Integrand size = 14, antiderivative size = 29 \[ \int (c+d x) \text {sech}^2(a+b x) \, dx=-\frac {d \log (\cosh (a+b x))}{b^2}+\frac {(c+d x) \tanh (a+b x)}{b} \]
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Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4269, 3556} \[ \int (c+d x) \text {sech}^2(a+b x) \, dx=\frac {(c+d x) \tanh (a+b x)}{b}-\frac {d \log (\cosh (a+b x))}{b^2} \]
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Rule 3556
Rule 4269
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x) \tanh (a+b x)}{b}-\frac {d \int \tanh (a+b x) \, dx}{b} \\ & = -\frac {d \log (\cosh (a+b x))}{b^2}+\frac {(c+d x) \tanh (a+b x)}{b} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int (c+d x) \text {sech}^2(a+b x) \, dx=-\frac {d \log (\cosh (a+b x))}{b^2}+\frac {d x \text {sech}(a) \text {sech}(a+b x) \sinh (b x)}{b}+\frac {d x \tanh (a)}{b}+\frac {c \tanh (a+b x)}{b} \]
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Time = 0.56 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97
method | result | size |
risch | \(\frac {2 d x}{b}+\frac {2 d a}{b^{2}}-\frac {2 \left (d x +c \right )}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )}-\frac {d \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b^{2}}\) | \(57\) |
parallelrisch | \(\frac {-d \ln \left (-\operatorname {sech}\left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+2\right ) \cosh \left (b x +a \right )+2 \ln \left (1-\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d \cosh \left (b x +a \right )+b \left (\cosh \left (b x +a \right ) d x +\sinh \left (b x +a \right ) \left (d x +c \right )\right )}{b^{2} \cosh \left (b x +a \right )}\) | \(86\) |
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Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (29) = 58\).
Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 5.55 \[ \int (c+d x) \text {sech}^2(a+b x) \, dx=\frac {2 \, b d x \cosh \left (b x + a\right )^{2} + 4 \, b d x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 2 \, b d x \sinh \left (b x + a\right )^{2} - 2 \, b c - {\left (d \cosh \left (b x + a\right )^{2} + 2 \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d \sinh \left (b x + a\right )^{2} + d\right )} \log \left (\frac {2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )^{2} + b^{2}} \]
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\[ \int (c+d x) \text {sech}^2(a+b x) \, dx=\int \left (c + d x\right ) \operatorname {sech}^{2}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (29) = 58\).
Time = 0.21 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.48 \[ \int (c+d x) \text {sech}^2(a+b x) \, dx=d {\left (\frac {2 \, x e^{\left (2 \, b x + 2 \, a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} + b} - \frac {\log \left ({\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-2 \, a\right )}\right )}{b^{2}}\right )} + \frac {2 \, c}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (29) = 58\).
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.69 \[ \int (c+d x) \text {sech}^2(a+b x) \, dx=\frac {2 \, b d x e^{\left (2 \, b x + 2 \, a\right )} - d e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) - 2 \, b c - d \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} \]
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Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int (c+d x) \text {sech}^2(a+b x) \, dx=\frac {2\,d\,x}{b}-\frac {2\,\left (c+d\,x\right )}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {d\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1\right )}{b^2} \]
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