Optimal. Leaf size=36 \[ \frac {2 \tanh (x)}{3 a \sqrt {a \text {sech}^2(x)}}+\frac {\tanh (x)}{3 \left (a \text {sech}^2(x)\right )^{3/2}} \]
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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} \[ \frac {2 \tanh (x)}{3 a \sqrt {a \text {sech}^2(x)}}+\frac {\tanh (x)}{3 \left (a \text {sech}^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\left (a \text {sech}^2(x)\right )^{3/2}} \, dx &=a \operatorname {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{3 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{\left (a-a x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh (x)}{3 \left (a \text {sech}^2(x)\right )^{3/2}}+\frac {2 \tanh (x)}{3 a \sqrt {a \text {sech}^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 0.75 \[ \frac {(9 \sinh (x)+\sinh (3 x)) \text {sech}^3(x)}{12 \left (a \text {sech}^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 277, normalized size = 7.69 \[ \frac {{\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{6} + \cosh \relax (x)^{6} + 6 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x)^{5} + 3 \, {\left (5 \, \cosh \relax (x)^{2} + {\left (5 \, \cosh \relax (x)^{2} + 3\right )} e^{\left (2 \, x\right )} + 3\right )} \sinh \relax (x)^{4} + 9 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} + {\left (5 \, \cosh \relax (x)^{3} + 9 \, \cosh \relax (x)\right )} e^{\left (2 \, x\right )} + 9 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} + 18 \, \cosh \relax (x)^{2} + {\left (5 \, \cosh \relax (x)^{4} + 18 \, \cosh \relax (x)^{2} - 3\right )} e^{\left (2 \, x\right )} - 3\right )} \sinh \relax (x)^{2} - 9 \, \cosh \relax (x)^{2} + {\left (\cosh \relax (x)^{6} + 9 \, \cosh \relax (x)^{4} - 9 \, \cosh \relax (x)^{2} - 1\right )} e^{\left (2 \, x\right )} + 6 \, {\left (\cosh \relax (x)^{5} + 6 \, \cosh \relax (x)^{3} + {\left (\cosh \relax (x)^{5} + 6 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} e^{\left (2 \, x\right )} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x) - 1\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{24 \, {\left (a^{2} \cosh \relax (x)^{3} e^{x} + 3 \, a^{2} \cosh \relax (x)^{2} e^{x} \sinh \relax (x) + 3 \, a^{2} \cosh \relax (x) e^{x} \sinh \relax (x)^{2} + a^{2} e^{x} \sinh \relax (x)^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 29, normalized size = 0.81 \[ -\frac {{\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} - e^{\left (3 \, x\right )} - 9 \, e^{x}}{24 \, a^{\frac {3}{2}}} \]
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 \[ \frac {{\mathrm e}^{4 x}}{24 a \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}+\frac {3 \,{\mathrm e}^{2 x}}{8 a \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}-\frac {3}{8 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right ) a}-\frac {{\mathrm e}^{-2 x}}{24 a \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, 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 \[ \int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\relax (x)}^2}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.25, size = 37, normalized size = 1.03 \[ - \frac {2 \tanh ^{3}{\relax (x )}}{3 a^{\frac {3}{2}} \left (\operatorname {sech}^{2}{\relax (x )}\right )^{\frac {3}{2}}} + \frac {\tanh {\relax (x )}}{a^{\frac {3}{2}} \left (\operatorname {sech}^{2}{\relax (x )}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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