Optimal. Leaf size=16 \[ \frac {\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt {a \cosh ^2(x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3207, 3770} \[ \frac {\cosh (x) \tan ^{-1}(\sinh (x))}{\sqrt {a \cosh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 3770
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \cosh ^2(x)}} \, dx &=\frac {\cosh (x) \int \text {sech}(x) \, dx}{\sqrt {a \cosh ^2(x)}}\\ &=\frac {\tan ^{-1}(\sinh (x)) \cosh (x)}{\sqrt {a \cosh ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.31 \[ \frac {2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )}{\sqrt {a \cosh ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.32, size = 186, normalized size = 11.62 \[ \left [-\frac {\sqrt {-a} \log \left (\frac {a \cosh \relax (x)^{2} - 2 \, \sqrt {a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a} {\left (\cosh \relax (x) e^{x} + e^{x} \sinh \relax (x)\right )} \sqrt {-a} e^{\left (-x\right )} + {\left (a e^{\left (2 \, x\right )} + a\right )} \sinh \relax (x)^{2} + {\left (a \cosh \relax (x)^{2} - a\right )} e^{\left (2 \, x\right )} + 2 \, {\left (a \cosh \relax (x) e^{\left (2 \, x\right )} + a \cosh \relax (x)\right )} \sinh \relax (x) - a}{{\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x) + 1}\right )}{a}, \frac {2 \, \sqrt {a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right )}{a e^{\left (2 \, x\right )} + a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.22, size = 55, normalized size = 3.44 \[ -\frac {\cosh \relax (x ) \sqrt {a \left (\sinh ^{2}\relax (x )\right )}\, \ln \left (\frac {2 \sqrt {-a}\, \sqrt {a \left (\sinh ^{2}\relax (x )\right )}-2 a}{\cosh \relax (x )}\right )}{\sqrt {-a}\, \sinh \relax (x ) \sqrt {a \left (\cosh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 8, normalized size = 0.50 \[ \frac {2 \, \arctan \left (e^{x}\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {a\,{\mathrm {cosh}\relax (x)}^2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \cosh ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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