Optimal. Leaf size=16 \[ -\frac{\sinh (x) \cosh (x)}{\sqrt{a \sinh ^4(x)}} \]
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Rubi [A] time = 0.0136318, antiderivative size = 16, 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} \[ -\frac{\sinh (x) \cosh (x)}{\sqrt{a \sinh ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \sinh ^4(x)}} \, dx &=\frac{\sinh ^2(x) \int \text{csch}^2(x) \, dx}{\sqrt{a \sinh ^4(x)}}\\ &=-\frac{\left (i \sinh ^2(x)\right ) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (x))}{\sqrt{a \sinh ^4(x)}}\\ &=-\frac{\cosh (x) \sinh (x)}{\sqrt{a \sinh ^4(x)}}\\ \end{align*}
Mathematica [A] time = 0.0056078, size = 16, normalized size = 1. \[ -\frac{\sinh (x) \cosh (x)}{\sqrt{a \sinh ^4(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.083, size = 56, normalized size = 3.5 \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 ( -1+\cosh \left ( 2\,x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78469, size = 24, normalized size = 1.5 \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 = 1.68028, size = 338, normalized size = 21.12 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \sinh ^{4}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31557, size = 18, normalized size = 1.12 \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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