Integrand size = 15, antiderivative size = 34 \[ \int \text {sech}(a+b x) \tanh ^2(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{2 b}-\frac {\text {sech}(a+b x) \tanh (a+b x)}{2 b} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2691, 3855} \[ \int \text {sech}(a+b x) \tanh ^2(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{2 b}-\frac {\tanh (a+b x) \text {sech}(a+b x)}{2 b} \]
[In]
[Out]
Rule 2691
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {sech}(a+b x) \tanh (a+b x)}{2 b}+\frac {1}{2} \int \text {sech}(a+b x) \, dx \\ & = \frac {\arctan (\sinh (a+b x))}{2 b}-\frac {\text {sech}(a+b x) \tanh (a+b x)}{2 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \text {sech}(a+b x) \tanh ^2(a+b x) \, dx=\frac {\arctan (\sinh (a+b x))}{2 b}-\frac {\text {sech}(a+b x) \tanh (a+b x)}{2 b} \]
[In]
[Out]
Time = 1.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26
method | result | size |
derivativedivides | \(\frac {-\frac {\sinh \left (b x +a \right )}{\cosh \left (b x +a \right )^{2}}+\frac {\operatorname {sech}\left (b x +a \right ) \tanh \left (b x +a \right )}{2}+\arctan \left ({\mathrm e}^{b x +a}\right )}{b}\) | \(43\) |
default | \(\frac {-\frac {\sinh \left (b x +a \right )}{\cosh \left (b x +a \right )^{2}}+\frac {\operatorname {sech}\left (b x +a \right ) \tanh \left (b x +a \right )}{2}+\arctan \left ({\mathrm e}^{b x +a}\right )}{b}\) | \(43\) |
risch | \(-\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{b x +a}+i\right )}{2 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i\right )}{2 b}\) | \(69\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (30) = 60\).
Time = 0.26 (sec) , antiderivative size = 269, normalized size of antiderivative = 7.91 \[ \int \text {sech}(a+b x) \tanh ^2(a+b x) \, dx=-\frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} - {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \]
[In]
[Out]
\[ \int \text {sech}(a+b x) \tanh ^2(a+b x) \, dx=\int \sinh ^{2}{\left (a + b x \right )} \operatorname {sech}^{3}{\left (a + b x \right )}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (30) = 60\).
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.94 \[ \int \text {sech}(a+b x) \tanh ^2(a+b x) \, dx=-\frac {\arctan \left (e^{\left (-b x - a\right )}\right )}{b} - \frac {e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} + e^{\left (-4 \, b x - 4 \, a\right )} + 1\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (30) = 60\).
Time = 0.27 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.24 \[ \int \text {sech}(a+b x) \tanh ^2(a+b x) \, dx=\frac {\pi - \frac {4 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}}{{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4} + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{4 \, b} \]
[In]
[Out]
Time = 2.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.41 \[ \int \text {sech}(a+b x) \tanh ^2(a+b x) \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}}+\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
[In]
[Out]