Optimal. Leaf size=15 \[ \frac{\sinh (x) \cosh (x)}{\sqrt{a \cosh ^4(x)}} \]
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Rubi [A] time = 0.0166324, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 3767, 8} \[ \frac{\sinh (x) \cosh (x)}{\sqrt{a \cosh ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 3767
Rule 8
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*}
Mathematica [A] time = 0.0048388, size = 15, normalized size = 1. \[ \frac{\sinh (x) \cosh (x)}{\sqrt{a \cosh ^4(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.085, size = 56, normalized size = 3.7 \begin{align*}{\frac{\sqrt{8}\sqrt{2}}{4\,a\sinh \left ( 2\,x \right ) }\sqrt{a \left ( -1+\cosh \left ( 2\,x \right ) \right ) \left ( \cosh \left ( 2\,x \right ) +1 \right ) }\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}{\frac{1}{\sqrt{a \left ( \cosh \left ( 2\,x \right ) +1 \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.72424, size = 22, normalized size = 1.47 \begin{align*} \frac{2}{\sqrt{a} e^{\left (-2 \, x\right )} + \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88053, size = 338, normalized size = 22.53 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16102, size = 18, normalized size = 1.2 \begin{align*} -\frac{2}{\sqrt{a}{\left (e^{\left (2 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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