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 = 0.0487565, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {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 6129
Rule 43
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx &=c^4 \int \frac{(1-a x)^5}{1+a x} \, dx\\ &=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\\ &=-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 = 0.0191269, 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 = 0.028, 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 = 0.941401, 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 = 1.54748, size = 155, normalized size = 1.7 \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 = 0.346247, 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 = 1.19463, size = 127, normalized size = 1.4 \begin{align*} -\frac{{\left (6 \, c^{4} - \frac{75 \, c^{4}}{a x + 1} + \frac{400 \, c^{4}}{{\left (a x + 1\right )}^{2}} - \frac{1200 \, c^{4}}{{\left (a x + 1\right )}^{3}} + \frac{2400 \, c^{4}}{{\left (a x + 1\right )}^{4}}\right )}{\left (a x + 1\right )}^{5}}{30 \, a} - \frac{32 \, c^{4} \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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