Optimal. Leaf size=69 \[ \frac {1}{2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^2}+\frac {1}{4} \log \left (1-\tanh \left (\frac {x}{2}\right )\right )+\frac {3}{4} \log \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]
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Rubi [A] time = 0.20, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4397, 12, 894} \[ \frac {1}{2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^2}+\frac {1}{4} \log \left (1-\tanh \left (\frac {x}{2}\right )\right )+\frac {3}{4} \log \left (\tanh \left (\frac {x}{2}\right )+1\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 894
Rule 4397
Rubi steps
\begin {align*} \int \frac {1+\sinh ^2(x)}{1+\cosh (x)+\sinh (x)} \, dx &=\int \frac {\cosh ^2(x)}{1+\cosh (x)+\sinh (x)} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{2 (1-x)^2 (1+x)^3} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\\ &=\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{(1-x)^2 (1+x)^3} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {1}{2 (-1+x)^2}+\frac {1}{4 (-1+x)}+\frac {1}{(1+x)^3}-\frac {1}{(1+x)^2}+\frac {3}{4 (1+x)}\right ) \, dx,x,\tanh \left (\frac {x}{2}\right )\right )\\ &=\frac {1}{4} \log \left (1-\tanh \left (\frac {x}{2}\right )\right )+\frac {3}{4} \log \left (1+\tanh \left (\frac {x}{2}\right )\right )+\frac {1}{2 \left (1-\tanh \left (\frac {x}{2}\right )\right )}-\frac {1}{2 \left (1+\tanh \left (\frac {x}{2}\right )\right )^2}+\frac {1}{1+\tanh \left (\frac {x}{2}\right )}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 37, normalized size = 0.54 \[ \frac {x}{4}+\frac {1}{8} \sinh (2 x)+\frac {\cosh (x)}{2}-\frac {1}{8} \cosh (2 x)-\log \left (\cosh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 95, normalized size = 1.38 \[ \frac {6 \, x \cosh \relax (x)^{2} + 2 \, \cosh \relax (x)^{3} + 6 \, {\left (x + \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 2 \, \sinh \relax (x)^{3} - 8 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + 2 \, {\left (6 \, x \cosh \relax (x) + 3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x) + 2 \, \cosh \relax (x) - 1}{8 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 27, normalized size = 0.39 \[ \frac {1}{8} \, {\left (2 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )} + \frac {3}{4} \, x + \frac {1}{4} \, e^{x} - \log \left (e^{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 48, normalized size = 0.70 \[ -\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{4}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 29, normalized size = 0.42 \[ -\frac {1}{4} \, x + \frac {1}{4} \, e^{\left (-x\right )} - \frac {1}{8} \, e^{\left (-2 \, x\right )} + \frac {1}{4} \, e^{x} - \log \left (e^{\left (-x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.72, size = 27, normalized size = 0.39 \[ \frac {3\,x}{4}+\frac {{\mathrm {e}}^{-x}}{4}-\frac {{\mathrm {e}}^{-2\,x}}{8}-\ln \left ({\mathrm {e}}^x+1\right )+\frac {{\mathrm {e}}^x}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.18, size = 381, normalized size = 5.52 \[ - \frac {x \tanh ^{3}{\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} - \frac {x \tanh ^{2}{\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} + \frac {x \tanh {\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} + \frac {x}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} + \frac {4 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} \tanh ^{3}{\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} + \frac {4 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} \tanh ^{2}{\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} - \frac {4 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )} \tanh {\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} - \frac {4 \log {\left (\tanh {\left (\frac {x}{2} \right )} + 1 \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} + \frac {2 \tanh ^{2}{\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} - \frac {6 \tanh {\left (\frac {x}{2} \right )}}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} - \frac {4}{4 \tanh ^{3}{\left (\frac {x}{2} \right )} + 4 \tanh ^{2}{\left (\frac {x}{2} \right )} - 4 \tanh {\left (\frac {x}{2} \right )} - 4} \]
Verification of antiderivative is not currently implemented for this CAS.
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