Optimal. Leaf size=42 \[ \frac {2}{b \left (1-e^{2 a+2 b x}\right )}+\frac {2 \log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A] time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2282, 12, 266, 43} \[ \frac {2}{b \left (1-e^{2 a+2 b x}\right )}+\frac {2 \log \left (1-e^{2 a+2 b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 266
Rule 2282
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \text {csch}^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {4 x^3}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {x^3}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {x}{(1-x)^2} \, dx,x,e^{2 a+2 b x}\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {1}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx,x,e^{2 a+2 b x}\right )}{b}\\ &=\frac {2}{b \left (1-e^{2 a+2 b x}\right )}+\frac {2 \log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 37, normalized size = 0.88 \[ \frac {2 \left (\frac {1}{1-e^{2 a+2 b x}}+\log \left (1-e^{2 a+2 b x}\right )\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 104, normalized size = 2.48 \[ \frac {2 \, {\left ({\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) - 1\right )}}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 48, normalized size = 1.14 \[ \frac {2 \, {\left (e^{\left (-2 \, a\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right ) - \frac {e^{\left (2 \, b x\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1}\right )} e^{\left (2 \, a\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 43, normalized size = 1.02 \[ -\frac {4 a}{b}-\frac {2}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {2 \ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 62, normalized size = 1.48 \[ 4 \, x + \frac {4 \, a}{b} + \frac {2 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {2 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{b} + \frac {2}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.79, size = 37, normalized size = 0.88 \[ \frac {2\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
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 \[ e^{2 a} \int e^{2 b x} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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