Optimal. Leaf size=37 \[ \frac {c^2 (1-a x)^5}{5 a}-\frac {c^2 (1-a x)^4}{2 a} \]
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Rubi [A] time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6140, 43} \[ \frac {c^2 (1-a x)^5}{5 a}-\frac {c^2 (1-a x)^4}{2 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int (1-a x)^3 (1+a x) \, dx\\ &=c^2 \int \left (2 (1-a x)^3-(1-a x)^4\right ) \, dx\\ &=-\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 = 32, normalized size = 0.86 \[ c^2 \left (-\frac {1}{5} a^4 x^5+\frac {a^3 x^4}{2}-a x^2+x\right ) \]
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 = 0.15, size = 54, normalized size = 1.46 \[ -\frac {{\left (2 \, c^{2} - \frac {15 \, c^{2}}{a x + 1} + \frac {40 \, c^{2}}{{\left (a x + 1\right )}^{2}} - \frac {40 \, c^{2}}{{\left (a x + 1\right )}^{3}}\right )} {\left (a x + 1\right )}^{5}}{10 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 29, normalized size = 0.78 \[ c^{2} \left (-\frac {1}{5} a^{4} x^{5}+\frac {1}{2} x^{4} a^{3}-a \,x^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, 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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