Optimal. Leaf size=66 \[ -\frac {c^4}{3 a^4 x^3}+\frac {3 c^4}{a^3 x^2}-\frac {16 c^4}{a^2 x}-\frac {26 c^4 \log (x)}{a}+\frac {32 c^4 \log (a x+1)}{a}+c^4 (-x) \]
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Rubi [A] time = 0.12, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 88
Rule 6129
Rule 6131
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*}
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Mathematica [A] time = 0.18, size = 68, 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 = 0.45, size = 71, normalized size = 1.08 \[ -\frac {3 \, a^{4} c^{4} x^{4} - 96 \, a^{3} c^{4} x^{3} \log \left (a x + 1\right ) + 78 \, a^{3} c^{4} x^{3} \log \relax (x) + 48 \, a^{2} c^{4} x^{2} - 9 \, a c^{4} x + c^{4}}{3 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 112, normalized size = 1.70 \[ -\frac {6 \, c^{4} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} - \frac {26 \, c^{4} \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a} + \frac {{\left (3 \, c^{4} + \frac {49 \, c^{4}}{a x + 1} - \frac {117 \, c^{4}}{{\left (a x + 1\right )}^{2}} + \frac {66 \, c^{4}}{{\left (a x + 1\right )}^{3}}\right )} {\left (a x + 1\right )}}{3 \, a {\left (\frac {1}{a x + 1} - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 65, 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 = 0.31, size = 61, 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 = 0.11, size = 63, normalized size = 0.95 \[ \frac {32\,c^4\,\ln \left (a\,x+1\right )}{a}-\frac {16\,a^2\,c^4\,x^2-3\,a\,c^4\,x+\frac {c^4}{3}}{a^4\,x^3}-\frac {26\,c^4\,\ln \relax (x)}{a}-c^4\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 58, normalized size = 0.88 \[ - c^{4} x - \frac {2 c^{4} \left (13 \log {\relax (x )} - 16 \log {\left (x + \frac {1}{a} \right )}\right )}{a} - \frac {48 a^{2} c^{4} x^{2} - 9 a c^{4} x + c^{4}}{3 a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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