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 = 0.05, antiderivative size = 91, normalized size of antiderivative = 1.00, 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 43
Rule 6129
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*}
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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 = 0.58, 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 = 0.21, size = 94, normalized size = 1.03 \[ -\frac {{\left (6 \, c^{4} - \frac {75 \, c^{4}}{a x + 1} + \frac {400 \, c^{4}}{{\left (a x + 1\right )}^{2}} - \frac {1200 \, c^{4}}{{\left (a x + 1\right )}^{3}} + \frac {2400 \, c^{4}}{{\left (a x + 1\right )}^{4}}\right )} {\left (a x + 1\right )}^{5}}{30 \, a} - \frac {32 \, c^{4} \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, 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 = 0.33, 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 = 0.05, size = 63, normalized size = 0.69 \[ 13\,a\,c^4\,x^2-31\,c^4\,x-\frac {16\,a^2\,c^4\,x^3}{3}+\frac {3\,a^3\,c^4\,x^4}{2}-\frac {a^4\,c^4\,x^5}{5}+\frac {32\,c^4\,\ln \left (a\,x+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, 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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