Optimal. Leaf size=37 \[ \frac {c^2 (1-a x)^4}{2 a}-\frac {c^2 (1-a x)^5}{5 a} \]
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Rubi [A] time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6167, 6140, 43} \[ \frac {c^2 (1-a x)^4}{2 a}-\frac {c^2 (1-a x)^5}{5 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx\\ &=-\left (c^2 \int (1-a x)^3 (1+a x) \, dx\right )\\ &=-\left (c^2 \int \left (2 (1-a x)^3-(1-a x)^4\right ) \, dx\right )\\ &=\frac {c^2 (1-a x)^4}{2 a}-\frac {c^2 (1-a x)^5}{5 a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.81 \[ \frac {1}{10} c^2 x \left (2 a^4 x^4-5 a^3 x^3+10 a x-10\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 37, normalized size = 1.00 \[ \frac {1}{5} \, a^{4} c^{2} x^{5} - \frac {1}{2} \, a^{3} c^{2} x^{4} + a c^{2} x^{2} - c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 37, normalized size = 1.00 \[ \frac {1}{5} \, a^{4} c^{2} x^{5} - \frac {1}{2} \, a^{3} c^{2} x^{4} + a c^{2} x^{2} - c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 30, normalized size = 0.81 \[ c^{2} \left (\frac {1}{5} a^{4} x^{5}-\frac {1}{2} x^{4} a^{3}+a \,x^{2}-x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 37, normalized size = 1.00 \[ \frac {1}{5} \, a^{4} c^{2} x^{5} - \frac {1}{2} \, a^{3} c^{2} x^{4} + a c^{2} x^{2} - c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 37, normalized size = 1.00 \[ \frac {a^4\,c^2\,x^5}{5}-\frac {a^3\,c^2\,x^4}{2}+a\,c^2\,x^2-c^2\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 36, normalized size = 0.97 \[ \frac {a^{4} c^{2} x^{5}}{5} - \frac {a^{3} c^{2} x^{4}}{2} + a c^{2} x^{2} - c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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