Optimal. Leaf size=63 \[ \frac {\sqrt {1-a^2 x^2}}{a c^2}+\frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)}-\frac {\text {ArcSin}(a x)}{a c^2} \]
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Rubi [A]
time = 0.12, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6266, 6263,
1653, 12, 807, 222} \begin {gather*} \frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)}+\frac {\sqrt {1-a^2 x^2}}{a c^2}-\frac {\text {ArcSin}(a x)}{a c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 222
Rule 807
Rule 1653
Rule 6263
Rule 6266
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx &=\frac {a^2 \int \frac {e^{-\tanh ^{-1}(a x)} x^2}{(1-a x)^2} \, dx}{c^2}\\ &=\frac {a^2 \int \frac {x^2}{(1-a x) \sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{a c^2}+\frac {\int \frac {a^3 x}{(1-a x) \sqrt {1-a^2 x^2}} \, dx}{a^2 c^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{a c^2}+\frac {a \int \frac {x}{(1-a x) \sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{a c^2}+\frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{a c^2}+\frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)}-\frac {\sin ^{-1}(a x)}{a c^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 47, normalized size = 0.75 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{c^2}-\frac {1}{c^2 (-1+a x)}\right )}{a}-\frac {\text {ArcSin}(a x)}{a c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(270\) vs.
\(2(59)=118\).
time = 0.77, size = 271, normalized size = 4.30
method | result | size |
risch | \(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {\left (-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{4} \left (x -\frac {1}{a}\right )}\right ) a^{2}}{c^{2}}\) | \(110\) |
default | \(\frac {a^{2} \left (\frac {\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a \left (x -\frac {1}{a}\right )^{2}}+a \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{2 a^{4}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}}{4 a^{3}}+\frac {\frac {3 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{4}-\frac {3 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{4 \sqrt {a^{2}}}}{a^{3}}\right )}{c^{2}}\) | \(271\) |
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 = 0.35, size = 71, normalized size = 1.13 \begin {gather*} \frac {2 \, a x + 2 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x - 2\right )} - 2}{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 {a^{2} \int \frac {x^{2} \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 [A]
time = 0.42, size = 72, normalized size = 1.14 \begin {gather*} -\frac {\arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{c^{2} {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a c^{2}} + \frac {2}{c^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.86, size = 89, normalized size = 1.41 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}}{a\,c^2}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}}+\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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