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 = 0.0706506, antiderivative size = 91, normalized size of antiderivative = 1., 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 6167
Rule 6129
Rule 43
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*}
Mathematica [A] time = 0.0205326, 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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Maple [A] time = 0.04, size = 64, normalized size = 0.7 \begin{align*}{\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-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 = 1.00712, size = 85, normalized size = 0.93 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51959, size = 154, normalized size = 1.69 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.670782, size = 68, normalized size = 0.75 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19164, size = 101, normalized size = 1.11 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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