Optimal. Leaf size=65 \[ \frac {3}{8} a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a \text {sech}^2(x)}}\right )+\frac {3}{8} a^2 \tanh (x) \sqrt {a \text {sech}^2(x)}+\frac {1}{4} a \tanh (x) \left (a \text {sech}^2(x)\right )^{3/2} \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4122, 195, 217, 203} \[ \frac {3}{8} a^2 \tanh (x) \sqrt {a \text {sech}^2(x)}+\frac {3}{8} a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a \text {sech}^2(x)}}\right )+\frac {1}{4} a \tanh (x) \left (a \text {sech}^2(x)\right )^{3/2} \]
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 )^{5/2} \, dx &=a \operatorname {Subst}\left (\int \left (a-a x^2\right )^{3/2} \, dx,x,\tanh (x)\right )\\ &=\frac {1}{4} a \left (a \text {sech}^2(x)\right )^{3/2} \tanh (x)+\frac {1}{4} \left (3 a^2\right ) \operatorname {Subst}\left (\int \sqrt {a-a x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {3}{8} a^2 \sqrt {a \text {sech}^2(x)} \tanh (x)+\frac {1}{4} a \left (a \text {sech}^2(x)\right )^{3/2} \tanh (x)+\frac {1}{8} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x^2}} \, dx,x,\tanh (x)\right )\\ &=\frac {3}{8} a^2 \sqrt {a \text {sech}^2(x)} \tanh (x)+\frac {1}{4} a \left (a \text {sech}^2(x)\right )^{3/2} \tanh (x)+\frac {1}{8} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a \text {sech}^2(x)}}\right )\\ &=\frac {3}{8} a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a \text {sech}^2(x)}}\right )+\frac {3}{8} a^2 \sqrt {a \text {sech}^2(x)} \tanh (x)+\frac {1}{4} a \left (a \text {sech}^2(x)\right )^{3/2} \tanh (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 42, normalized size = 0.65 \[ \frac {1}{8} \cosh (x) \left (a \text {sech}^2(x)\right )^{5/2} \left (2 \sinh (x)+3 \sinh (x) \cosh ^2(x)+6 \cosh ^4(x) \tan ^{-1}\left (\tanh \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 1082, normalized size = 16.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 65, normalized size = 1.00 \[ \frac {1}{16} \, {\left (3 \, \pi - \frac {4 \, {\left (3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 20 \, e^{\left (-x\right )} - 20 \, e^{x}\right )}}{{\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2}} + 6 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} a^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.24, size = 127, normalized size = 1.95 \[ \frac {a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (3 \,{\mathrm e}^{6 x}+11 \,{\mathrm e}^{4 x}-11 \,{\mathrm e}^{2 x}-3\right )}{4 \left (1+{\mathrm e}^{2 x}\right )^{3}}+\frac {3 i a^{2} {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \ln \left ({\mathrm e}^{x}+i\right )}{8}-\frac {3 i a^{2} {\mathrm e}^{-x} \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \ln \left ({\mathrm e}^{x}-i\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 72, normalized size = 1.11 \[ \frac {3}{4} \, a^{\frac {5}{2}} \arctan \left (e^{x}\right ) + \frac {3 \, a^{\frac {5}{2}} e^{\left (7 \, x\right )} + 11 \, a^{\frac {5}{2}} e^{\left (5 \, x\right )} - 11 \, a^{\frac {5}{2}} e^{\left (3 \, x\right )} - 3 \, a^{\frac {5}{2}} e^{x}}{4 \, {\left (e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1\right )}} \]
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 \[ \int {\left (\frac {a}{{\mathrm {cosh}\relax (x)}^2}\right )}^{5/2} \,d x \]
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 \[ \int \left (a \operatorname {sech}^{2}{\relax (x )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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