Optimal. Leaf size=15 \[ \frac {\sinh (x) \cosh (x)}{\sqrt {a \cosh ^4(x)}} \]
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Rubi [A] time = 0.02, 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 = {3207, 3767, 8} \[ \frac {\sinh (x) \cosh (x)}{\sqrt {a \cosh ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3207
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \cosh ^4(x)}} \, dx &=\frac {\cosh ^2(x) \int \text {sech}^2(x) \, dx}{\sqrt {a \cosh ^4(x)}}\\ &=\frac {\left (i \cosh ^2(x)\right ) \operatorname {Subst}(\int 1 \, dx,x,-i \tanh (x))}{\sqrt {a \cosh ^4(x)}}\\ &=\frac {\cosh (x) \sinh (x)}{\sqrt {a \cosh ^4(x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\sinh (x) \cosh (x)}{\sqrt {a \cosh ^4(x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.33, size = 116, normalized size = 7.73 \[ -\frac {2 \, \sqrt {a e^{\left (8 \, x\right )} + 4 \, a e^{\left (6 \, x\right )} + 6 \, a e^{\left (4 \, x\right )} + 4 \, a e^{\left (2 \, x\right )} + a}}{a \cosh \relax (x)^{2} + {\left (a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \relax (x)^{2} + {\left (a \cosh \relax (x)^{2} + a\right )} e^{\left (4 \, x\right )} + 2 \, {\left (a \cosh \relax (x)^{2} + a\right )} e^{\left (2 \, x\right )} + 2 \, {\left (a \cosh \relax (x) e^{\left (4 \, x\right )} + 2 \, a \cosh \relax (x) e^{\left (2 \, x\right )} + a \cosh \relax (x)\right )} \sinh \relax (x) + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 13, normalized size = 0.87 \[ -\frac {2}{\sqrt {a} {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 56, normalized size = 3.73 \[ \frac {\sqrt {8}\, \sqrt {2}\, \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (\cosh \left (2 x \right )+1\right )}\, \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}}{4 a \sinh \left (2 x \right ) \sqrt {\left (\cosh \left (2 x \right )+1\right )^{2} a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 16, normalized size = 1.07 \[ \frac {2}{\sqrt {a} e^{\left (-2 \, x\right )} + \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 39, normalized size = 2.60 \[ -\frac {{\mathrm {e}}^{-x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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