Optimal. Leaf size=25 \[ \frac {e^{a+b x}}{b}-\frac {2 \text {ArcTan}\left (e^{a+b x}\right )}{b} \]
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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 = {2320, 396, 209}
\begin {gather*} \frac {e^{a+b x}}{b}-\frac {2 \text {ArcTan}\left (e^{a+b x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 396
Rule 2320
Rubi steps
\begin {align*} \int e^{a+b x} \tanh (a+b x) \, dx &=\frac {\text {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 \text {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*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.88 \begin {gather*} \frac {e^{a+b x}-2 \text {ArcTan}\left (e^{a+b x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 27, normalized size = 1.08
method | result | size |
derivativedivides | \(\frac {\sinh \left (b x +a \right )-2 \arctan \left ({\mathrm e}^{b x +a}\right )+\cosh \left (b x +a \right )}{b}\) | \(27\) |
default | \(\frac {\sinh \left (b x +a \right )-2 \arctan \left ({\mathrm e}^{b x +a}\right )+\cosh \left (b x +a \right )}{b}\) | \(27\) |
risch | \(\frac {{\mathrm e}^{b x +a}}{b}+\frac {i \ln \left ({\mathrm e}^{b x +a}-i\right )}{b}-\frac {i \ln \left ({\mathrm e}^{b x +a}+i\right )}{b}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 23, normalized size = 0.92 \begin {gather*} -\frac {2 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} + \frac {e^{\left (b x + a\right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 38, normalized size = 1.52 \begin {gather*} -\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 {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*} e^{a} \int e^{b x} \tanh {\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 23, normalized size = 0.92 \begin {gather*} -\frac {2 \, \arctan \left (e^{\left (b x + a\right )}\right ) - e^{\left (b x + a\right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 34, normalized size = 1.36 \begin {gather*} \frac {{\mathrm {e}}^{a+b\,x}}{b}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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