Optimal. Leaf size=60 \[ \frac {1-a x}{a c \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c}+\frac {\sin ^{-1}(a x)}{a c} \]
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Rubi [A] time = 0.14, antiderivative size = 60, 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 = {6157, 6149, 797, 641, 216, 637} \[ \frac {1-a x}{a c \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c}+\frac {\sin ^{-1}(a x)}{a c} \]
Antiderivative was successfully verified.
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Rule 216
Rule 637
Rule 641
Rule 797
Rule 6149
Rule 6157
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\frac {a^2 \int \frac {e^{-\tanh ^{-1}(a x)} x^2}{1-a^2 x^2} \, dx}{c}\\ &=-\frac {a^2 \int \frac {x^2 (1-a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}+\frac {\int \frac {1-a x}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {1-a x}{a c \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {1-a x}{a c \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c}+\frac {\sin ^{-1}(a x)}{a c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 54, normalized size = 0.90 \[ \frac {-a^2 x^2+\sqrt {1-a^2 x^2} \sin ^{-1}(a x)-a x+2}{a c \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 66, normalized size = 1.10 \[ \frac {2 \, a x - 2 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x + 2\right )} + 2}{a^{2} c x + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 71, normalized size = 1.18 \[ \frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a c} - \frac {2}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 192, normalized size = 3.20 \[ \frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a c}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{4 c \sqrt {a^{2}}}+\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{2 a^{3} c \left (x +\frac {1}{a}\right )^{2}}+\frac {5 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a c}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{4 c \sqrt {a^{2}}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 88, normalized size = 1.47 \[ \frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}}{a\,c}-\frac {\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}} \]
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 \[ \frac {a^{2} \int \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + a^{2} x^{2} - a x - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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