Optimal. Leaf size=26 \[ \frac {2 e^{a+b x}}{b}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2282, 12, 321, 207} \[ \frac {2 e^{a+b x}}{b}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 207
Rule 321
Rule 2282
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \text {csch}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {2 x^2}{-1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {2 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 {2 e^{a+b x}}{b}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.88 \[ \frac {2 \left (e^{a+b x}-\tanh ^{-1}\left (e^{a+b x}\right )\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 53, normalized size = 2.04 \[ \frac {2 \, \cosh \left (b x + a\right ) - \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, \sinh \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 50, normalized size = 1.92 \[ -\frac {{\left (e^{\left (-2 \, a\right )} \log \left (e^{\left (b x + a\right )} + 1\right ) - e^{\left (-2 \, a\right )} \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right ) - 2 \, e^{\left (b x - a\right )}\right )} e^{\left (2 \, a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 40, normalized size = 1.54 \[ \frac {2 \,{\mathrm e}^{b x +a}}{b}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 45, normalized size = 1.73 \[ \frac {2 \, e^{\left (b x + a\right )}}{b} - \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 39, normalized size = 1.50 \[ \frac {2\,{\mathrm {e}}^{a+b\,x}}{b}-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}} \]
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 \[ e^{2 a} \int e^{2 b x} \operatorname {csch}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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