx}{2 a^2}+\frac{1}{30} \left (a^2 c\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac{c x \tan ^{-1}(a x)}{6 a^3}-\frac{c x^3 \tan ^{-1}(a x)}{18 a}-\frac{1}{15} a c x^5 \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)^2}{12 a^4}+\frac{1}{4} c x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^2-\frac{c \log \left (1+a^2 x^2\right )}{4 a^4}-\frac{1}{18} c \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )+\frac{1}{12} c \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 \int \frac{x}{1+a^2 x^2} \, dx}{3 a^2}+\frac{1}{30} \left (a^2 c\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{c x^2}{20 a^2}+\frac{c x^4}{60}+\frac{c x \tan ^{-1}(a x)}{6 a^3}-\frac{c x^3 \tan ^{-1}(a x)}{18 a}-\frac{1}{15} a c x^5 \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)^2}{12 a^4}+\frac{1}{4} c x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^2-\frac{2 c \log \left (1+a^2 x^2\right )}{15 a^4}-\frac{1}{18} c \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 x^2}{180 a^2}+\frac{c x^4}{60}+\frac{c x \tan ^{-1}(a x)}{6 a^3}-\frac{c x^3 \tan ^{-1}(a x)}{18 a}-\frac{1}{15} a c x^5 \tan ^{-1}(a x)-\frac{c \tan ^{-1}(a x)^2}{12 a^4}+\frac{1}{4} c x^4 \tan ^{-1}(a x)^2+\frac{1}{6} a^2 c x^6 \tan ^{-1}(a x)^2-\frac{7 c \log \left (1+a^2 x^2\right )}{90 a^4}\\ \end{align*}
Mathematica [A] time = 0.0382577, size = 89, normalized size = 0.72 \[ \frac{c \left (3 a^4 x^4-a^2 x^2-14 \log \left (a^2 x^2+1\right )-2 a x \left (6 a^4 x^4+5 a^2 x^2-15\right ) \tan ^{-1}(a x)+15 \left (2 a^6 x^6+3 a^4 x^4-1\right ) \tan ^{-1}(a x)^2\right )}{180 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 107, normalized size = 0.9 \begin{align*} -{\frac{c{x}^{2}}{180\,{a}^{2}}}+{\frac{c{x}^{4}}{60}}+{\frac{cx\arctan \left ( ax \right ) }{6\,{a}^{3}}}-{\frac{c{x}^{3}\arctan \left ( ax \right ) }{18\,a}}-{\frac{ac{x}^{5}\arctan \left ( ax \right ) }{15}}-{\frac{c \left ( \arctan \left ( ax \right ) \right ) ^{2}}{12\,{a}^{4}}}+{\frac{c{x}^{4} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4}}+{\frac{{a}^{2}c{x}^{6} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{6}}-{\frac{7\,c\ln \left ({a}^{2}{x}^{2}+1 \right ) }{90\,{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52954, size = 157, normalized size = 1.27 \begin{align*} -\frac{1}{90} \, a{\left (\frac{6 \, a^{4} c x^{5} + 5 \, a^{2} c x^{3} - 15 \, c x}{a^{4}} + \frac{15 \, c \arctan \left (a x\right )}{a^{5}}\right )} \arctan \left (a x\right ) + \frac{1}{12} \,{\left (2 \, a^{2} c x^{6} + 3 \, c x^{4}\right )} \arctan \left (a x\right )^{2} + \frac{3 \, a^{4} c x^{4} - a^{2} c x^{2} + 15 \, c \arctan \left (a x\right )^{2} - 14 \, c \log \left (a^{2} x^{2} + 1\right )}{180 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.2027, size = 225, normalized size = 1.81 \begin{align*} \frac{3 \, a^{4} c x^{4} - a^{2} c x^{2} + 15 \,{\left (2 \, a^{6} c x^{6} + 3 \, a^{4} c x^{4} - c\right )} \arctan \left (a x\right )^{2} - 2 \,{\left (6 \, a^{5} c x^{5} + 5 \, a^{3} c x^{3} - 15 \, a c x\right )} \arctan \left (a x\right ) - 14 \, c \log \left (a^{2} x^{2} + 1\right )}{180 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.808, size = 121, normalized size = 0.98 \begin{align*} \begin{cases} \frac{a^{2} c x^{6} \operatorname{atan}^{2}{\left (a x \right )}}{6} - \frac{a c x^{5} \operatorname{atan}{\left (a x \right )}}{15} + \frac{c x^{4} \operatorname{atan}^{2}{\left (a x \right )}}{4} + \frac{c x^{4}}{60} - \frac{c x^{3} \operatorname{atan}{\left (a x \right )}}{18 a} - \frac{c x^{2}}{180 a^{2}} + \frac{c x \operatorname{atan}{\left (a x \right )}}{6 a^{3}} - \frac{7 c \log{\left (x^{2} + \frac{1}{a^{2}} \right )}}{90 a^{4}} - \frac{c \operatorname{atan}^{2}{\left (a x \right )}}{12 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 = 1.15108, size = 142, normalized size = 1.15 \begin{align*} \frac{1}{12} \,{\left (2 \, a^{2} c x^{6} + 3 \, c x^{4}\right )} \arctan \left (a x\right )^{2} - \frac{12 \, a^{5} c x^{5} \arctan \left (a x\right ) - 3 \, a^{4} c x^{4} + 10 \, a^{3} c x^{3} \arctan \left (a x\right ) + a^{2} c x^{2} - 30 \, a c x \arctan \left (a x\right ) + 15 \, c \arctan \left (a x\right )^{2} + 14 \, c \log \left (a^{2} x^{2} + 1\right )}{180 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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