Optimal. Leaf size=43 \[ \frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}-\frac {\sin ^{-1}(a x)}{a c} \]
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Rubi [A] time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6127, 663, 216} \[ \frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}-\frac {\sin ^{-1}(a x)}{a c} \]
Antiderivative was successfully verified.
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Rule 216
Rule 663
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{c-a c x} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^2} \, dx\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}-\frac {\sin ^{-1}(a x)}{a c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 1.07 \[ \frac {2 \left (\frac {\sqrt {a x+1}}{\sqrt {1-a x}}+\sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{a c} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 62, normalized size = 1.44 \[ \frac {2 \, {\left (a x + {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} - 1\right )}}{a^{2} c x - a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 53, normalized size = 1.23 \[ -\frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c {\left | a \right |}} + \frac {4}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 76, normalized size = 1.77 \[ -\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c \sqrt {a^{2}}}-\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{c \,a^{2} \left (x -\frac {1}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 40, normalized size = 0.93 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2} c x - a c} - \frac {\arcsin \left (a x\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 71, normalized size = 1.65 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}} \]
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 \[ - \frac {\int \frac {a x}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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