Optimal. Leaf size=15 \[ \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4208, 3852, 8}
\begin {gather*} \sinh (x) \cosh (x) \sqrt {a \text {sech}^4(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 4208
Rubi steps
\begin {align*} \int \sqrt {a \text {sech}^4(x)} \, dx &=\left (\cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \int \text {sech}^2(x) \, dx\\ &=\left (i \cosh ^2(x) \sqrt {a \text {sech}^4(x)}\right ) \text {Subst}(\int 1 \, dx,x,-i \tanh (x))\\ &=\cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (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. \(28\) vs.
\(2(13)=26\).
time = 0.99, size = 29, normalized size = 1.93
method | result | size |
risch | \(-2 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, {\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 13, normalized size = 0.87 \begin {gather*} \frac {2 \, \sqrt {a}}{e^{\left (-2 \, x\right )} + 1} \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. 81 vs.
\(2 (13) = 26\).
time = 0.42, size = 81, normalized size = 5.40 \begin {gather*} -\frac {2 \, \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (2 \, x\right )}}{2 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} \sinh \left (x\right ) + e^{\left (2 \, x\right )} \sinh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )}} \end {gather*}
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 \begin {gather*} \int \sqrt {a \operatorname {sech}^{4}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 13, normalized size = 0.87 \begin {gather*} -\frac {2 \, \sqrt {a}}{e^{\left (2 \, x\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 71, normalized size = 4.73 \begin {gather*} -\frac {\sqrt {a}\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (2\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+2\,{\mathrm {e}}^{6\,x}+\frac {{\mathrm {e}}^{8\,x}}{2}+\frac {1}{2}\right )}{\left ({\mathrm {e}}^{2\,x}+1\right )\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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