Optimal. Leaf size=28 \[ \frac {\sin ^{-1}(a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6123, 641, 216} \[ \frac {\sin ^{-1}(a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 6123
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} \, dx &=\int \frac {1+a x}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{a}+\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{a}+\frac {\sin ^{-1}(a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.89 \[ \frac {\sin ^{-1}(a x)-\sqrt {1-a^2 x^2}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 42, normalized size = 1.50 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} + 2 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 29, normalized size = 1.04 \[ \frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 45, normalized size = 1.61 \[ -\frac {\sqrt {-a^{2} x^{2}+1}}{a}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 26, normalized size = 0.93 \[ \frac {\arcsin \left (a x\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 36, normalized size = 1.29 \[ \frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.62, size = 19, normalized size = 0.68 \[ \begin {cases} \frac {- \sqrt {- a^{2} x^{2} + 1} + \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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