Optimal. Leaf size=29 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^n(x)}}{\sqrt {a}}\right )}{\sqrt {a} n} \]
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Rubi [A] time = 0.09, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 = {3230, 266, 63, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^n(x)}}{\sqrt {a}}\right )}{\sqrt {a} n} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 3230
Rubi steps
\begin {align*} \int \frac {\tanh (x)}{\sqrt {a+b \cosh ^n(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^n}} \, dx,x,\cosh (x)\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\cosh ^n(x)\right )}{n}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^n(x)}\right )}{b n}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^n(x)}}{\sqrt {a}}\right )}{\sqrt {a} n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh ^n(x)}}{\sqrt {a}}\right )}{\sqrt {a} n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 113, normalized size = 3.90 \[ \left [\frac {\log \left (\frac {b \cosh \left (n \log \left (\cosh \relax (x)\right )\right ) + b \sinh \left (n \log \left (\cosh \relax (x)\right )\right ) - 2 \, \sqrt {b \cosh \left (n \log \left (\cosh \relax (x)\right )\right ) + b \sinh \left (n \log \left (\cosh \relax (x)\right )\right ) + a} \sqrt {a} + 2 \, a}{\cosh \left (n \log \left (\cosh \relax (x)\right )\right ) + \sinh \left (n \log \left (\cosh \relax (x)\right )\right )}\right )}{\sqrt {a} n}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b \cosh \left (n \log \left (\cosh \relax (x)\right )\right ) + b \sinh \left (n \log \left (\cosh \relax (x)\right )\right ) + a} \sqrt {-a}}{a}\right )}{a n}\right ] \]
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 \[ \int \frac {\tanh \relax (x)}{\sqrt {b \cosh \relax (x)^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 24, normalized size = 0.83 \[ -\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (\cosh ^{n}\relax (x )\right )}}{\sqrt {a}}\right )}{n \sqrt {a}} \]
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 \[ \int \frac {\tanh \relax (x)}{\sqrt {b \cosh \relax (x)^{n} + a}}\,{d x} \]
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 \[ \int \frac {\mathrm {tanh}\relax (x)}{\sqrt {a+b\,{\mathrm {cosh}\relax (x)}^n}} \,d x \]
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 \[ \int \frac {\tanh {\relax (x )}}{\sqrt {a + b \cosh ^{n}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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