Optimal. Leaf size=36 \[ \frac {1}{2} \tanh (x) \sqrt {a \cosh ^4(x)}+\frac {1}{2} x \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 36, 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, 2635, 8} \[ \frac {1}{2} \tanh (x) \sqrt {a \cosh ^4(x)}+\frac {1}{2} x \text {sech}^2(x) \sqrt {a \cosh ^4(x)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3207
Rubi steps
\begin {align*} \int \sqrt {a \cosh ^4(x)} \, dx &=\left (\sqrt {a \cosh ^4(x)} \text {sech}^2(x)\right ) \int \cosh ^2(x) \, dx\\ &=\frac {1}{2} \sqrt {a \cosh ^4(x)} \tanh (x)+\frac {1}{2} \left (\sqrt {a \cosh ^4(x)} \text {sech}^2(x)\right ) \int 1 \, dx\\ &=\frac {1}{2} x \sqrt {a \cosh ^4(x)} \text {sech}^2(x)+\frac {1}{2} \sqrt {a \cosh ^4(x)} \tanh (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.69 \[ \frac {1}{2} \text {sech}^2(x) \sqrt {a \cosh ^4(x)} (x+\sinh (x) \cosh (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 180, normalized size = 5.00 \[ \frac {{\left (4 \, \cosh \relax (x) e^{\left (2 \, x\right )} \sinh \relax (x)^{3} + e^{\left (2 \, x\right )} \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 2 \, x\right )} e^{\left (2 \, x\right )} \sinh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + 2 \, x \cosh \relax (x)\right )} e^{\left (2 \, x\right )} \sinh \relax (x) + {\left (\cosh \relax (x)^{4} + 4 \, x \cosh \relax (x)^{2} - 1\right )} e^{\left (2 \, x\right )}\right )} \sqrt {a e^{\left (8 \, x\right )} + 4 \, a e^{\left (6 \, x\right )} + 6 \, a e^{\left (4 \, x\right )} + 4 \, a e^{\left (2 \, x\right )} + a} e^{\left (-2 \, x\right )}}{8 \, {\left (\cosh \relax (x)^{2} e^{\left (4 \, x\right )} + 2 \, \cosh \relax (x)^{2} e^{\left (2 \, x\right )} + {\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{2} + \cosh \relax (x)^{2} + 2 \, {\left (\cosh \relax (x) e^{\left (4 \, x\right )} + 2 \, \cosh \relax (x) e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 28, normalized size = 0.78 \[ -\frac {1}{8} \, {\left ({\left (2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} - 4 \, x - e^{\left (2 \, x\right )}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.45, size = 89, normalized size = 2.47 \[ \frac {\sqrt {8}\, \left (\cosh \left (2 x \right )+1\right ) \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (\cosh \left (2 x \right )+1\right )}\, \sqrt {2}\, \left (\sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}+\ln \left (\sqrt {a}\, \cosh \left (2 x \right )+\sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\right ) a \right )}{16 \sqrt {a}\, \sinh \left (2 x \right ) \sqrt {\left (\cosh \left (2 x \right )+1\right )^{2} a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 27, normalized size = 0.75 \[ -\frac {1}{8} \, {\left (\sqrt {a} e^{\left (-4 \, x\right )} - \sqrt {a}\right )} e^{\left (2 \, x\right )} + \frac {1}{2} \, \sqrt {a} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \sqrt {a\,{\mathrm {cosh}\relax (x)}^4} \,d x \]
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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