Optimal. Leaf size=55 \[ \frac{8 \coth (x)}{15 a^2 \sqrt{a \text{csch}^2(x)}}-\frac{4 \coth (x)}{15 a \left (a \text{csch}^2(x)\right )^{3/2}}+\frac{\coth (x)}{5 \left (a \text{csch}^2(x)\right )^{5/2}} \]
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Rubi [A] time = 0.0303103, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4122, 192, 191} \[ \frac{8 \coth (x)}{15 a^2 \sqrt{a \text{csch}^2(x)}}-\frac{4 \coth (x)}{15 a \left (a \text{csch}^2(x)\right )^{3/2}}+\frac{\coth (x)}{5 \left (a \text{csch}^2(x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 4122
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\left (a \text{csch}^2(x)\right )^{5/2}} \, dx &=-\left (a \operatorname{Subst}\left (\int \frac{1}{\left (-a+a x^2\right )^{7/2}} \, dx,x,\coth (x)\right )\right )\\ &=\frac{\coth (x)}{5 \left (a \text{csch}^2(x)\right )^{5/2}}+\frac{4}{5} \operatorname{Subst}\left (\int \frac{1}{\left (-a+a x^2\right )^{5/2}} \, dx,x,\coth (x)\right )\\ &=\frac{\coth (x)}{5 \left (a \text{csch}^2(x)\right )^{5/2}}-\frac{4 \coth (x)}{15 a \left (a \text{csch}^2(x)\right )^{3/2}}-\frac{8 \operatorname{Subst}\left (\int \frac{1}{\left (-a+a x^2\right )^{3/2}} \, dx,x,\coth (x)\right )}{15 a}\\ &=\frac{\coth (x)}{5 \left (a \text{csch}^2(x)\right )^{5/2}}-\frac{4 \coth (x)}{15 a \left (a \text{csch}^2(x)\right )^{3/2}}+\frac{8 \coth (x)}{15 a^2 \sqrt{a \text{csch}^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.027251, size = 36, normalized size = 0.65 \[ \frac{\sinh (x) (150 \cosh (x)-25 \cosh (3 x)+3 \cosh (5 x)) \sqrt{a \text{csch}^2(x)}}{240 a^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 196, normalized size = 3.6 \begin{align*}{\frac{{{\rm e}^{6\,x}}}{160\,{a}^{2} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{5\,{{\rm e}^{4\,x}}}{96\,{a}^{2} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{5\,{{\rm e}^{2\,x}}}{16\,{a}^{2} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{5}{16\,{a}^{2} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}-{\frac{5\,{{\rm e}^{-2\,x}}}{96\,{a}^{2} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}}+{\frac{{{\rm e}^{-4\,x}}}{160\,{a}^{2} \left ({{\rm e}^{2\,x}}-1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65226, size = 72, normalized size = 1.31 \begin{align*} -\frac{e^{\left (5 \, x\right )}}{160 \, a^{\frac{5}{2}}} + \frac{5 \, e^{\left (3 \, x\right )}}{96 \, a^{\frac{5}{2}}} - \frac{5 \, e^{\left (-x\right )}}{16 \, a^{\frac{5}{2}}} + \frac{5 \, e^{\left (-3 \, x\right )}}{96 \, a^{\frac{5}{2}}} - \frac{e^{\left (-5 \, x\right )}}{160 \, a^{\frac{5}{2}}} - \frac{5 \, e^{x}}{16 \, a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.91041, size = 1885, normalized size = 34.27 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{csch}^{2}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15165, size = 90, normalized size = 1.64 \begin{align*} \frac{\frac{{\left (150 \, e^{\left (4 \, x\right )} - 25 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )}}{\mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} + \frac{3 \, e^{\left (5 \, x\right )} - 25 \, e^{\left (3 \, x\right )} + 150 \, e^{x}}{\mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )}}{480 \, a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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