Optimal. Leaf size=32 \[ \frac {1}{5} a^4 c^4 x^5-\frac {2}{3} a^2 c^4 x^3+c^4 x \]
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Rubi [A] time = 0.06, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6167, 6129, 41, 194} \[ \frac {1}{5} a^4 c^4 x^5-\frac {2}{3} a^2 c^4 x^3+c^4 x \]
Antiderivative was successfully verified.
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Rule 41
Rule 194
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} (c-a c x)^4 \, dx &=\int e^{4 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx\\ &=c^4 \int (1-a x)^2 (1+a x)^2 \, dx\\ &=c^4 \int \left (1-a^2 x^2\right )^2 \, dx\\ &=c^4 \int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx\\ &=c^4 x-\frac {2}{3} a^2 c^4 x^3+\frac {1}{5} a^4 c^4 x^5\\ \end {align*}
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Mathematica [A] time = 0.02, size = 26, normalized size = 0.81 \[ c^4 \left (\frac {a^4 x^5}{5}-\frac {2 a^2 x^3}{3}+x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 28, normalized size = 0.88 \[ \frac {1}{5} \, a^{4} c^{4} x^{5} - \frac {2}{3} \, a^{2} c^{4} x^{3} + c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 42, normalized size = 1.31 \[ \frac {{\left (3 \, c^{4} + \frac {15 \, c^{4}}{a x - 1} + \frac {20 \, c^{4}}{{\left (a x - 1\right )}^{2}}\right )} {\left (a x - 1\right )}^{5}}{15 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 23, normalized size = 0.72 \[ c^{4} \left (\frac {1}{5} a^{4} x^{5}-\frac {2}{3} x^{3} a^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 28, normalized size = 0.88 \[ \frac {1}{5} \, a^{4} c^{4} x^{5} - \frac {2}{3} \, a^{2} c^{4} x^{3} + c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 24, normalized size = 0.75 \[ \frac {c^4\,x\,\left (3\,a^4\,x^4-10\,a^2\,x^2+15\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 29, normalized size = 0.91 \[ \frac {a^{4} c^{4} x^{5}}{5} - \frac {2 a^{2} c^{4} x^{3}}{3} + c^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
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