Optimal. Leaf size=65 \[ -\frac{3 c^4}{a^3 x^2}+\frac{c^4}{3 a^4 x^3}+\frac{16 c^4}{a^2 x}+\frac{26 c^4 \log (x)}{a}-\frac{32 c^4 \log (a x+1)}{a}+c^4 x \]
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Rubi [A] time = 0.145399, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6167, 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 6167
Rule 6131
Rule 6129
Rule 88
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^4 \, dx &=-\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 = 0.17904, size = 67, 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 = 0.048, size = 64, 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 = 1.06117, size = 81, normalized size = 1.25 \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 = 1.85412, size = 162, normalized size = 2.49 \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 = 0.661916, size = 56, normalized size = 0.86 \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 = 1.13839, size = 84, normalized size = 1.29 \begin{align*} c^{4} x - \frac{32 \, c^{4} \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac{26 \, c^{4} \log \left ({\left | x \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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