Optimal. Leaf size=37 \[ 2 \sqrt {a+b \cosh (x)}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2721, 50, 63, 207} \[ 2 \sqrt {a+b \cosh (x)}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 207
Rule 2721
Rubi steps
\begin {align*} \int \sqrt {a+b \cosh (x)} \tanh (x) \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {a+x}}{x} \, dx,x,b \cosh (x)\right )\\ &=2 \sqrt {a+b \cosh (x)}+a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+x}} \, dx,x,b \cosh (x)\right )\\ &=2 \sqrt {a+b \cosh (x)}+(2 a) \operatorname {Subst}\left (\int \frac {1}{-a+x^2} \, dx,x,\sqrt {a+b \cosh (x)}\right )\\ &=-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right )+2 \sqrt {a+b \cosh (x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 1.00 \[ 2 \sqrt {a+b \cosh (x)}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 376, normalized size = 10.16 \[ \left [\frac {1}{2} \, \sqrt {a} \log \left (-\frac {b^{2} \cosh \relax (x)^{4} + b^{2} \sinh \relax (x)^{4} + 16 \, a b \cosh \relax (x)^{3} + 4 \, {\left (b^{2} \cosh \relax (x) + 4 \, a b\right )} \sinh \relax (x)^{3} + 16 \, a b \cosh \relax (x) + 2 \, {\left (16 \, a^{2} + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b^{2} \cosh \relax (x)^{2} + 24 \, a b \cosh \relax (x) + 16 \, a^{2} + b^{2}\right )} \sinh \relax (x)^{2} - 8 \, {\left (b \cosh \relax (x)^{3} + b \sinh \relax (x)^{3} + 4 \, a \cosh \relax (x)^{2} + {\left (3 \, b \cosh \relax (x) + 4 \, a\right )} \sinh \relax (x)^{2} + b \cosh \relax (x) + {\left (3 \, b \cosh \relax (x)^{2} + 8 \, a \cosh \relax (x) + b\right )} \sinh \relax (x)\right )} \sqrt {b \cosh \relax (x) + a} \sqrt {a} + b^{2} + 4 \, {\left (b^{2} \cosh \relax (x)^{3} + 12 \, a b \cosh \relax (x)^{2} + 4 \, a b + {\left (16 \, a^{2} + b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1}\right ) + 2 \, \sqrt {b \cosh \relax (x) + a}, \sqrt {-a} \arctan \left (\frac {{\left (b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 4 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + 2 \, a\right )} \sinh \relax (x) + b\right )} \sqrt {b \cosh \relax (x) + a} \sqrt {-a}}{2 \, {\left (a b \cosh \relax (x)^{2} + a b \sinh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) + a b + 2 \, {\left (a b \cosh \relax (x) + a^{2}\right )} \sinh \relax (x)\right )}}\right ) + 2 \, \sqrt {b \cosh \relax (x) + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cosh \relax (x) + a} \tanh \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 30, normalized size = 0.81 \[ -2 \arctanh \left (\frac {\sqrt {a +b \cosh \relax (x )}}{\sqrt {a}}\right ) \sqrt {a}+2 \sqrt {a +b \cosh \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \cosh \relax (x) + a} \tanh \relax (x)\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {tanh}\relax (x)\,\sqrt {a+b\,\mathrm {cosh}\relax (x)} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \cosh {\relax (x )}} \tanh {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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