Optimal. Leaf size=19 \[ \sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )-x \]
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Rubi [A] time = 0.0415577, 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} \[ \sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )-x \]
Antiderivative was successfully verified.
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Rule 3171
Rule 3181
Rule 206
Rubi steps
\begin{align*} \int \frac{1+\sinh ^2(x)}{1-\sinh ^2(x)} \, dx &=-x+2 \int \frac{1}{1-\sinh ^2(x)} \, dx\\ &=-x+2 \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\tanh (x)\right )\\ &=-x+\sqrt{2} \tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )\\ \end{align*}
Mathematica [A] time = 0.031268, size = 24, normalized size = 1.26 \[ -2 \left (\frac{x}{2}-\frac{\tanh ^{-1}\left (\sqrt{2} \tanh (x)\right )}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 54, normalized size = 2.8 \begin{align*} -\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) +\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) +2 \right ) } \right ) +\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) +\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{4} \left ( 2\,\tanh \left ( x/2 \right ) -2 \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.61228, size = 86, normalized size = 4.53 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} + 1}{\sqrt{2} + e^{\left (-x\right )} - 1}\right ) - \frac{1}{2} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - e^{\left (-x\right )} - 1}{\sqrt{2} + e^{\left (-x\right )} + 1}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.4234, 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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15389, size = 55, normalized size = 2.89 \begin{align*} -\frac{1}{2} \, \sqrt{2} \log \left (\frac{{\left | -4 \, \sqrt{2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}{{\left | 4 \, \sqrt{2} + 2 \, e^{\left (2 \, x\right )} - 6 \right |}}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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