Optimal. Leaf size=34 \[ \frac {\text {ArcTan}(\sinh (a+b x))}{2 b}-\frac {\text {sech}(a+b x) \tanh (a+b x)}{2 b} \]
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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 = {2691, 3855}
\begin {gather*} \frac {\text {ArcTan}(\sinh (a+b x))}{2 b}-\frac {\tanh (a+b x) \text {sech}(a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2691
Rule 3855
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 = 0.01, size = 34, normalized size = 1.00 \begin {gather*} \frac {\text {ArcTan}(\sinh (a+b x))}{2 b}-\frac {\text {sech}(a+b x) \tanh (a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.19, size = 69, normalized size = 2.03
method | result | size |
risch | \(-\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b \left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{b x +a}+i\right )}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i\right )}{2 b}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (30) = 60\).
time = 0.47, size = 66, normalized size = 1.94 \begin {gather*} -\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 269 vs.
\(2 (30) = 60\).
time = 0.37, size = 269, normalized size = 7.91 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \int \tanh ^{2}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (30) = 60\).
time = 0.40, size = 76, normalized size = 2.24 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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