Optimal. Leaf size=15 \[ \frac {\cosh (x) \sinh (x)}{\sqrt {a \cosh ^4(x)}} \]
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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 = {3286, 3852, 8}
\begin {gather*} \frac {\sinh (x) \cosh (x)}{\sqrt {a \cosh ^4(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3286
Rule 3852
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 ) \text {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 = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\cosh (x) \sinh (x)}{\sqrt {a \cosh ^4(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. \(55\) vs.
\(2(13)=26\).
time = 1.49, size = 56, normalized size = 3.73
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )}{\sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}\) | \(29\) |
default | \(\frac {\sqrt {8}\, \sqrt {2}\, \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (1+\cosh \left (2 x \right )\right )}\, \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}}{4 a \sinh \left (2 x \right ) \sqrt {a \left (1+\cosh \left (2 x \right )\right )^{2}}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 16, normalized size = 1.07 \begin {gather*} \frac {2}{\sqrt {a} e^{\left (-2 \, x\right )} + \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. 116 vs.
\(2 (13) = 26\).
time = 0.36, size = 116, normalized size = 7.73 \begin {gather*} -\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 \left (x\right )^{2} + {\left (a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} + {\left (a \cosh \left (x\right )^{2} + a\right )} e^{\left (4 \, x\right )} + 2 \, {\left (a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + 2 \, {\left (a \cosh \left (x\right ) e^{\left (4 \, x\right )} + 2 \, a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 13, normalized size = 0.87 \begin {gather*} -\frac {2}{\sqrt {a} {\left (e^{\left (2 \, x\right )} + 1\right )}} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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