Optimal. Leaf size=21 \[ 2 a c x+\frac {1}{2} a^2 c x^2+c \log (x) \]
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Rubi [A]
time = 0.03, 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 = {6285, 45}
\begin {gather*} \frac {1}{2} a^2 c x^2+2 a c x+c \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6285
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 \begin {gather*} c \left (2 a x+\frac {a^2 x^2}{2}+\log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 18, normalized size = 0.86
method | result | size |
default | \(c \left (\frac {a^{2} x^{2}}{2}+2 a x +\ln \left (x \right )\right )\) | \(18\) |
norman | \(2 a c x +\frac {a^{2} c \,x^{2}}{2}+c \ln \left (x \right )\) | \(20\) |
risch | \(2 a c x +\frac {a^{2} c \,x^{2}}{2}+c \ln \left (x \right )\) | \(20\) |
meijerg | \(-\frac {c \left (-a^{2} x^{2}-\ln \left (-a^{2} x^{2}+1\right )\right )}{2}+\frac {a c \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{\sqrt {-a^{2}}}+2 c \arctanh \left (a x \right )+\frac {c \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{2}\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 19, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 19, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 20, normalized size = 0.95 \begin {gather*} \frac {a^{2} c x^{2}}{2} + 2 a c x + c \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 20, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, a^{2} c x^{2} + 2 \, a c x + c \log \left ({\left | x \right |}\right ) \end {gather*}
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 \begin {gather*} \frac {c\,\left (2\,\ln \left (x\right )+4\,a\,x+a^2\,x^2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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