Optimal. Leaf size=29 \[ \frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6262, 665}
\begin {gather*} \frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 665
Rule 6262
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac {\int \frac {1}{(c-a c x) \sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.90 \begin {gather*} \frac {\sqrt {1+a x}}{a c^2 \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 1.09, size = 268, normalized size = 9.24
method | result | size |
gosper | \(-\frac {\sqrt {-a^{2} x^{2}+1}}{a \left (a x -1\right ) c^{2}}\) | \(28\) |
trager | \(-\frac {\sqrt {-a^{2} x^{2}+1}}{a \left (a x -1\right ) c^{2}}\) | \(28\) |
default | \(\frac {\frac {\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a \left (x -\frac {1}{a}\right )^{2}}+a \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{2 a^{2}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}}{4 a}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}}{4 a}}{c^{2}}\) | \(268\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 37, normalized size = 1.28 \begin {gather*} \frac {a x - \sqrt {-a^{2} x^{2} + 1} - 1}{a^{2} c^{2} x - a c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.42, size = 70, normalized size = 2.41 \begin {gather*} -c^{2} {\left (\frac {\sqrt {-\frac {2 \, c}{a c x - c} - 1} \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}{a^{2} c^{4}} - \frac {i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}{a^{2} c^{4}}\right )} {\left | a \right |} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 47, normalized size = 1.62 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}}{c^2\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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