Optimal. Leaf size=91 \[ -\frac {c^4 (1-a x)^5}{5 a}-\frac {c^4 (1-a x)^4}{2 a}-\frac {4 c^4 (1-a x)^3}{3 a}-\frac {4 c^4 (1-a x)^2}{a}-\frac {32 c^4 \log (a x+1)}{a}+16 c^4 x \]
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Rubi [A] time = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6167, 6129, 43} \[ -\frac {c^4 (1-a x)^5}{5 a}-\frac {c^4 (1-a x)^4}{2 a}-\frac {4 c^4 (1-a x)^3}{3 a}-\frac {4 c^4 (1-a x)^2}{a}-\frac {32 c^4 \log (a x+1)}{a}+16 c^4 x \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} (c-a c x)^4 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx\\ &=-\left (c^4 \int \frac {(1-a x)^5}{1+a x} \, dx\right )\\ &=-\left (c^4 \int \left (-16-8 (1-a x)-4 (1-a x)^2-2 (1-a x)^3-(1-a x)^4+\frac {32}{1+a x}\right ) \, dx\right )\\ &=16 c^4 x-\frac {4 c^4 (1-a x)^2}{a}-\frac {4 c^4 (1-a x)^3}{3 a}-\frac {c^4 (1-a x)^4}{2 a}-\frac {c^4 (1-a x)^5}{5 a}-\frac {32 c^4 \log (1+a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 56, normalized size = 0.62 \[ \frac {c^4 \left (6 a^5 x^5-45 a^4 x^4+160 a^3 x^3-390 a^2 x^2+930 a x-960 \log (a x+1)-181\right )}{30 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 68, normalized size = 0.75 \[ \frac {6 \, a^{5} c^{4} x^{5} - 45 \, a^{4} c^{4} x^{4} + 160 \, a^{3} c^{4} x^{3} - 390 \, a^{2} c^{4} x^{2} + 930 \, a c^{4} x - 960 \, c^{4} \log \left (a x + 1\right )}{30 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 75, normalized size = 0.82 \[ -\frac {32 \, c^{4} \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac {6 \, a^{9} c^{4} x^{5} - 45 \, a^{8} c^{4} x^{4} + 160 \, a^{7} c^{4} x^{3} - 390 \, a^{6} c^{4} x^{2} + 930 \, a^{5} c^{4} x}{30 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 64, normalized size = 0.70 \[ \frac {a^{4} c^{4} x^{5}}{5}-\frac {3 c^{4} x^{4} a^{3}}{2}+\frac {16 a^{2} c^{4} x^{3}}{3}-13 c^{4} x^{2} a +31 c^{4} x -\frac {32 c^{4} \ln \left (a x +1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 63, normalized size = 0.69 \[ \frac {1}{5} \, a^{4} c^{4} x^{5} - \frac {3}{2} \, a^{3} c^{4} x^{4} + \frac {16}{3} \, a^{2} c^{4} x^{3} - 13 \, a c^{4} x^{2} + 31 \, c^{4} x - \frac {32 \, c^{4} \log \left (a x + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 63, normalized size = 0.69 \[ 31\,c^4\,x-13\,a\,c^4\,x^2+\frac {16\,a^2\,c^4\,x^3}{3}-\frac {3\,a^3\,c^4\,x^4}{2}+\frac {a^4\,c^4\,x^5}{5}-\frac {32\,c^4\,\ln \left (a\,x+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 68, normalized size = 0.75 \[ \frac {a^{4} c^{4} x^{5}}{5} - \frac {3 a^{3} c^{4} x^{4}}{2} + \frac {16 a^{2} c^{4} x^{3}}{3} - 13 a c^{4} x^{2} + 31 c^{4} x - \frac {32 c^{4} \log {\left (a x + 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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