Optimal. Leaf size=18 \[ c^2 (-x)-\frac {c^2}{a^2 x} \]
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Rubi [A] time = 0.09, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6131, 6129, 73, 14} \[ c^2 (-x)-\frac {c^2}{a^2 x} \]
Antiderivative was successfully verified.
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Rule 14
Rule 73
Rule 6129
Rule 6131
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx &=\frac {c^2 \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \frac {(1-a x) (1+a x)}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \frac {1-a^2 x^2}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \left (-a^2+\frac {1}{x^2}\right ) \, dx}{a^2}\\ &=-\frac {c^2}{a^2 x}-c^2 x\\ \end {align*}
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Mathematica [A] time = 0.10, size = 18, normalized size = 1.00 \[ c^2 (-x)-\frac {c^2}{a^2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 22, normalized size = 1.22 \[ -\frac {a^{2} c^{2} x^{2} + c^{2}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 18, normalized size = 1.00 \[ -c^{2} x - \frac {c^{2}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 20, normalized size = 1.11 \[ \frac {c^{2} \left (-a^{2} x -\frac {1}{x}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 18, normalized size = 1.00 \[ -c^{2} x - \frac {c^{2}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 20, normalized size = 1.11 \[ -\frac {c^2\,\left (a^2\,x^2+1\right )}{a^2\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 17, normalized size = 0.94 \[ \frac {- a^{2} c^{2} x - \frac {c^{2}}{x}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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