Optimal. Leaf size=65 \[ \frac {3 \sin ^{-1}(\tanh (a+b x))}{8 b}+\frac {\tanh (a+b x) \text {sech}^2(a+b x)^{3/2}}{4 b}+\frac {3 \tanh (a+b x) \sqrt {\text {sech}^2(a+b x)}}{8 b} \]
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Rubi [A] time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4122, 195, 216} \[ \frac {3 \sin ^{-1}(\tanh (a+b x))}{8 b}+\frac {\tanh (a+b x) \text {sech}^2(a+b x)^{3/2}}{4 b}+\frac {3 \tanh (a+b x) \sqrt {\text {sech}^2(a+b x)}}{8 b} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 4122
Rubi steps
\begin {align*} \int \text {sech}^2(a+b x)^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (1-x^2\right )^{3/2} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac {\text {sech}^2(a+b x)^{3/2} \tanh (a+b x)}{4 b}+\frac {3 \operatorname {Subst}\left (\int \sqrt {1-x^2} \, dx,x,\tanh (a+b x)\right )}{4 b}\\ &=\frac {3 \sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{8 b}+\frac {\text {sech}^2(a+b x)^{3/2} \tanh (a+b x)}{4 b}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\tanh (a+b x)\right )}{8 b}\\ &=\frac {3 \sin ^{-1}(\tanh (a+b x))}{8 b}+\frac {3 \sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{8 b}+\frac {\text {sech}^2(a+b x)^{3/2} \tanh (a+b x)}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 55, normalized size = 0.85 \[ \frac {\text {sech}^2(a+b x)^{3/2} \left (3 \sinh (2 (a+b x))+4 \tanh (a+b x)+6 \cosh ^3(a+b x) \tan ^{-1}(\sinh (a+b x))\right )}{16 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 812, normalized size = 12.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 102, normalized size = 1.57 \[ \frac {3 \, \pi + \frac {4 \, {\left (3 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3} + 20 \, e^{\left (b x + a\right )} - 20 \, e^{\left (-b x - a\right )}\right )}}{{\left ({\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}^{2}} + 6 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.42, size = 208, normalized size = 3.20 \[ \frac {\sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (3 \,{\mathrm e}^{6 b x +6 a}+11 \,{\mathrm e}^{4 b x +4 a}-11 \,{\mathrm e}^{2 b x +2 a}-3\right )}{4 \left (1+{\mathrm e}^{2 b x +2 a}\right )^{3} b}+\frac {3 i \ln \left ({\mathrm e}^{b x}+i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{8 b}-\frac {3 i \ln \left ({\mathrm e}^{b x}-i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 112, normalized size = 1.72 \[ -\frac {3 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{4 \, b} + \frac {3 \, e^{\left (-b x - a\right )} + 11 \, e^{\left (-3 \, b x - 3 \, a\right )} - 11 \, e^{\left (-5 \, b x - 5 \, a\right )} - 3 \, e^{\left (-7 \, b x - 7 \, a\right )}}{4 \, b {\left (4 \, e^{\left (-2 \, b x - 2 \, a\right )} + 6 \, e^{\left (-4 \, b x - 4 \, a\right )} + 4 \, e^{\left (-6 \, b x - 6 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} + 1\right )}} \]
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 \[ \int {\left (\frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^2}\right )}^{5/2} \,d x \]
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 \[ \int \left (\operatorname {sech}^{2}{\left (a + b x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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