Optimal. Leaf size=46 \[ \frac {1}{2} a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a \text {sech}^2(x)}}\right )+\frac {1}{2} a \tanh (x) \sqrt {a \text {sech}^2(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4122, 195, 217, 203} \[ \frac {1}{2} a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a \text {sech}^2(x)}}\right )+\frac {1}{2} a \tanh (x) \sqrt {a \text {sech}^2(x)} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 4122
Rubi steps
\begin {align*} \int \left (a \text {sech}^2(x)\right )^{3/2} \, dx &=a \operatorname {Subst}\left (\int \sqrt {a-a x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{2} a \sqrt {a \text {sech}^2(x)} \tanh (x)+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x^2}} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{2} a \sqrt {a \text {sech}^2(x)} \tanh (x)+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a \text {sech}^2(x)}}\right )\\ &=\frac {1}{2} a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a \text {sech}^2(x)}}\right )+\frac {1}{2} a \sqrt {a \text {sech}^2(x)} \tanh (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 0.63 \[ \frac {1}{2} a \sqrt {a \text {sech}^2(x)} \left (\tanh (x)+2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 310, normalized size = 6.74 \[ \frac {{\left (a \cosh \relax (x)^{3} + {\left (a e^{\left (2 \, x\right )} + a\right )} \sinh \relax (x)^{3} + 3 \, {\left (a \cosh \relax (x) e^{\left (2 \, x\right )} + a \cosh \relax (x)\right )} \sinh \relax (x)^{2} + {\left (a \cosh \relax (x)^{4} + {\left (a e^{\left (2 \, x\right )} + a\right )} \sinh \relax (x)^{4} + 4 \, {\left (a \cosh \relax (x) e^{\left (2 \, x\right )} + a \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 2 \, a \cosh \relax (x)^{2} + 2 \, {\left (3 \, a \cosh \relax (x)^{2} + {\left (3 \, a \cosh \relax (x)^{2} + a\right )} e^{\left (2 \, x\right )} + a\right )} \sinh \relax (x)^{2} + {\left (a \cosh \relax (x)^{4} + 2 \, a \cosh \relax (x)^{2} + a\right )} e^{\left (2 \, x\right )} + 4 \, {\left (a \cosh \relax (x)^{3} + a \cosh \relax (x) + {\left (a \cosh \relax (x)^{3} + a \cosh \relax (x)\right )} e^{\left (2 \, x\right )}\right )} \sinh \relax (x) + a\right )} \arctan \left (\cosh \relax (x) + \sinh \relax (x)\right ) - a \cosh \relax (x) + {\left (a \cosh \relax (x)^{3} - a \cosh \relax (x)\right )} e^{\left (2 \, x\right )} + {\left (3 \, a \cosh \relax (x)^{2} + {\left (3 \, a \cosh \relax (x)^{2} - a\right )} e^{\left (2 \, x\right )} - a\right )} \sinh \relax (x)\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{4 \, \cosh \relax (x) e^{x} \sinh \relax (x)^{3} + e^{x} \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} e^{x} \sinh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} e^{x} \sinh \relax (x) + {\left (\cosh \relax (x)^{4} + 2 \, \cosh \relax (x)^{2} + 1\right )} e^{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 48, normalized size = 1.04 \[ \frac {1}{4} \, {\left (\pi - \frac {4 \, {\left (e^{\left (-x\right )} - e^{x}\right )}}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4} + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} a^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 106, normalized size = 2.30 \[ \frac {a \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}+\frac {i a \,{\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \ln \left ({\mathrm e}^{x}+i\right )}{2}-\frac {i a \,{\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \ln \left ({\mathrm e}^{x}-i\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 39, normalized size = 0.85 \[ a^{\frac {3}{2}} \arctan \left (e^{x}\right ) + \frac {a^{\frac {3}{2}} e^{\left (3 \, x\right )} - a^{\frac {3}{2}} e^{x}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {a}{{\mathrm {cosh}\relax (x)}^2}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \operatorname {sech}^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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