Optimal. Leaf size=105 \[ \frac{23}{16 a^6 c^3 (1-a x)}+\frac{1}{16 a^6 c^3 (a x+1)}-\frac{1}{2 a^6 c^3 (1-a x)^2}+\frac{1}{12 a^6 c^3 (1-a x)^3}+\frac{13 \log (1-a x)}{16 a^6 c^3}+\frac{3 \log (a x+1)}{16 a^6 c^3} \]
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Rubi [A] time = 0.133259, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 88} \[ \frac{23}{16 a^6 c^3 (1-a x)}+\frac{1}{16 a^6 c^3 (a x+1)}-\frac{1}{2 a^6 c^3 (1-a x)^2}+\frac{1}{12 a^6 c^3 (1-a x)^3}+\frac{13 \log (1-a x)}{16 a^6 c^3}+\frac{3 \log (a x+1)}{16 a^6 c^3} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{x^5}{(1-a x)^4 (1+a x)^2} \, dx}{c^3}\\ &=\frac{\int \left (\frac{1}{4 a^5 (-1+a x)^4}+\frac{1}{a^5 (-1+a x)^3}+\frac{23}{16 a^5 (-1+a x)^2}+\frac{13}{16 a^5 (-1+a x)}-\frac{1}{16 a^5 (1+a x)^2}+\frac{3}{16 a^5 (1+a x)}\right ) \, dx}{c^3}\\ &=\frac{1}{12 a^6 c^3 (1-a x)^3}-\frac{1}{2 a^6 c^3 (1-a x)^2}+\frac{23}{16 a^6 c^3 (1-a x)}+\frac{1}{16 a^6 c^3 (1+a x)}+\frac{13 \log (1-a x)}{16 a^6 c^3}+\frac{3 \log (1+a x)}{16 a^6 c^3}\\ \end{align*}
Mathematica [A] time = 0.053934, size = 87, normalized size = 0.83 \[ \frac{-66 a^3 x^3+36 a^2 x^2+74 a x+39 (a x-1)^3 (a x+1) \log (1-a x)+9 (a x-1)^3 (a x+1) \log (a x+1)-52}{48 a^6 c^3 (a x-1)^3 (a x+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 90, normalized size = 0.9 \begin{align*}{\frac{1}{16\,{a}^{6}{c}^{3} \left ( ax+1 \right ) }}+{\frac{3\,\ln \left ( ax+1 \right ) }{16\,{a}^{6}{c}^{3}}}-{\frac{1}{2\,{a}^{6}{c}^{3} \left ( ax-1 \right ) ^{2}}}-{\frac{1}{12\,{a}^{6}{c}^{3} \left ( ax-1 \right ) ^{3}}}-{\frac{23}{16\,{a}^{6}{c}^{3} \left ( ax-1 \right ) }}+{\frac{13\,\ln \left ( ax-1 \right ) }{16\,{a}^{6}{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959164, size = 127, normalized size = 1.21 \begin{align*} -\frac{33 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 37 \, a x + 26}{24 \,{\left (a^{10} c^{3} x^{4} - 2 \, a^{9} c^{3} x^{3} + 2 \, a^{7} c^{3} x - a^{6} c^{3}\right )}} + \frac{3 \, \log \left (a x + 1\right )}{16 \, a^{6} c^{3}} + \frac{13 \, \log \left (a x - 1\right )}{16 \, a^{6} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37255, size = 277, normalized size = 2.64 \begin{align*} -\frac{66 \, a^{3} x^{3} - 36 \, a^{2} x^{2} - 74 \, a x - 9 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x + 1\right ) - 39 \,{\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \log \left (a x - 1\right ) + 52}{48 \,{\left (a^{10} c^{3} x^{4} - 2 \, a^{9} c^{3} x^{3} + 2 \, a^{7} c^{3} x - a^{6} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.786879, size = 92, normalized size = 0.88 \begin{align*} - \frac{33 a^{3} x^{3} - 18 a^{2} x^{2} - 37 a x + 26}{24 a^{10} c^{3} x^{4} - 48 a^{9} c^{3} x^{3} + 48 a^{7} c^{3} x - 24 a^{6} c^{3}} + \frac{\frac{13 \log{\left (x - \frac{1}{a} \right )}}{16} + \frac{3 \log{\left (x + \frac{1}{a} \right )}}{16}}{a^{6} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15004, size = 101, normalized size = 0.96 \begin{align*} \frac{3 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{6} c^{3}} + \frac{13 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{6} c^{3}} - \frac{33 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 37 \, a x + 26}{24 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{3} a^{6} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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