Optimal. Leaf size=63 \[ \frac {e^{2 a+2 b x}}{4 b}+\frac {2}{b \left (1-e^{2 a+2 b x}\right )}+\frac {x}{2}+\frac {\log \left (1-e^{2 a+2 b x}\right )}{b} \]
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Rubi [A]
time = 0.04, antiderivative size = 63, 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, 457,
90} \begin {gather*} \frac {e^{2 a+2 b x}}{4 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}+\frac {x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 90
Rule 457
Rule 2320
Rubi steps
\begin {align*} \int e^{a+b x} \cosh (a+b x) \coth ^2(a+b x) \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{2 x \left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x \left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{2 b}\\ &=\frac {\text {Subst}\left (\int \frac {(1+x)^3}{(1-x)^2 x} \, dx,x,e^{2 a+2 b x}\right )}{4 b}\\ &=\frac {\text {Subst}\left (\int \left (1+\frac {8}{(-1+x)^2}+\frac {4}{-1+x}+\frac {1}{x}\right ) \, dx,x,e^{2 a+2 b x}\right )}{4 b}\\ &=\frac {e^{2 a+2 b x}}{4 b}+\frac {2}{b \left (1-e^{2 a+2 b x}\right )}+\frac {x}{2}+\frac {\log \left (1-e^{2 a+2 b x}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 52, normalized size = 0.83 \begin {gather*} \frac {e^{2 (a+b x)}-\frac {8}{-1+e^{2 (a+b x)}}+2 b x+4 \log \left (1-e^{2 (a+b x)}\right )}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.93, size = 56, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{2}+\ln \left (\sinh \left (b x +a \right )\right )+\frac {\cosh ^{3}\left (b x +a \right )}{2 \sinh \left (b x +a \right )}+\frac {3 b x}{2}+\frac {3 a}{2}-\frac {3 \coth \left (b x +a \right )}{2}}{b}\) | \(56\) |
default | \(\frac {\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{2}+\ln \left (\sinh \left (b x +a \right )\right )+\frac {\cosh ^{3}\left (b x +a \right )}{2 \sinh \left (b x +a \right )}+\frac {3 b x}{2}+\frac {3 a}{2}-\frac {3 \coth \left (b x +a \right )}{2}}{b}\) | \(56\) |
risch | \(\frac {x}{2}+\frac {{\mathrm e}^{2 b x +2 a}}{4 b}-\frac {2 a}{b}-\frac {2}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 68, normalized size = 1.08 \begin {gather*} \frac {1}{2} \, x + \frac {a}{2 \, b} + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{4 \, b} + \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. 213 vs.
\(2 (54) = 108\).
time = 0.36, size = 213, normalized size = 3.38 \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 (2 \, b x - 1\right )} \cosh \left (b x + a\right )^{2} + {\left (2 \, b x + 6 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, b x + 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} + {\left (2 \, b x - 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 8}{4 \, {\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(-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 = 0.42, size = 63, normalized size = 1.00 \begin {gather*} \frac {2 \, b x + 2 \, a - \frac {4 \, {\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\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 )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.82, size = 53, normalized size = 0.84 \begin {gather*} \frac {x}{2}+\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 )}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}}{4\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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