Optimal. Leaf size=54 \[ \frac {2 e^{a+b x}}{b}+\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {4 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2320, 12, 466,
396, 212} \begin {gather*} \frac {2 e^{a+b x}}{b}+\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {4 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 396
Rule 466
Rule 2320
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \coth (a+b x) \text {csch}(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {2 x^2 \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^2 \left (1+x^2\right )}{\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 {\text {Subst}\left (\int \frac {2+2 x^2}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {2 e^{a+b x}}{b}+\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {4 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {2 e^{a+b x}}{b}+\frac {2 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {4 \tanh ^{-1}\left (e^{a+b x}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 62, normalized size = 1.15 \begin {gather*} \frac {2 \left (\frac {e^{a+b x} \left (-2+e^{2 (a+b x)}\right )}{-1+e^{2 (a+b x)}}+\log \left (1-e^{a+b x}\right )-\log \left (1+e^{a+b x}\right )\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.88, size = 65, normalized size = 1.20
method | result | size |
risch | \(\frac {2 \,{\mathrm e}^{b x +a}}{b}-\frac {2 \,{\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}-\frac {2 \ln \left ({\mathrm e}^{b x +a}+1\right )}{b}+\frac {2 \ln \left ({\mathrm e}^{b x +a}-1\right )}{b}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 76, normalized size = 1.41 \begin {gather*} -\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 (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}}{b {\left (e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}\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. 200 vs.
\(2 (48) = 96\).
time = 0.35, size = 200, normalized size = 3.70 \begin {gather*} \frac {2 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} - {\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 ) + {\left (3 \, \cosh \left (b x + a\right )^{2} - 2\right )} \sinh \left (b x + a\right ) - 2 \, \cosh \left (b x + a\right )\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} \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 = 0.42, size = 55, normalized size = 1.02 \begin {gather*} -\frac {2 \, {\left (\frac {e^{\left (b x + a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - e^{\left (b x + a\right )} + \log \left (e^{\left (b x + a\right )} + 1\right ) - \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )\right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.76, size = 63, normalized size = 1.17 \begin {gather*} \frac {2\,{\mathrm {e}}^{a+b\,x}}{b}-\frac {4\,\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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