Optimal. Leaf size=35 \[ \frac {c^2 (a x+1)^5}{5 a}-\frac {c^2 (a x+1)^4}{2 a} \]
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Rubi [A] time = 0.07, antiderivative size = 35, 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} \[ \frac {c^2 (a x+1)^5}{5 a}-\frac {c^2 (a x+1)^4}{2 a} \]
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) (1+a x)^3 \, 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 = 23, normalized size = 0.66 \[ \frac {c^2 (a x+1)^4 (2 a x-3)}{10 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 38, normalized size = 1.09 \[ \frac {1}{5} \, a^{4} c^{2} x^{5} + \frac {1}{2} \, a^{3} c^{2} x^{4} - a c^{2} x^{2} - c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 38, normalized size = 1.09 \[ \frac {1}{5} \, a^{4} c^{2} x^{5} + \frac {1}{2} \, a^{3} c^{2} x^{4} - a c^{2} x^{2} - c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 31, normalized size = 0.89 \[ c^{2} \left (\frac {1}{5} a^{4} x^{5}+\frac {1}{2} x^{4} a^{3}-a \,x^{2}-x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 38, normalized size = 1.09 \[ \frac {1}{5} \, a^{4} c^{2} x^{5} + \frac {1}{2} \, a^{3} c^{2} x^{4} - a c^{2} x^{2} - c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 38, normalized size = 1.09 \[ \frac {a^4\,c^2\,x^5}{5}+\frac {a^3\,c^2\,x^4}{2}-a\,c^2\,x^2-c^2\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 36, normalized size = 1.03 \[ \frac {a^{4} c^{2} x^{5}}{5} + \frac {a^{3} c^{2} x^{4}}{2} - a c^{2} x^{2} - c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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