Optimal. Leaf size=36 \[ \frac {\coth (x)}{3 \left (a \text {csch}^2(x)\right )^{3/2}}-\frac {2 \coth (x)}{3 a \sqrt {a \text {csch}^2(x)}} \]
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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 = {4122, 192, 191} \[ \frac {\coth (x)}{3 \left (a \text {csch}^2(x)\right )^{3/2}}-\frac {2 \coth (x)}{3 a \sqrt {a \text {csch}^2(x)}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\left (a \text {csch}^2(x)\right )^{3/2}} \, dx &=-\left (a \operatorname {Subst}\left (\int \frac {1}{\left (-a+a x^2\right )^{5/2}} \, dx,x,\coth (x)\right )\right )\\ &=\frac {\coth (x)}{3 \left (a \text {csch}^2(x)\right )^{3/2}}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\left (-a+a x^2\right )^{3/2}} \, dx,x,\coth (x)\right )\\ &=\frac {\coth (x)}{3 \left (a \text {csch}^2(x)\right )^{3/2}}-\frac {2 \coth (x)}{3 a \sqrt {a \text {csch}^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 0.75 \[ \frac {(\cosh (3 x)-9 \cosh (x)) \text {csch}^3(x)}{12 \left (a \text {csch}^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.31, size = 285, normalized size = 7.92 \[ \frac {{\left ({\left (e^{\left (2 \, x\right )} - 1\right )} \sinh \relax (x)^{6} - \cosh \relax (x)^{6} + 6 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} - \cosh \relax (x)\right )} \sinh \relax (x)^{5} - 3 \, {\left (5 \, \cosh \relax (x)^{2} - {\left (5 \, \cosh \relax (x)^{2} - 3\right )} e^{\left (2 \, x\right )} - 3\right )} \sinh \relax (x)^{4} + 9 \, \cosh \relax (x)^{4} - 4 \, {\left (5 \, \cosh \relax (x)^{3} - {\left (5 \, \cosh \relax (x)^{3} - 9 \, \cosh \relax (x)\right )} e^{\left (2 \, x\right )} - 9 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} - 3 \, {\left (5 \, \cosh \relax (x)^{4} - 18 \, \cosh \relax (x)^{2} - {\left (5 \, \cosh \relax (x)^{4} - 18 \, \cosh \relax (x)^{2} - 3\right )} e^{\left (2 \, x\right )} - 3\right )} \sinh \relax (x)^{2} + 9 \, \cosh \relax (x)^{2} + {\left (\cosh \relax (x)^{6} - 9 \, \cosh \relax (x)^{4} - 9 \, \cosh \relax (x)^{2} + 1\right )} e^{\left (2 \, x\right )} - 6 \, {\left (\cosh \relax (x)^{5} - 6 \, \cosh \relax (x)^{3} - {\left (\cosh \relax (x)^{5} - 6 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} e^{\left (2 \, x\right )} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x) - 1\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{24 \, {\left (a^{2} \cosh \relax (x)^{3} e^{x} + 3 \, a^{2} \cosh \relax (x)^{2} e^{x} \sinh \relax (x) + 3 \, a^{2} \cosh \relax (x) e^{x} \sinh \relax (x)^{2} + a^{2} e^{x} \sinh \relax (x)^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 54, normalized size = 1.50 \[ -\frac {\frac {{\left (9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-3 \, x\right )}}{\mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} - \frac {e^{\left (3 \, x\right )} - 9 \, e^{x}}{\mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )}}{24 \, a^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 130, normalized size = 3.61 \[ \frac {{\mathrm e}^{4 x}}{24 a \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {3 \,{\mathrm e}^{2 x}}{8 a \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {3}{8 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right ) a}+\frac {{\mathrm e}^{-2 x}}{24 a \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 35, normalized size = 0.97 \[ -\frac {e^{\left (3 \, x\right )}}{24 \, a^{\frac {3}{2}}} + \frac {3 \, e^{\left (-x\right )}}{8 \, a^{\frac {3}{2}}} - \frac {e^{\left (-3 \, x\right )}}{24 \, a^{\frac {3}{2}}} + \frac {3 \, e^{x}}{8 \, a^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\left (\frac {a}{{\mathrm {sinh}\relax (x)}^2}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \operatorname {csch}^{2}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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