Optimal. Leaf size=20 \[ e^x x-e^x+\frac {e^{2 x}}{2} \]
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Rubi [A] time = 0.07, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5648, 6742, 2176, 2194, 2282, 12, 14} \[ e^x x-e^x+\frac {e^{2 x}}{2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2176
Rule 2194
Rule 2282
Rule 5648
Rule 6742
Rubi steps
\begin {align*} \int \frac {x+\cosh (x)+\sinh (x)}{\cosh (x)-\sinh (x)} \, dx &=\int e^x (x+\cosh (x)+\sinh (x)) \, dx\\ &=\int \left (e^x x+e^x \cosh (x)+e^x \sinh (x)\right ) \, dx\\ &=\int e^x x \, dx+\int e^x \cosh (x) \, dx+\int e^x \sinh (x) \, dx\\ &=e^x x-\int e^x \, dx+\operatorname {Subst}\left (\int \frac {-1+x^2}{2 x} \, dx,x,e^x\right )+\operatorname {Subst}\left (\int \frac {1+x^2}{2 x} \, dx,x,e^x\right )\\ &=-e^x+e^x x+\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+x^2}{x} \, dx,x,e^x\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{x} \, dx,x,e^x\right )\\ &=-e^x+e^x x+\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{x}+x\right ) \, dx,x,e^x\right )+\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x}+x\right ) \, dx,x,e^x\right )\\ &=-e^x+\frac {e^{2 x}}{2}+e^x x\\ \end {align*}
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Mathematica [A] time = 0.10, size = 23, normalized size = 1.15 \[ (x-1) \sinh (x)+\frac {1}{2} \cosh (2 x)+(x+\sinh (x)-1) \cosh (x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x+\cosh (x)+\sinh (x)}{\cosh (x)-\sinh (x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.36, size = 20, normalized size = 1.00 \[ \frac {2 \, x + \cosh \relax (x) + \sinh \relax (x) - 2}{2 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 11, normalized size = 0.55 \[ \frac {1}{2} \, {\left (2 \, x + e^{x} - 2\right )} e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 14, normalized size = 0.70
method | result | size |
risch | \(\left (-1+x \right ) {\mathrm e}^{x}+\frac {{\mathrm e}^{2 x}}{2}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 13, normalized size = 0.65 \[ {\left (x - 1\right )} e^{x} + \frac {1}{2} \, e^{\left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 16, normalized size = 0.80 \[ {\mathrm {e}}^x\,\left (x+\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 26, normalized size = 1.30 \[ \frac {x}{- \sinh {\relax (x )} + \cosh {\relax (x )}} + \frac {\sinh {\relax (x )}}{- \sinh {\relax (x )} + \cosh {\relax (x )}} - \frac {1}{- \sinh {\relax (x )} + \cosh {\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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