Optimal. Leaf size=42 \[ \frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 42, 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, 288, 206} \[ \frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 288
Rule 2282
Rubi steps
\begin {align*} \int e^{a+b x} \text {csch}^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {4 x^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\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 \left (1-e^{2 a+2 b x}\right )}-\frac {2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 38, normalized size = 0.90 \[ \frac {2 \left (-\frac {e^{a+b x}}{e^{2 (a+b x)}-1}-\tanh ^{-1}\left (e^{a+b x}\right )\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 157, normalized size = 3.74 \[ -\frac {{\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, \cosh \left (b x + a\right ) + 2 \, \sinh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 48, normalized size = 1.14 \[ -\frac {\frac {2 \, e^{\left (b x + a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1} + \log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 25, normalized size = 0.60 \[ \frac {-2 \arctanh \left ({\mathrm e}^{b x +a}\right )-\frac {1}{\sinh \left (b x +a \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 52, normalized size = 1.24 \[ -\frac {\log \left (e^{\left (b x + a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (b x + a\right )} - 1\right )}{b} - \frac {2 \, e^{\left (b x + a\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 52, normalized size = 1.24 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
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 \[ e^{a} \int e^{b x} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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