Optimal. Leaf size=21 \[ \frac {1}{2} a^2 c x^2+2 a c x+c \log (x) \]
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Rubi [A] time = 0.05, antiderivative size = 21, 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} \[ \frac {1}{2} a^2 c x^2+2 a c x+c \log (x) \]
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} \, dx &=c \int \frac {(1+a x)^2}{x} \, dx\\ &=c \int \left (2 a+\frac {1}{x}+a^2 x\right ) \, dx\\ &=2 a c x+\frac {1}{2} a^2 c x^2+c \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 0.90 \[ c \left (\frac {a^2 x^2}{2}+2 a x+\log (x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 19, normalized size = 0.90 \[ \frac {1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 20, normalized size = 0.95 \[ \frac {1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 20, normalized size = 0.95 \[ 2 a c x +\frac {a^{2} c \,x^{2}}{2}+c \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 19, normalized size = 0.90 \[ \frac {1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 19, normalized size = 0.90 \[ \frac {c\,\left (2\,\ln \relax (x)+4\,a\,x+a^2\,x^2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 20, normalized size = 0.95 \[ \frac {a^{2} c x^{2}}{2} + 2 a c x + c \log {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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