Optimal. Leaf size=25 \[ \frac{e^{a+b x}}{b}-\frac{2 \tan ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.0148009, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2282, 388, 203} \[ \frac{e^{a+b x}}{b}-\frac{2 \tan ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 388
Rule 203
Rubi steps
\begin{align*} \int e^{a+b x} \tanh (a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{-1+x^2}{1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{e^{a+b x}}{b}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{e^{a+b x}}{b}-\frac{2 \tan ^{-1}\left (e^{a+b x}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0144078, size = 22, normalized size = 0.88 \[ \frac{e^{a+b x}-2 \tan ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 34, normalized size = 1.4 \begin{align*}{\frac{\sinh \left ( bx+a \right ) }{b}}-2\,{\frac{\arctan \left ({{\rm e}^{bx+a}} \right ) }{b}}+{\frac{\cosh \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61474, size = 31, normalized size = 1.24 \begin{align*} -\frac{2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} + \frac{e^{\left (b x + a\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92056, size = 105, normalized size = 4.2 \begin{align*} -\frac{2 \, \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - \cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{a} \int e^{b x} \tanh{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.52603, size = 31, normalized size = 1.24 \begin{align*} -\frac{2 \, \arctan \left (e^{\left (b x + a\right )}\right ) - e^{\left (b x + a\right )}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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