Optimal. Leaf size=59 \[ \frac {e^{2 a+2 b x}}{2 b}+\frac {2}{b \left (1-e^{2 a+2 b x}\right )}+\frac {2 \log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A]
time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2320, 455, 45}
\begin {gather*} \frac {e^{2 a+2 b x}}{2 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 {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 455
Rule 2320
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \coth ^2(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {x \left (1+x^2\right )^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {(1+x)^2}{(1-x)^2} \, dx,x,e^{2 a+2 b x}\right )}{2 b}\\ &=\frac {\text {Subst}\left (\int \left (1+\frac {4}{(-1+x)^2}+\frac {4}{-1+x}\right ) \, dx,x,e^{2 a+2 b x}\right )}{2 b}\\ &=\frac {e^{2 a+2 b x}}{2 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 = 48, normalized size = 0.81 \begin {gather*} \frac {e^{2 (a+b x)}-\frac {4}{-1+e^{2 (a+b x)}}+4 \log \left (1-e^{2 (a+b x)}\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.17, size = 57, normalized size = 0.97
method | result | size |
risch | \(\frac {{\mathrm e}^{2 b x +2 a}}{2 b}-\frac {4 a}{b}-\frac {2}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {2 \ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 86, normalized size = 1.46 \begin {gather*} \frac {4 \, {\left (b x + a\right )}}{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 {5 \, e^{\left (-2 \, b x - 2 \, a\right )} - 1}{2 \, b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, 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. 195 vs.
\(2 (52) = 104\).
time = 0.37, size = 195, normalized size = 3.31 \begin {gather*} \frac {\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} + {\left (6 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - \cosh \left (b x + a\right )^{2} + 4 \, {\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 \, {\left (2 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 4}{2 \, {\left (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\right )}} \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 = 0.41, size = 56, normalized size = 0.95 \begin {gather*} -\frac {\frac {4 \, e^{\left (2 \, b x + 2 \, a\right )}}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - e^{\left (2 \, b x + 2 \, a\right )} - 4 \, \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 = 1.85, size = 51, normalized size = 0.86 \begin {gather*} \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 )}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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