Optimal. Leaf size=65 \[ \frac {c^4}{3 a^4 x^3}-\frac {3 c^4}{a^3 x^2}+\frac {16 c^4}{a^2 x}+\frac {26 c^4 \log (x)}{a}-\frac {32 c^4 \log (a x+1)}{a}+c^4 x \]
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Rubi [A] time = 0.15, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 88
Rule 6129
Rule 6131
Rule 6167
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*}
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Mathematica [A] time = 0.19, size = 67, normalized size = 1.03 \[ \frac {c^4}{3 a^4 x^3}-\frac {3 c^4}{a^3 x^2}+\frac {16 c^4}{a^2 x}+\frac {26 c^4 \log (a x)}{a}-\frac {32 c^4 \log (a x+1)}{a}+c^4 x \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 71, normalized size = 1.09 \[ \frac {3 \, a^{4} c^{4} x^{4} - 96 \, a^{3} c^{4} x^{3} \log \left (a x + 1\right ) + 78 \, a^{3} c^{4} x^{3} \log \relax (x) + 48 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 62, normalized size = 0.95 \[ c^{4} x - \frac {32 \, c^{4} \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac {26 \, c^{4} \log \left ({\left | x \right |}\right )}{a} + \frac {48 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 64, normalized size = 0.98 \[ \frac {c^{4}}{3 a^{4} x^{3}}-\frac {3 c^{4}}{x^{2} a^{3}}+\frac {16 c^{4}}{a^{2} x}+c^{4} x +\frac {26 c^{4} \ln \relax (x )}{a}-\frac {32 c^{4} \ln \left (a x +1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 60, normalized size = 0.92 \[ c^{4} x - \frac {32 \, c^{4} \log \left (a x + 1\right )}{a} + \frac {26 \, c^{4} \log \relax (x)}{a} + \frac {48 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 61, normalized size = 0.94 \[ c^4\,x+\frac {16\,a^2\,c^4\,x^2-3\,a\,c^4\,x+\frac {c^4}{3}}{a^4\,x^3}+\frac {26\,c^4\,\ln \relax (x)}{a}-\frac {32\,c^4\,\ln \left (a\,x+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 56, normalized size = 0.86 \[ c^{4} x + \frac {2 c^{4} \left (13 \log {\relax (x )} - 16 \log {\left (x + \frac {1}{a} \right )}\right )}{a} + \frac {48 a^{2} c^{4} x^{2} - 9 a c^{4} x + c^{4}}{3 a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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