Optimal. Leaf size=39 \[ \sqrt{a+b \cosh ^2(x)}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cosh ^2(x)}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0687702, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3194, 50, 63, 208} \[ \sqrt{a+b \cosh ^2(x)}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cosh ^2(x)}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 3194
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \sqrt{a+b \cosh ^2(x)} \tanh (x) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\cosh ^2(x)\right )\\ &=\sqrt{a+b \cosh ^2(x)}+\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\cosh ^2(x)\right )\\ &=\sqrt{a+b \cosh ^2(x)}+\frac{a \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \cosh ^2(x)}\right )}{b}\\ &=-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cosh ^2(x)}}{\sqrt{a}}\right )+\sqrt{a+b \cosh ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.0248286, size = 39, normalized size = 1. \[ \sqrt{a+b \cosh ^2(x)}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \cosh ^2(x)}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 42, normalized size = 1.1 \begin{align*} \sqrt{a+b \left ( \cosh \left ( x \right ) \right ) ^{2}}-\sqrt{a}\ln \left ({\frac{1}{\cosh \left ( x \right ) } \left ( 2\,a+2\,\sqrt{a}\sqrt{a+b \left ( \cosh \left ( x \right ) \right ) ^{2}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cosh \left (x\right )^{2} + a} \tanh \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 4.66662, size = 1192, normalized size = 30.56 \begin{align*} \left [\frac{\sqrt{a}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\frac{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 (4 \, a + b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, b \cosh \left (x\right )^{2} + 4 \, a + b\right )} \sinh \left (x\right )^{2} - 4 \, \sqrt{2} \sqrt{a} \sqrt{\frac{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a + b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 4 \,{\left (b \cosh \left (x\right )^{3} +{\left (4 \, a + b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right ) + \sqrt{2} \sqrt{\frac{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a + b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}, \frac{2 \, \sqrt{-a}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a + b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{2 \,{\left (a \cosh \left (x\right ) + a \sinh \left (x\right )\right )}}\right ) + \sqrt{2} \sqrt{\frac{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a + b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{2 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \cosh ^{2}{\left (x \right )}} \tanh{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \cosh \left (x\right )^{2} + a} \tanh \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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