Optimal. Leaf size=16 \[ \frac {\tanh (x) \log (\sinh (x))}{\sqrt {a \tanh ^2(x)}} \]
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Rubi [A] time = 0.02, 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 = {3658, 3475} \[ \frac {\tanh (x) \log (\sinh (x))}{\sqrt {a \tanh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \tanh ^2(x)}} \, dx &=\frac {\tanh (x) \int \coth (x) \, dx}{\sqrt {a \tanh ^2(x)}}\\ &=\frac {\log (\sinh (x)) \tanh (x)}{\sqrt {a \tanh ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \frac {\tanh (x) \log (\sinh (x))}{\sqrt {a \tanh ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 76, normalized size = 4.75 \[ -\frac {{\left (x e^{\left (2 \, x\right )} - {\left (e^{\left (2 \, x\right )} + 1\right )} \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + x\right )} \sqrt {\frac {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}}{a e^{\left (2 \, x\right )} - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 1, normalized size = 0.06 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 29, normalized size = 1.81 \[ -\frac {\tanh \relax (x ) \left (\ln \left (\tanh \relax (x )-1\right )+\ln \left (1+\tanh \relax (x )\right )-2 \ln \left (\tanh \relax (x )\right )\right )}{2 \sqrt {a \left (\tanh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 31, normalized size = 1.94 \[ -\frac {x}{\sqrt {a}} - \frac {\log \left (e^{\left (-x\right )} + 1\right )}{\sqrt {a}} - \frac {\log \left (e^{\left (-x\right )} - 1\right )}{\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 14, normalized size = 0.88 \[ \frac {\mathrm {atanh}\left (\frac {\mathrm {tanh}\relax (x)}{\sqrt {{\mathrm {tanh}\relax (x)}^2}}\right )}{\sqrt {a}} \]
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 \[ \int \frac {1}{\sqrt {a \tanh ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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