Optimal. Leaf size=40 \[ \frac {\sin ^{-1}(\tanh (a+b x))}{2 b}+\frac {\tanh (a+b x) \sqrt {\text {sech}^2(a+b x)}}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4122, 195, 216} \[ \frac {\sin ^{-1}(\tanh (a+b x))}{2 b}+\frac {\tanh (a+b x) \sqrt {\text {sech}^2(a+b x)}}{2 b} \]
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)^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {1-x^2} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac {\sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\tanh (a+b x)\right )}{2 b}\\ &=\frac {\sin ^{-1}(\tanh (a+b x))}{2 b}+\frac {\sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 1.15 \[ \frac {\text {sech}(a+b x) \left (\tan ^{-1}(\sinh (a+b x))+\tanh (a+b x) \text {sech}(a+b x)\right )}{2 b \sqrt {\text {sech}^2(a+b x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 267, normalized size = 6.68 \[ \frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 76, normalized size = 1.90 \[ \frac {\pi + \frac {4 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}}{{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4} + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.43, size = 183, normalized size = 4.58 \[ \frac {\sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left ({\mathrm e}^{2 b x +2 a}-1\right )}{\left (1+{\mathrm e}^{2 b x +2 a}\right ) b}+\frac {i \ln \left ({\mathrm e}^{b x}+i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x}-i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 65, normalized size = 1.62 \[ -\frac {\arctan \left (e^{\left (-b x - a\right )}\right )}{b} + \frac {e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )}} \]
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 \[ \int {\left (\frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^2}\right )}^{3/2} \,d x \]
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 \[ \int \left (\operatorname {sech}^{2}{\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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