Optimal. Leaf size=21 \[ \frac {c}{a^2 x}+\frac {2 c \log (x)}{a}+c (-x) \]
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Rubi [A] time = 0.07, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6157, 6150, 43} \[ \frac {c}{a^2 x}+\frac {2 c \log (x)}{a}+c (-x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rule 6157
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx &=-\frac {c \int \frac {e^{-2 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )}{x^2} \, dx}{a^2}\\ &=-\frac {c \int \frac {(1-a x)^2}{x^2} \, dx}{a^2}\\ &=-\frac {c \int \left (a^2+\frac {1}{x^2}-\frac {2 a}{x}\right ) \, dx}{a^2}\\ &=\frac {c}{a^2 x}-c x+\frac {2 c \log (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ \frac {c}{a^2 x}+\frac {2 c \log (x)}{a}+c (-x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 27, normalized size = 1.29 \[ -\frac {a^{2} c x^{2} - 2 \, a c x \log \relax (x) - c}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 88, normalized size = 4.19 \[ -\frac {2 \, c \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} + \frac {2 \, c \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a} - \frac {c - \frac {2 \, c}{a x + 1}}{a^{2} {\left (\frac {1}{{\left (a x + 1\right )} a} - \frac {1}{{\left (a x + 1\right )}^{2} a}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 22, normalized size = 1.05 \[ \frac {c}{a^{2} x}-c x +\frac {2 c \ln \relax (x )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 21, normalized size = 1.00 \[ -c x + \frac {2 \, c \log \relax (x)}{a} + \frac {c}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 24, normalized size = 1.14 \[ \frac {c\,\left (2\,a\,x\,\ln \relax (x)-a^2\,x^2+1\right )}{a^2\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 20, normalized size = 0.95 \[ \frac {- a^{2} c x + 2 a c \log {\relax (x )} + \frac {c}{x}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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