Optimal. Leaf size=66 \[ \frac{3 c^4}{a^3 x^2}-\frac{c^4}{3 a^4 x^3}-\frac{16 c^4}{a^2 x}-\frac{26 c^4 \log (x)}{a}+\frac{32 c^4 \log (a x+1)}{a}+c^4 (-x) \]
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Rubi [A] time = 0.117514, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6131, 6129, 88} \[ \frac{3 c^4}{a^3 x^2}-\frac{c^4}{3 a^4 x^3}-\frac{16 c^4}{a^2 x}-\frac{26 c^4 \log (x)}{a}+\frac{32 c^4 \log (a x+1)}{a}+c^4 (-x) \]
Antiderivative was successfully verified.
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Rule 6131
Rule 6129
Rule 88
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^4 \, dx &=\frac{c^4 \int \frac{e^{-2 \tanh ^{-1}(a x)} (1-a x)^4}{x^4} \, dx}{a^4}\\ &=\frac{c^4 \int \frac{(1-a x)^5}{x^4 (1+a x)} \, dx}{a^4}\\ &=\frac{c^4 \int \left (-a^4+\frac{1}{x^4}-\frac{6 a}{x^3}+\frac{16 a^2}{x^2}-\frac{26 a^3}{x}+\frac{32 a^4}{1+a x}\right ) \, dx}{a^4}\\ &=-\frac{c^4}{3 a^4 x^3}+\frac{3 c^4}{a^3 x^2}-\frac{16 c^4}{a^2 x}-c^4 x-\frac{26 c^4 \log (x)}{a}+\frac{32 c^4 \log (1+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.164862, size = 68, normalized size = 1.03 \[ \frac{3 c^4}{a^3 x^2}-\frac{c^4}{3 a^4 x^3}-\frac{16 c^4}{a^2 x}-\frac{26 c^4 \log (a x)}{a}+\frac{32 c^4 \log (a x+1)}{a}+c^4 (-x) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.039, size = 65, normalized size = 1. \begin{align*} -{\frac{{c}^{4}}{3\,{a}^{4}{x}^{3}}}+3\,{\frac{{c}^{4}}{{x}^{2}{a}^{3}}}-16\,{\frac{{c}^{4}}{{a}^{2}x}}-{c}^{4}x-26\,{\frac{{c}^{4}\ln \left ( x \right ) }{a}}+32\,{\frac{{c}^{4}\ln \left ( ax+1 \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.957922, size = 82, normalized size = 1.24 \begin{align*} -c^{4} x + \frac{32 \, c^{4} \log \left (a x + 1\right )}{a} - \frac{26 \, c^{4} \log \left (x\right )}{a} - \frac{48 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17741, size = 163, normalized size = 2.47 \begin{align*} -\frac{3 \, a^{4} c^{4} x^{4} - 96 \, a^{3} c^{4} x^{3} \log \left (a x + 1\right ) + 78 \, a^{3} c^{4} x^{3} \log \left (x\right ) + 48 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.726072, size = 58, normalized size = 0.88 \begin{align*} - c^{4} x - \frac{2 c^{4} \left (13 \log{\left (x \right )} - 16 \log{\left (x + \frac{1}{a} \right )}\right )}{a} - \frac{48 a^{2} c^{4} x^{2} - 9 a c^{4} x + c^{4}}{3 a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13984, size = 151, normalized size = 2.29 \begin{align*} -\frac{6 \, c^{4} \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} - \frac{26 \, c^{4} \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right )}{a} + \frac{{\left (3 \, c^{4} + \frac{49 \, c^{4}}{a x + 1} - \frac{117 \, c^{4}}{{\left (a x + 1\right )}^{2}} + \frac{66 \, c^{4}}{{\left (a x + 1\right )}^{3}}\right )}{\left (a x + 1\right )}}{3 \, a{\left (\frac{1}{a x + 1} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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