Optimal. Leaf size=18 \[ -\frac {c^2}{a^2 x}-c^2 x \]
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Rubi [A]
time = 0.06, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6266, 6264, 74,
14} \begin {gather*} c^2 (-x)-\frac {c^2}{a^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 74
Rule 6264
Rule 6266
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^2 \, dx &=\frac {c^2 \int \frac {e^{2 \tanh ^{-1}(a x)} (1-a x)^2}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \frac {(1-a x) (1+a x)}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \frac {1-a^2 x^2}{x^2} \, dx}{a^2}\\ &=\frac {c^2 \int \left (-a^2+\frac {1}{x^2}\right ) \, dx}{a^2}\\ &=-\frac {c^2}{a^2 x}-c^2 x\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 18, normalized size = 1.00 \begin {gather*} -\frac {c^2}{a^2 x}-c^2 x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 20, normalized size = 1.11
method | result | size |
risch | \(-\frac {c^{2}}{a^{2} x}-x \,c^{2}\) | \(19\) |
default | \(\frac {c^{2} \left (-a^{2} x -\frac {1}{x}\right )}{a^{2}}\) | \(20\) |
gosper | \(-\frac {\left (a^{2} x^{2}+1\right ) c^{2}}{x \,a^{2}}\) | \(21\) |
norman | \(\frac {-\frac {c^{2}}{a}-a \,c^{2} x^{2}}{a x}\) | \(26\) |
meijerg | \(-\frac {c^{2} \left (-\frac {2 x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2}}+\frac {2 \left (-a^{2}\right )^{\frac {3}{2}} \arctanh \left (a x \right )}{a^{3}}\right )}{2 \sqrt {-a^{2}}}-\frac {2 c^{2} \arctanh \left (a x \right )}{a}-\frac {c^{2} \left (-\frac {2}{x \sqrt {-a^{2}}}+\frac {2 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(95\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 18, normalized size = 1.00 \begin {gather*} -c^{2} x - \frac {c^{2}}{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 = 22, normalized size = 1.22 \begin {gather*} -\frac {a^{2} c^{2} x^{2} + c^{2}}{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 = 17, normalized size = 0.94 \begin {gather*} \frac {- a^{2} c^{2} x - \frac {c^{2}}{x}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 18, normalized size = 1.00 \begin {gather*} -c^{2} x - \frac {c^{2}}{a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 20, normalized size = 1.11 \begin {gather*} -\frac {c^2\,\left (a^2\,x^2+1\right )}{a^2\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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