Optimal. Leaf size=29 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a+b}}\right )}{\sqrt{a} \sqrt{a+b}} \]
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Rubi [A] time = 0.0218575, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3181, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a+b}}\right )}{\sqrt{a} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 3181
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{a+b \cosh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a-(a+b) x^2} \, dx,x,\coth (x)\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a+b}}\right )}{\sqrt{a} \sqrt{a+b}}\\ \end{align*}
Mathematica [A] time = 0.0632053, size = 29, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tanh (x)}{\sqrt{a+b}}\right )}{\sqrt{a} \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 78, normalized size = 2.7 \begin{align*}{\frac{1}{2}\ln \left ( \sqrt{a+b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,\sqrt{a}\tanh \left ( x/2 \right ) +\sqrt{a+b} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{a+b}}}}-{\frac{1}{2}\ln \left ( \sqrt{a+b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}-2\,\sqrt{a}\tanh \left ( x/2 \right ) +\sqrt{a+b} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{a+b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98422, size = 828, normalized size = 28.55 \begin{align*} \left [\frac{\log \left (\frac{b^{2} \cosh \left (x\right )^{4} + 4 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{2} \sinh \left (x\right )^{4} + 2 \,{\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, b^{2} \cosh \left (x\right )^{2} + 2 \, a b + b^{2}\right )} \sinh \left (x\right )^{2} + 8 \, a^{2} + 8 \, a b + b^{2} + 4 \,{\left (b^{2} \cosh \left (x\right )^{3} +{\left (2 \, a b + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 4 \,{\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} + 2 \, a + b\right )} \sqrt{a^{2} + a b}}{b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \,{\left (2 \, a + b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, b \cosh \left (x\right )^{2} + 2 \, a + b\right )} \sinh \left (x\right )^{2} + 4 \,{\left (b \cosh \left (x\right )^{3} +{\left (2 \, a + b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b}\right )}{2 \, \sqrt{a^{2} + a b}}, \frac{\sqrt{-a^{2} - a b} \arctan \left (\frac{{\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} + 2 \, a + b\right )} \sqrt{-a^{2} - a b}}{2 \,{\left (a^{2} + a b\right )}}\right )}{a^{2} + a b}\right ] \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 [A] time = 1.2333, size = 53, normalized size = 1.83 \begin{align*} \frac{\arctan \left (\frac{b e^{\left (2 \, x\right )} + 2 \, a + b}{2 \, \sqrt{-a^{2} - a b}}\right )}{\sqrt{-a^{2} - a b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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