Optimal. Leaf size=37 \[ \frac{2 \cosh (x)}{243 \sqrt{4 \cosh ^2(x)-9}}-\frac{\cosh (x)}{27 \left (4 \cosh ^2(x)-9\right )^{3/2}} \]
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Rubi [A] time = 0.0452293, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3190, 192, 191} \[ \frac{2 \cosh (x)}{243 \sqrt{4 \cosh ^2(x)-9}}-\frac{\cosh (x)}{27 \left (4 \cosh ^2(x)-9\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3190
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (-9+4 x^2\right )^{5/2}} \, dx,x,\cosh (x)\right )\\ &=-\frac{\cosh (x)}{27 \left (-9+4 \cosh ^2(x)\right )^{3/2}}-\frac{2}{27} \operatorname{Subst}\left (\int \frac{1}{\left (-9+4 x^2\right )^{3/2}} \, dx,x,\cosh (x)\right )\\ &=-\frac{\cosh (x)}{27 \left (-9+4 \cosh ^2(x)\right )^{3/2}}+\frac{2 \cosh (x)}{243 \sqrt{-9+4 \cosh ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0709346, size = 26, normalized size = 0.7 \[ \frac{\cosh (x) (4 \cosh (2 x)-23)}{243 (2 \cosh (2 x)-7)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 30, normalized size = 0.8 \begin{align*} -{\frac{\cosh \left ( x \right ) }{27} \left ( -9+4\, \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,\cosh \left ( x \right ) }{243}{\frac{1}{\sqrt{-9+4\, \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03513, size = 169, normalized size = 4.57 \begin{align*} -\frac{1855 \, e^{\left (-2 \, x\right )} - 8485 \, e^{\left (-4 \, x\right )} + 5285 \, e^{\left (-6 \, x\right )} - 980 \, e^{\left (-8 \, x\right )} + 56 \, e^{\left (-10 \, x\right )} - 106}{12150 \,{\left (3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}{\left (-3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}} + \frac{980 \, e^{\left (-2 \, x\right )} - 5285 \, e^{\left (-4 \, x\right )} + 8485 \, e^{\left (-6 \, x\right )} - 1855 \, e^{\left (-8 \, x\right )} + 106 \, e^{\left (-10 \, x\right )} - 56}{12150 \,{\left (3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}{\left (-3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05894, size = 1585, normalized size = 42.84 \begin{align*} \text{result too large to display} \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 = 1.14142, size = 54, normalized size = 1.46 \begin{align*} \frac{{\left ({\left (2 \, e^{\left (2 \, x\right )} - 21\right )} e^{\left (2 \, x\right )} - 21\right )} e^{\left (2 \, x\right )} + 2}{486 \,{\left (e^{\left (4 \, x\right )} - 7 \, e^{\left (2 \, x\right )} + 1\right )}^{\frac{3}{2}}} - \frac{1}{243} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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