Optimal. Leaf size=13 \[ \frac {\tanh (x)}{\sqrt {a \text {sech}^2(x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 13, 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 = {4122, 191} \[ \frac {\tanh (x)}{\sqrt {a \text {sech}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \text {sech}^2(x)}} \, dx &=a \operatorname {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{\sqrt {a \text {sech}^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ \frac {\tanh (x)}{\sqrt {a \text {sech}^2(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 79, normalized size = 6.08 \[ \frac {{\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + {\left (\cosh \relax (x)^{2} - 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x) - 1\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \, {\left (a \cosh \relax (x) e^{x} + a e^{x} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 14, normalized size = 1.08 \[ -\frac {e^{\left (-x\right )} - e^{x}}{2 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 58, normalized size = 4.46 \[ \frac {{\mathrm e}^{2 x}}{2 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {1}{2 \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 17, normalized size = 1.31 \[ -\frac {e^{\left (-x\right )}}{2 \, \sqrt {a}} + \frac {e^{x}}{2 \, \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 33, normalized size = 2.54 \[ -\frac {\left (\frac {{\mathrm {e}}^{-2\,x}}{2}-\frac {{\mathrm {e}}^{2\,x}}{2}\right )\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^2}}}{2\,\sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 15, normalized size = 1.15 \[ \frac {\tanh {\relax (x )}}{\sqrt {a} \sqrt {\operatorname {sech}^{2}{\relax (x )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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