Integrand size = 19, antiderivative size = 40 \[ \int \text {sech}(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {(2 a+b) \arctan (\sinh (c+d x))}{2 d}-\frac {b \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3757, 393, 209} \[ \int \text {sech}(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {(2 a+b) \arctan (\sinh (c+d x))}{2 d}-\frac {b \tanh (c+d x) \text {sech}(c+d x)}{2 d} \]
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Rule 209
Rule 393
Rule 3757
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+(a+b) x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d} \\ & = -\frac {b \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {(2 a+b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d} \\ & = \frac {(2 a+b) \arctan (\sinh (c+d x))}{2 d}-\frac {b \text {sech}(c+d x) \tanh (c+d x)}{2 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.20 \[ \int \text {sech}(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {a \arctan (\sinh (c+d x))}{d}+\frac {b \arctan (\sinh (c+d x))}{2 d}-\frac {b \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]
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Time = 0.62 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.40
method | result | size |
derivativedivides | \(\frac {2 a \arctan \left ({\mathrm e}^{d x +c}\right )+b \left (-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(56\) |
default | \(\frac {2 a \arctan \left ({\mathrm e}^{d x +c}\right )+b \left (-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(56\) |
risch | \(-\frac {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{d}+\frac {i b \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{d}-\frac {i b \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d}\) | \(106\) |
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Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (36) = 72\).
Time = 0.25 (sec) , antiderivative size = 323, normalized size of antiderivative = 8.08 \[ \int \text {sech}(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-\frac {b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{3} - {\left ({\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (2 \, a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, a + b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 2 \, a + b\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - b \cosh \left (d x + c\right ) + {\left (3 \, b \cosh \left (d x + c\right )^{2} - b\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} + 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \]
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\[ \int \text {sech}(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \operatorname {sech}{\left (c + d x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (36) = 72\).
Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.00 \[ \int \text {sech}(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=-b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {a \arctan \left (\sinh \left (d x + c\right )\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (36) = 72\).
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.10 \[ \int \text {sech}(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (2 \, a + b\right )} - \frac {4 \, b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4}}{4 \, d} \]
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Time = 0.12 (sec) , antiderivative size = 125, normalized size of antiderivative = 3.12 \[ \int \text {sech}(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (2\,a\,\sqrt {d^2}+b\,\sqrt {d^2}\right )}{d\,\sqrt {4\,a^2+4\,a\,b+b^2}}\right )\,\sqrt {4\,a^2+4\,a\,b+b^2}}{\sqrt {d^2}}-\frac {b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
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