Optimal. Leaf size=62 \[ -\frac {2}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {4}{b \left (1-e^{2 a+2 b x}\right )}+\frac {\log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A]
time = 0.05, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2320, 12, 455,
45} \begin {gather*} \frac {4}{b \left (1-e^{2 a+2 b x}\right )}-\frac {2}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {\log \left (1-e^{2 a+2 b x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 455
Rule 2320
Rubi steps
\begin {align*} \int e^{a+b x} \coth ^2(a+b x) \text {csch}(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {2 x \left (1+x^2\right )^2}{\left (-1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {2 \text {Subst}\left (\int \frac {x \left (1+x^2\right )^2}{\left (-1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {(1+x)^2}{(-1+x)^3} \, dx,x,e^{2 a+2 b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \left (\frac {4}{(-1+x)^3}+\frac {4}{(-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 {4}{b \left (1-e^{2 a+2 b x}\right )}+\frac {\log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 46, normalized size = 0.74 \begin {gather*} \frac {\frac {2-4 e^{2 (a+b x)}}{\left (-1+e^{2 (a+b x)}\right )^2}+\log \left (1-e^{2 (a+b x)}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.18, size = 35, normalized size = 0.56
method | result | size |
derivativedivides | \(\frac {b x +a -\coth \left (b x +a \right )+\ln \left (\sinh \left (b x +a \right )\right )-\frac {\left (\coth ^{2}\left (b x +a \right )\right )}{2}}{b}\) | \(35\) |
default | \(\frac {b x +a -\coth \left (b x +a \right )+\ln \left (\sinh \left (b x +a \right )\right )-\frac {\left (\coth ^{2}\left (b x +a \right )\right )}{2}}{b}\) | \(35\) |
risch | \(-\frac {2 a}{b}-\frac {2 \left (2 \,{\mathrm e}^{2 b x +2 a}-1\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 69, normalized size = 1.11 \begin {gather*} \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 \, {\left (2 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}}{b {\left (e^{\left (4 \, b x + 4 \, a\right )} - 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 262 vs.
\(2 (55) = 110\).
time = 0.33, size = 262, normalized size = 4.23 \begin {gather*} -\frac {4 \, \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 ) + 8 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 4 \, \sinh \left (b x + a\right )^{2} - 2}{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 59, normalized size = 0.95 \begin {gather*} -\frac {\frac {3 \, e^{\left (4 \, b x + 4 \, a\right )} + 2 \, e^{\left (2 \, b x + 2 \, 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 )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.84, size = 65, normalized size = 1.05 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-\frac {4}{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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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