Optimal. Leaf size=87 \[ \frac{a^3 x^7}{56}+\frac{1}{8} a^4 x^8 \tanh ^{-1}(a x)-\frac{1}{3} a^2 x^6 \tanh ^{-1}(a x)+\frac{x}{24 a^3}-\frac{\tanh ^{-1}(a x)}{24 a^4}-\frac{a x^5}{24}+\frac{x^3}{72 a}+\frac{1}{4} x^4 \tanh ^{-1}(a x) \]
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Rubi [A] time = 0.137494, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6012, 5916, 302, 206} \[ \frac{a^3 x^7}{56}+\frac{1}{8} a^4 x^8 \tanh ^{-1}(a x)-\frac{1}{3} a^2 x^6 \tanh ^{-1}(a x)+\frac{x}{24 a^3}-\frac{\tanh ^{-1}(a x)}{24 a^4}-\frac{a x^5}{24}+\frac{x^3}{72 a}+\frac{1}{4} x^4 \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6012
Rule 5916
Rule 302
Rule 206
Rubi steps
\begin{align*} \int x^3 \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x) \, dx &=\int \left (x^3 \tanh ^{-1}(a x)-2 a^2 x^5 \tanh ^{-1}(a x)+a^4 x^7 \tanh ^{-1}(a x)\right ) \, dx\\ &=-\left (\left (2 a^2\right ) \int x^5 \tanh ^{-1}(a x) \, dx\right )+a^4 \int x^7 \tanh ^{-1}(a x) \, dx+\int x^3 \tanh ^{-1}(a x) \, dx\\ &=\frac{1}{4} x^4 \tanh ^{-1}(a x)-\frac{1}{3} a^2 x^6 \tanh ^{-1}(a x)+\frac{1}{8} a^4 x^8 \tanh ^{-1}(a x)-\frac{1}{4} a \int \frac{x^4}{1-a^2 x^2} \, dx+\frac{1}{3} a^3 \int \frac{x^6}{1-a^2 x^2} \, dx-\frac{1}{8} a^5 \int \frac{x^8}{1-a^2 x^2} \, dx\\ &=\frac{1}{4} x^4 \tanh ^{-1}(a x)-\frac{1}{3} a^2 x^6 \tanh ^{-1}(a x)+\frac{1}{8} a^4 x^8 \tanh ^{-1}(a x)-\frac{1}{4} a \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}{3} a^3 \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{1}{8} a^5 \int \left (-\frac{1}{a^8}-\frac{x^2}{a^6}-\frac{x^4}{a^4}-\frac{x^6}{a^2}+\frac{1}{a^8 \left (1-a^2 x^2\right )}\right ) \, dx\\ &=\frac{x}{24 a^3}+\frac{x^3}{72 a}-\frac{a x^5}{24}+\frac{a^3 x^7}{56}+\frac{1}{4} x^4 \tanh ^{-1}(a x)-\frac{1}{3} a^2 x^6 \tanh ^{-1}(a x)+\frac{1}{8} a^4 x^8 \tanh ^{-1}(a x)-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{8 a^3}-\frac{\int \frac{1}{1-a^2 x^2} \, dx}{4 a^3}+\frac{\int \frac{1}{1-a^2 x^2} \, dx}{3 a^3}\\ &=\frac{x}{24 a^3}+\frac{x^3}{72 a}-\frac{a x^5}{24}+\frac{a^3 x^7}{56}-\frac{\tanh ^{-1}(a x)}{24 a^4}+\frac{1}{4} x^4 \tanh ^{-1}(a x)-\frac{1}{3} a^2 x^6 \tanh ^{-1}(a x)+\frac{1}{8} a^4 x^8 \tanh ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0283403, size = 103, normalized size = 1.18 \[ \frac{a^3 x^7}{56}+\frac{1}{8} a^4 x^8 \tanh ^{-1}(a x)-\frac{1}{3} a^2 x^6 \tanh ^{-1}(a x)+\frac{x}{24 a^3}+\frac{\log (1-a x)}{48 a^4}-\frac{\log (a x+1)}{48 a^4}-\frac{a x^5}{24}+\frac{x^3}{72 a}+\frac{1}{4} x^4 \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 85, normalized size = 1. \begin{align*}{\frac{{a}^{4}{x}^{8}{\it Artanh} \left ( ax \right ) }{8}}-{\frac{{a}^{2}{x}^{6}{\it Artanh} \left ( ax \right ) }{3}}+{\frac{{x}^{4}{\it Artanh} \left ( ax \right ) }{4}}+{\frac{{a}^{3}{x}^{7}}{56}}-{\frac{a{x}^{5}}{24}}+{\frac{{x}^{3}}{72\,a}}+{\frac{x}{24\,{a}^{3}}}+{\frac{\ln \left ( ax-1 \right ) }{48\,{a}^{4}}}-{\frac{\ln \left ( ax+1 \right ) }{48\,{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943079, size = 119, normalized size = 1.37 \begin{align*} \frac{1}{1008} \, a{\left (\frac{2 \,{\left (9 \, a^{6} x^{7} - 21 \, a^{4} x^{5} + 7 \, a^{2} x^{3} + 21 \, x\right )}}{a^{4}} - \frac{21 \, \log \left (a x + 1\right )}{a^{5}} + \frac{21 \, \log \left (a x - 1\right )}{a^{5}}\right )} + \frac{1}{24} \,{\left (3 \, a^{4} x^{8} - 8 \, a^{2} x^{6} + 6 \, x^{4}\right )} \operatorname{artanh}\left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95539, size = 177, normalized size = 2.03 \begin{align*} \frac{18 \, a^{7} x^{7} - 42 \, a^{5} x^{5} + 14 \, a^{3} x^{3} + 42 \, a x + 21 \,{\left (3 \, a^{8} x^{8} - 8 \, a^{6} x^{6} + 6 \, a^{4} x^{4} - 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{1008 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.00134, size = 76, normalized size = 0.87 \begin{align*} \begin{cases} \frac{a^{4} x^{8} \operatorname{atanh}{\left (a x \right )}}{8} + \frac{a^{3} x^{7}}{56} - \frac{a^{2} x^{6} \operatorname{atanh}{\left (a x \right )}}{3} - \frac{a x^{5}}{24} + \frac{x^{4} \operatorname{atanh}{\left (a x \right )}}{4} + \frac{x^{3}}{72 a} + \frac{x}{24 a^{3}} - \frac{\operatorname{atanh}{\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 = 1.18255, size = 135, normalized size = 1.55 \begin{align*} \frac{1}{48} \,{\left (3 \, a^{4} x^{8} - 8 \, a^{2} x^{6} + 6 \, x^{4}\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - \frac{\log \left ({\left | a x + 1 \right |}\right )}{48 \, a^{4}} + \frac{\log \left ({\left | a x - 1 \right |}\right )}{48 \, a^{4}} + \frac{9 \, a^{17} x^{7} - 21 \, a^{15} x^{5} + 7 \, a^{13} x^{3} + 21 \, a^{11} x}{504 \, a^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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