Optimal. Leaf size=19 \[ a^2 c x+2 a c \log (x)-\frac {c}{x} \]
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Rubi [A] time = 0.05, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6150, 43} \[ a^2 c x+2 a c \log (x)-\frac {c}{x} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )}{x^2} \, dx &=c \int \frac {(1+a x)^2}{x^2} \, dx\\ &=c \int \left (a^2+\frac {1}{x^2}+\frac {2 a}{x}\right ) \, dx\\ &=-\frac {c}{x}+a^2 c x+2 a c \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.95 \[ c \left (a^2 x+2 a \log (x)-\frac {1}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 23, normalized size = 1.21 \[ \frac {a^{2} c x^{2} + 2 \, a c x \log \relax (x) - c}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 20, normalized size = 1.05 \[ a^{2} c x + 2 \, a c \log \left ({\left | x \right |}\right ) - \frac {c}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 20, normalized size = 1.05 \[ -\frac {c}{x}+a^{2} c x +2 a c \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 19, normalized size = 1.00 \[ a^{2} c x + 2 \, a c \log \relax (x) - \frac {c}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 20, normalized size = 1.05 \[ \frac {c\,\left (a^2\,x^2+2\,a\,x\,\ln \relax (x)-1\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 17, normalized size = 0.89 \[ a^{2} c x + 2 a c \log {\relax (x )} - \frac {c}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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