Optimal. Leaf size=18 \[ \tanh ^{-1}\left (\frac {e^x}{\sqrt {a^2+e^{2 x}}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2281, 223, 212}
\begin {gather*} \tanh ^{-1}\left (\frac {e^x}{\sqrt {a^2+e^{2 x}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 2281
Rubi steps
\begin {align*} \int \frac {e^x}{\sqrt {a^2+e^{2 x}}} \, dx &=\text {Subst}\left (\int \frac {1}{\sqrt {a^2+x^2}} \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {e^x}{\sqrt {a^2+e^{2 x}}}\right )\\ &=\tanh ^{-1}\left (\frac {e^x}{\sqrt {a^2+e^{2 x}}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 18, normalized size = 1.00 \begin {gather*} \tanh ^{-1}\left (\frac {e^x}{\sqrt {a^2+e^{2 x}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 15, normalized size = 0.83
method | result | size |
default | \(\ln \left ({\mathrm e}^{x}+\sqrt {a^{2}+{\mathrm e}^{2 x}}\right )\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.44, size = 7, normalized size = 0.39 \begin {gather*} \operatorname {arsinh}\left (\frac {e^{x}}{a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 18, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt {a^{2} + e^{\left (2 \, x\right )}} - e^{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (15) = 30\).
time = 0.34, size = 31, normalized size = 1.72 \begin {gather*} \begin {cases} \operatorname {asinh}{\left (\sqrt {\frac {1}{a^{2}}} e^{x} \right )} & \text {for}\: a^{2} > 0 \\\operatorname {acosh}{\left (\sqrt {- \frac {1}{a^{2}}} e^{x} \right )} & \text {for}\: a^{2} < 0 \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.31, size = 18, normalized size = 1.00 \begin {gather*} -\log \left (\sqrt {a^{2} + e^{\left (2 \, x\right )}} - e^{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.41, size = 14, normalized size = 0.78 \begin {gather*} \ln \left ({\mathrm {e}}^x+\sqrt {a^2+{\mathrm {e}}^{2\,x}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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