Optimal. Leaf size=25 \[ \frac {2}{7} (\text {sech}(x)+1)^{7/2}-\frac {4}{5} (\text {sech}(x)+1)^{5/2} \]
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Rubi [A] time = 0.10, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4373, 1570, 1469, 627, 43} \[ \frac {2}{7} (\text {sech}(x)+1)^{7/2}-\frac {4}{5} (\text {sech}(x)+1)^{5/2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 627
Rule 1469
Rule 1570
Rule 4373
Rubi steps
\begin {align*} \int \text {sech}(x) \sqrt {1+\text {sech}(x)} \tanh ^3(x) \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {1}{x}} \left (1-x^2\right )}{x^4} \, dx,x,\cosh (x)\right )\\ &=-\operatorname {Subst}\left (\int \frac {\left (-1+\frac {1}{x^2}\right ) \sqrt {1+\frac {1}{x}}}{x^2} \, dx,x,\cosh (x)\right )\\ &=\operatorname {Subst}\left (\int \sqrt {1+x} \left (-1+x^2\right ) \, dx,x,\text {sech}(x)\right )\\ &=\operatorname {Subst}\left (\int (-1+x) (1+x)^{3/2} \, dx,x,\text {sech}(x)\right )\\ &=\operatorname {Subst}\left (\int \left (-2 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,\text {sech}(x)\right )\\ &=-\frac {4}{5} (1+\text {sech}(x))^{5/2}+\frac {2}{7} (1+\text {sech}(x))^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 30, normalized size = 1.20 \[ -\frac {8}{35} \cosh ^4\left (\frac {x}{2}\right ) (9 \cosh (x)-5) \text {sech}^3(x) \sqrt {\text {sech}(x)+1} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 431, normalized size = 17.24 \[ -\frac {2 \, {\left (9 \, \cosh \relax (x)^{6} + 54 \, \cosh \relax (x) \sinh \relax (x)^{5} + 9 \, \sinh \relax (x)^{6} + 27 \, {\left (5 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{4} + 27 \, \cosh \relax (x)^{4} + 36 \, {\left (5 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 27 \, {\left (5 \, \cosh \relax (x)^{4} + 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 27 \, \cosh \relax (x)^{2} + 54 \, {\left (\cosh \relax (x)^{5} + 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + \frac {9 \, \cosh \relax (x)^{7} + 7 \, {\left (9 \, \cosh \relax (x) + 5\right )} \sinh \relax (x)^{6} + 9 \, \sinh \relax (x)^{7} + 35 \, \cosh \relax (x)^{6} + 7 \, {\left (27 \, \cosh \relax (x)^{2} + 30 \, \cosh \relax (x) + 7\right )} \sinh \relax (x)^{5} + 49 \, \cosh \relax (x)^{5} + 35 \, {\left (9 \, \cosh \relax (x)^{3} + 15 \, \cosh \relax (x)^{2} + 7 \, \cosh \relax (x) + 1\right )} \sinh \relax (x)^{4} + 35 \, \cosh \relax (x)^{4} + 35 \, {\left (9 \, \cosh \relax (x)^{4} + 20 \, \cosh \relax (x)^{3} + 14 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) + 1\right )} \sinh \relax (x)^{3} + 35 \, \cosh \relax (x)^{3} + 7 \, {\left (27 \, \cosh \relax (x)^{5} + 75 \, \cosh \relax (x)^{4} + 70 \, \cosh \relax (x)^{3} + 30 \, \cosh \relax (x)^{2} + 15 \, \cosh \relax (x) + 7\right )} \sinh \relax (x)^{2} + 49 \, \cosh \relax (x)^{2} + 7 \, {\left (9 \, \cosh \relax (x)^{6} + 30 \, \cosh \relax (x)^{5} + 35 \, \cosh \relax (x)^{4} + 20 \, \cosh \relax (x)^{3} + 15 \, \cosh \relax (x)^{2} + 14 \, \cosh \relax (x) + 5\right )} \sinh \relax (x) + 35 \, \cosh \relax (x) + 9}{\sqrt {\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1}} + 9\right )}}{35 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{4} + 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} + 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} + 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)^{2} + 6 \, {\left (\cosh \relax (x)^{5} + 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 46, normalized size = 1.84 \[ -\frac {2 \, {\left ({\left ({\left ({\left ({\left ({\left ({\left (9 \, e^{x} + 35\right )} e^{x} + 49\right )} e^{x} + 35\right )} e^{x} + 35\right )} e^{x} + 49\right )} e^{x} + 35\right )} e^{x} + 9\right )}}{35 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \mathrm {sech}\relax (x ) \sqrt {1+\mathrm {sech}\relax (x )}\, \left (\tanh ^{3}\relax (x )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\operatorname {sech}\relax (x) + 1} \operatorname {sech}\relax (x) \tanh \relax (x)^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 148, normalized size = 5.92 \[ \frac {\left (\frac {72\,{\mathrm {e}}^x}{35}-\frac {24}{5}\right )\,\sqrt {\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}+1}}{\left ({\mathrm {e}}^x+1\right )\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^2}-\frac {\left (\frac {16\,{\mathrm {e}}^x}{7}-\frac {16}{7}\right )\,\sqrt {\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}+1}}{\left ({\mathrm {e}}^x+1\right )\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3}-\frac {\left (\frac {44\,{\mathrm {e}}^x}{35}-4\right )\,\sqrt {\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}+1}}{\left ({\mathrm {e}}^x+1\right )\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {\left (\frac {18\,{\mathrm {e}}^x}{35}+2\right )\,\sqrt {\frac {1}{\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}+1}}{{\mathrm {e}}^x+1} \]
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 \[ \int \sqrt {\operatorname {sech}{\relax (x )} + 1} \tanh ^{3}{\relax (x )} \operatorname {sech}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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