Optimal. Leaf size=41 \[ \frac {2}{b \left (1-e^{2 a+2 b x}\right )}+\frac {\log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2320, 12, 455,
45} \begin {gather*} \frac {2}{b \left (1-e^{2 a+2 b x}\right )}+\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 (a+b x) \text {csch}(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {2 x \left (1+x^2\right )}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {2 \text {Subst}\left (\int \frac {x \left (1+x^2\right )}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {1+x}{(1-x)^2} \, dx,x,e^{2 a+2 b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \left (\frac {2}{(-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 {\log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 34, normalized size = 0.83 \begin {gather*} \frac {-\frac {2}{-1+e^{2 (a+b x)}}+\log \left (1-e^{2 (a+b x)}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.93, size = 25, normalized size = 0.61
method | result | size |
derivativedivides | \(\frac {\ln \left (\sinh \left (b x +a \right )\right )+b x +a -\coth \left (b x +a \right )}{b}\) | \(25\) |
default | \(\frac {\ln \left (\sinh \left (b x +a \right )\right )+b x +a -\coth \left (b x +a \right )}{b}\) | \(25\) |
risch | \(-\frac {2 a}{b}-\frac {2}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 45, normalized size = 1.10 \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}{b {\left (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. 103 vs.
\(2 (37) = 74\).
time = 0.34, size = 103, normalized size = 2.51 \begin {gather*} \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 (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) - 2}{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 46, normalized size = 1.12 \begin {gather*} -\frac {\frac {e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.77, size = 36, normalized size = 0.88 \begin {gather*} \frac {\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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