Optimal. Leaf size=63 \[ \frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)}+\frac {\sqrt {1-a^2 x^2}}{a c^2}-\frac {\sin ^{-1}(a x)}{a c^2} \]
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Rubi [A] time = 0.18, 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 = {6131, 6128, 1639, 12, 793, 216} \[ \frac {\sqrt {1-a^2 x^2}}{a c^2 (1-a x)}+\frac {\sqrt {1-a^2 x^2}}{a c^2}-\frac {\sin ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 216
Rule 793
Rule 1639
Rule 6128
Rule 6131
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 = 0.10, size = 47, normalized size = 0.75 \[ \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{c^2}-\frac {1}{c^2 (a x-1)}\right )}{a}-\frac {\sin ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 71, normalized size = 1.13 \[ \frac {2 \, a x + 2 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x - 2\right )} - 2}{a^{2} c^{2} x - a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.11, size = 72, normalized size = 1.14 \[ -\frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c^{2} {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a c^{2}} + \frac {2}{c^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 198, normalized size = 3.14 \[ \frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{2 a^{3} c^{2} \left (x -\frac {1}{a}\right )^{2}}+\frac {5 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a \,c^{2}}-\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{4 c^{2} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a \,c^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{4 c^{2} \sqrt {a^{2}}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{2}}\,{d x} \]
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 \[ \frac {\sqrt {1-a^2\,x^2}}{a\,c^2}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}}{c^2\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
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 \[ \frac {a^{2} \int \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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