Optimal. Leaf size=18 \[ \tanh ^{-1}\left (\frac {e^x}{\sqrt {a^2+e^{2 x}}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.03, 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 = {2249, 217, 206} \[ \tanh ^{-1}\left (\frac {e^x}{\sqrt {a^2+e^{2 x}}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 2249
Rubi steps
\begin {align*} \int \frac {e^x}{\sqrt {a^2+e^{2 x}}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {a^2+x^2}} \, dx,x,e^x\right )\\ &=\operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \[ \tanh ^{-1}\left (\frac {e^x}{\sqrt {a^2+e^{2 x}}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^x}{\sqrt {a^2+e^{2 x}}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.26, size = 18, normalized size = 1.00 \[ -\log \left (\sqrt {a^{2} + e^{\left (2 \, x\right )}} - e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.58, size = 18, normalized size = 1.00 \[ -\log \left (\sqrt {a^{2} + e^{\left (2 \, x\right )}} - e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, 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.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.55, size = 7, normalized size = 0.39 \[ \operatorname {arsinh}\left (\frac {e^{x}}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.41, size = 14, normalized size = 0.78 \[ \ln \left ({\mathrm {e}}^x+\sqrt {a^2+{\mathrm {e}}^{2\,x}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.73, size = 31, normalized size = 1.72 \[ \begin {cases} \operatorname {asinh}{\left (\sqrt {\frac {1}{a^{2}}} e^{x} \right )} & \text {for}\: a^{2} > 0 \\\operatorname {acosh}{\left (\sqrt {- \frac {1}{a^{2}}} e^{x} \right )} & \text {for}\: a^{2} < 0 \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________