Optimal. Leaf size=34 \[ \frac {\tan ^{-1}(\sinh (a+b x))}{2 b}-\frac {\tanh (a+b x) \text {sech}(a+b x)}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2611, 3770} \[ \frac {\tan ^{-1}(\sinh (a+b x))}{2 b}-\frac {\tanh (a+b x) \text {sech}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3770
Rubi steps
\begin {align*} \int \text {sech}(a+b x) \tanh ^2(a+b x) \, dx &=-\frac {\text {sech}(a+b x) \tanh (a+b x)}{2 b}+\frac {1}{2} \int \text {sech}(a+b x) \, dx\\ &=\frac {\tan ^{-1}(\sinh (a+b x))}{2 b}-\frac {\text {sech}(a+b x) \tanh (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 1.00 \[ \frac {\tan ^{-1}(\sinh (a+b x))}{2 b}-\frac {\tanh (a+b x) \text {sech}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 269, normalized size = 7.91 \[ -\frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} - {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 47, normalized size = 1.38 \[ -\frac {\frac {e^{\left (3 \, b x + 3 \, a\right )} - e^{\left (b x + a\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{2}} - \arctan \left (e^{\left (b x + a\right )}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 49, normalized size = 1.44 \[ -\frac {\sinh \left (b x +a \right )}{b \cosh \left (b x +a \right )^{2}}+\frac {\mathrm {sech}\left (b x +a \right ) \tanh \left (b x +a \right )}{2 b}+\frac {\arctan \left ({\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 66, normalized size = 1.94 \[ -\frac {\arctan \left (e^{\left (-b x - a\right )}\right )}{b} - \frac {e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 82, normalized size = 2.41 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}}+\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
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 \[ \int \tanh ^{2}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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