Optimal. Leaf size=86 \[ \frac{3}{16 a^4 c^3 (1-a x)}+\frac{1}{16 a^4 c^3 (a x+1)}-\frac{1}{4 a^4 c^3 (1-a x)^2}+\frac{1}{12 a^4 c^3 (1-a x)^3}+\frac{\tanh ^{-1}(a x)}{8 a^4 c^3} \]
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Rubi [A] time = 0.12337, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6150, 88, 207} \[ \frac{3}{16 a^4 c^3 (1-a x)}+\frac{1}{16 a^4 c^3 (a x+1)}-\frac{1}{4 a^4 c^3 (1-a x)^2}+\frac{1}{12 a^4 c^3 (1-a x)^3}+\frac{\tanh ^{-1}(a x)}{8 a^4 c^3} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 88
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^3}{(1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=\frac{\int \left (\frac{1}{4 a^3 (-1+a x)^4}+\frac{1}{2 a^3 (-1+a x)^3}+\frac{3}{16 a^3 (-1+a x)^2}-\frac{1}{16 a^3 (1+a x)^2}-\frac{1}{8 a^3 \left (-1+a^2 x^2\right )}\right ) \, dx}{c^3}\\ &=\frac{1}{12 a^4 c^3 (1-a x)^3}-\frac{1}{4 a^4 c^3 (1-a x)^2}+\frac{3}{16 a^4 c^3 (1-a x)}+\frac{1}{16 a^4 c^3 (1+a x)}-\frac{\int \frac{1}{-1+a^2 x^2} \, dx}{8 a^3 c^3}\\ &=\frac{1}{12 a^4 c^3 (1-a x)^3}-\frac{1}{4 a^4 c^3 (1-a x)^2}+\frac{3}{16 a^4 c^3 (1-a x)}+\frac{1}{16 a^4 c^3 (1+a x)}+\frac{\tanh ^{-1}(a x)}{8 a^4 c^3}\\ \end{align*}
Mathematica [A] time = 0.0423958, size = 64, normalized size = 0.74 \[ \frac{-3 a^3 x^3-6 a^2 x^2+7 a x+3 (a x-1)^3 (a x+1) \tanh ^{-1}(a x)-2}{24 a^4 c^3 (a x-1)^3 (a x+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 90, normalized size = 1.1 \begin{align*}{\frac{1}{16\,{a}^{4}{c}^{3} \left ( ax+1 \right ) }}+{\frac{\ln \left ( ax+1 \right ) }{16\,{a}^{4}{c}^{3}}}-{\frac{1}{12\,{a}^{4}{c}^{3} \left ( ax-1 \right ) ^{3}}}-{\frac{1}{4\,{a}^{4}{c}^{3} \left ( ax-1 \right ) ^{2}}}-{\frac{3}{16\,{a}^{4}{c}^{3} \left ( ax-1 \right ) }}-{\frac{\ln \left ( ax-1 \right ) }{16\,{a}^{4}{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965609, size = 127, normalized size = 1.48 \begin{align*} -\frac{3 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 7 \, a x + 2}{24 \,{\left (a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{5} c^{3} x - a^{4} c^{3}\right )}} + \frac{\log \left (a x + 1\right )}{16 \, a^{4} c^{3}} - \frac{\log \left (a x - 1\right )}{16 \, a^{4} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25539, size = 271, normalized size = 3.15 \begin{align*} -\frac{6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 14 \, a x - 3 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x - 1\right ) + 4}{48 \,{\left (a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{5} c^{3} x - a^{4} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.669184, size = 88, normalized size = 1.02 \begin{align*} - \frac{3 a^{3} x^{3} + 6 a^{2} x^{2} - 7 a x + 2}{24 a^{8} c^{3} x^{4} - 48 a^{7} c^{3} x^{3} + 48 a^{5} c^{3} x - 24 a^{4} c^{3}} + \frac{- \frac{\log{\left (x - \frac{1}{a} \right )}}{16} + \frac{\log{\left (x + \frac{1}{a} \right )}}{16}}{a^{4} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16654, size = 101, normalized size = 1.17 \begin{align*} \frac{\log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{4} c^{3}} - \frac{\log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{4} c^{3}} - \frac{3 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 7 \, a x + 2}{24 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{3} a^{4} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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