Optimal. Leaf size=27 \[ \frac {\sqrt {1-a^2 x^2}}{a}+\frac {\text {ArcSin}(a x)}{a} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6258, 655, 222}
\begin {gather*} \frac {\sqrt {1-a^2 x^2}}{a}+\frac {\text {ArcSin}(a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 655
Rule 6258
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 = 23, normalized size = 0.85 \begin {gather*} \frac {\sqrt {1-a^2 x^2}+\text {ArcSin}(a x)}{a} \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. \(66\) vs.
\(2(25)=50\).
time = 0.83, size = 67, normalized size = 2.48
method | result | size |
risch | \(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\) | \(54\) |
default | \(\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}}{a}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 25, normalized size = 0.93 \begin {gather*} \frac {\arcsin \left (a x\right )}{a} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 41, normalized size = 1.52 \begin {gather*} \frac {\sqrt {-a^{2} x^{2} + 1} - 2 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{a} \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*} \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 28, normalized size = 1.04 \begin {gather*} \frac {\arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 35, normalized size = 1.30 \begin {gather*} \frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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