Optimal. Leaf size=21 \[ \frac {c}{a^2 x}-c x-\frac {2 c \log (x)}{a} \]
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Rubi [A]
time = 0.05, 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 = {6292, 6285, 45}
\begin {gather*} \frac {c}{a^2 x}-\frac {2 c \log (x)}{a}+c (-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6285
Rule 6292
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 \begin {gather*} \frac {c}{a^2 x}-c x-\frac {2 c \log (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.86, size = 21, normalized size = 1.00
method | result | size |
default | \(\frac {c \left (-a^{2} x +\frac {1}{x}-2 a \ln \left (x \right )\right )}{a^{2}}\) | \(21\) |
risch | \(\frac {c}{a^{2} x}-c x -\frac {2 c \ln \left (x \right )}{a}\) | \(22\) |
norman | \(\frac {\frac {c}{a}-a c \,x^{2}}{a x}-\frac {2 c \ln \left (x \right )}{a}\) | \(30\) |
meijerg | \(-\frac {c \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}-\frac {c \ln \left (-a^{2} x^{2}+1\right )}{a}-\frac {c \left (-\ln \left (-a^{2} x^{2}+1\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right )}{a}+\frac {c \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(126\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 21, normalized size = 1.00 \begin {gather*} -c x - \frac {2 \, c \log \left (x\right )}{a} + \frac {c}{a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 27, normalized size = 1.29 \begin {gather*} -\frac {a^{2} c x^{2} + 2 \, a c x \log \left (x\right ) - c}{a^{2} x} \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 a c \log {\left (x \right )} + \frac {c}{x}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 22, normalized size = 1.05 \begin {gather*} -c x - \frac {2 \, c \log \left ({\left | x \right |}\right )}{a} + \frac {c}{a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.84, size = 24, normalized size = 1.14 \begin {gather*} -\frac {c\,\left (a^2\,x^2+2\,a\,x\,\ln \left (x\right )-1\right )}{a^2\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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