Optimal. Leaf size=13 \[ \frac {\tanh (x)}{\sqrt {a \text {sech}^2(x)}} \]
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Rubi [A]
time = 0.02, 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 = {4207, 197}
\begin {gather*} \frac {\tanh (x)}{\sqrt {a \text {sech}^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 4207
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \text {sech}^2(x)}} \, dx &=a \text {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 \begin {gather*} \frac {\tanh (x)}{\sqrt {a \text {sech}^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs.
\(2(11)=22\).
time = 1.04, size = 58, normalized size = 4.46
method | result | size |
risch | \(\frac {{\mathrm e}^{2 x}}{2 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {1}{2 \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 17, normalized size = 1.31 \begin {gather*} -\frac {e^{\left (-x\right )}}{2 \, \sqrt {a}} + \frac {e^{x}}{2 \, \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs.
\(2 (11) = 22\).
time = 0.40, size = 79, normalized size = 6.08 \begin {gather*} \frac {{\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \, {\left (a \cosh \left (x\right ) e^{x} + a e^{x} \sinh \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 12, normalized size = 0.92 \begin {gather*} \frac {\tanh {\left (x \right )}}{\sqrt {a \operatorname {sech}^{2}{\left (x \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 14, normalized size = 1.08 \begin {gather*} -\frac {e^{\left (-x\right )} - e^{x}}{2 \, \sqrt {a}} \end {gather*}
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 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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