Optimal. Leaf size=19 \[ \sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )-x \]
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Rubi [A] time = 0.0394022, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3171, 3181, 206} \[ \sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )-x \]
Antiderivative was successfully verified.
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Rule 3171
Rule 3181
Rule 206
Rubi steps
\begin{align*} \int \frac{1-\cosh ^2(x)}{1+\cosh ^2(x)} \, dx &=-x+2 \int \frac{1}{1+\cosh ^2(x)} \, dx\\ &=-x+2 \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\coth (x)\right )\\ &=-x+\sqrt{2} \tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0302714, size = 24, normalized size = 1.26 \[ -2 \left (\frac{x}{2}-\frac{\tanh ^{-1}\left (\frac{\tanh (x)}{\sqrt{2}}\right )}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 102, normalized size = 5.4 \begin{align*} -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +{\frac{\sqrt{2}}{4}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{4}\ln \left ({ \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+\sqrt{2}\tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}} \right ) }+\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.6124, size = 138, normalized size = 7.26 \begin{align*} \frac{3}{16} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (2 \, x\right )} + 3}\right ) - \frac{5}{16} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (-2 \, x\right )} + 3}\right ) - 2 \, x + \frac{1}{4} \, \log \left (e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1\right ) - \frac{1}{4} \, \log \left (6 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.27978, size = 220, normalized size = 11.58 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{3 \,{\left (2 \, \sqrt{2} - 3\right )} \cosh \left (x\right )^{2} - 4 \,{\left (3 \, \sqrt{2} - 4\right )} \cosh \left (x\right ) \sinh \left (x\right ) + 3 \,{\left (2 \, \sqrt{2} - 3\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt{2} - 3}{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} + 3}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.58405, size = 61, normalized size = 3.21 \begin{align*} - x - \frac{\sqrt{2} \log{\left (4 \tanh ^{2}{\left (\frac{x}{2} \right )} - 4 \sqrt{2} \tanh{\left (\frac{x}{2} \right )} + 4 \right )}}{2} + \frac{\sqrt{2} \log{\left (4 \tanh ^{2}{\left (\frac{x}{2} \right )} + 4 \sqrt{2} \tanh{\left (\frac{x}{2} \right )} + 4 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15681, size = 51, normalized size = 2.68 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt{2} + e^{\left (2 \, x\right )} + 3}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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