Optimal. Leaf size=63 \[ \frac {6}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {2 \log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A] time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2282, 12, 446, 77} \[ \frac {6}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {2 \log \left (1-e^{2 a+2 b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 12
Rule 77
Rule 446
Rule 2282
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \coth (a+b x) \text {csch}^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {4 x^3 \left (-1-x^2\right )}{\left (1-x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {x^3 \left (-1-x^2\right )}{\left (1-x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {(-1-x) x}{(1-x)^3} \, dx,x,e^{2 a+2 b x}\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (\frac {2}{(-1+x)^3}+\frac {3}{(-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 )^2}+\frac {6}{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 = 0.08, size = 47, normalized size = 0.75 \[ \frac {2 \left (\frac {2-3 e^{2 (a+b x)}}{\left (e^{2 (a+b x)}-1\right )^2}+\log \left (1-e^{2 (a+b x)}\right )\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 262, normalized size = 4.16 \[ -\frac {2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 3 \, \sinh \left (b x + a\right )^{2} - 2\right )}}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 48, normalized size = 0.76 \[ -\frac {\frac {3 \, e^{\left (4 \, b x + 4 \, a\right )} - 1}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} - 2 \, \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 56, normalized size = 0.89 \[ -\frac {4 a}{b}-\frac {2 \left (3 \,{\mathrm e}^{2 b x +2 a}-2\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {2 \ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 86, normalized size = 1.37 \[ 4 \, x + \frac {4 \, a}{b} + \frac {2 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {2 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{b} - \frac {2 \, {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 2\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.86, size = 66, normalized size = 1.05 \[ \frac {2\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-\frac {6}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}-\frac {2}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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