Optimal. Leaf size=59 \[ \frac {e^{2 a+2 b x}}{2 b}+\frac {e^{4 a+4 b x}}{16 b}+\frac {\log \left (1-e^{2 a+2 b x}\right )}{b}-\frac {x}{4} \]
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Rubi [A] time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2282, 12, 446, 72} \[ \frac {e^{2 a+2 b x}}{2 b}+\frac {e^{4 a+4 b x}}{16 b}+\frac {\log \left (1-e^{2 a+2 b x}\right )}{b}-\frac {x}{4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 72
Rule 446
Rule 2282
Rubi steps
\begin {align*} \int e^{2 (a+b x)} \cosh ^2(a+b x) \coth (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{4 x \left (-1+x^2\right )} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x \left (-1+x^2\right )} \, dx,x,e^{a+b x}\right )}{4 b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(1+x)^3}{(-1+x) x} \, dx,x,e^{2 a+2 b x}\right )}{8 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (4+\frac {8}{-1+x}-\frac {1}{x}+x\right ) \, dx,x,e^{2 a+2 b x}\right )}{8 b}\\ &=\frac {e^{2 a+2 b x}}{2 b}+\frac {e^{4 a+4 b x}}{16 b}-\frac {x}{4}+\frac {\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 \[ \frac {8 e^{2 (a+b x)}+e^{4 (a+b x)}+16 \log \left (1-e^{2 (a+b x)}\right )-4 b x}{16 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 127, normalized size = 2.15 \[ \frac {\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 4\right )} \sinh \left (b x + a\right )^{2} - 4 \, b x + 8 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 16 \, \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 52, normalized size = 0.88 \[ -\frac {4 \, b x - {\left (e^{\left (4 \, b x + 8 \, a\right )} + 8 \, e^{\left (2 \, b x + 6 \, a\right )}\right )} e^{\left (-4 \, a\right )} - 16 \, \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 55, normalized size = 0.93 \[ -\frac {x}{4}+\frac {{\mathrm e}^{4 b x +4 a}}{16 b}+\frac {{\mathrm e}^{2 b x +2 a}}{2 b}-\frac {2 a}{b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 70, normalized size = 1.19 \[ \frac {{\left (8 \, e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{16 \, b} + \frac {7 \, {\left (b x + a\right )}}{4 \, b} + \frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 49, normalized size = 0.83 \[ \frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-1\right )}{b}-\frac {x}{4}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}}{2\,b}+\frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{16\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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