Optimal. Leaf size=23 \[ a^2 c \log (x)-\frac {2 a c}{x}-\frac {c}{2 x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 23, 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} \[ a^2 c \log (x)-\frac {2 a c}{x}-\frac {c}{2 x^2} \]
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^3} \, dx &=c \int \frac {(1+a x)^2}{x^3} \, dx\\ &=c \int \left (\frac {1}{x^3}+\frac {2 a}{x^2}+\frac {a^2}{x}\right ) \, dx\\ &=-\frac {c}{2 x^2}-\frac {2 a c}{x}+a^2 c \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 0.96 \[ c \left (a^2 \log (x)-\frac {2 a}{x}-\frac {1}{2 x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 25, normalized size = 1.09 \[ \frac {2 \, a^{2} c x^{2} \log \relax (x) - 4 \, a c x - c}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 21, normalized size = 0.91 \[ a^{2} c \log \left ({\left | x \right |}\right ) - \frac {4 \, a c x + c}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 22, normalized size = 0.96 \[ -\frac {c}{2 x^{2}}-\frac {2 a c}{x}+a^{2} c \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 20, normalized size = 0.87 \[ a^{2} c \log \relax (x) - \frac {4 \, a c x + c}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 22, normalized size = 0.96 \[ a^2\,c\,\ln \relax (x)-\frac {\frac {c}{2}+2\,a\,c\,x}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 22, normalized size = 0.96 \[ a^{2} c \log {\relax (x )} + \frac {- 4 a c x - c}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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