Integrand size = 25, antiderivative size = 44 \[ \int e^{c (a+b x)} \sqrt {\text {sech}^2(a c+b c x)} \, dx=\frac {\cosh (a c+b c x) \log \left (1+e^{2 c (a+b x)}\right ) \sqrt {\text {sech}^2(a c+b c x)}}{b c} \]
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Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6852, 2320, 12, 266} \[ \int e^{c (a+b x)} \sqrt {\text {sech}^2(a c+b c x)} \, dx=\frac {\log \left (e^{2 c (a+b x)}+1\right ) \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c} \]
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Rule 12
Rule 266
Rule 2320
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \left (\cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \int e^{c (a+b x)} \text {sech}(a c+b c x) \, dx \\ & = \frac {\left (\cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \text {Subst}\left (\int \frac {2 x}{1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c} \\ & = \frac {\left (2 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,e^{c (a+b x)}\right )}{b c} \\ & = \frac {\cosh (a c+b c x) \log \left (1+e^{2 c (a+b x)}\right ) \sqrt {\text {sech}^2(a c+b c x)}}{b c} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int e^{c (a+b x)} \sqrt {\text {sech}^2(a c+b c x)} \, dx=\frac {\cosh (c (a+b x)) \log \left (1+e^{2 c (a+b x)}\right ) \sqrt {\text {sech}^2(c (a+b x))}}{b c} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.38 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.66
method | result | size |
default | \(\operatorname {csgn}\left (\operatorname {sech}\left (c \left (b x +a \right )\right )\right ) \left (x +\frac {\ln \left (\cosh \left (c \left (b x +a \right )\right )\right )}{c b}\right )\) | \(29\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 b c x}+{\mathrm e}^{-2 a c}\right ) \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, {\mathrm e}^{-c \left (b x +a \right )}}{b c}\) | \(66\) |
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int e^{c (a+b x)} \sqrt {\text {sech}^2(a c+b c x)} \, dx=\frac {\log \left (\frac {2 \, \cosh \left (b c x + a c\right )}{\cosh \left (b c x + a c\right ) - \sinh \left (b c x + a c\right )}\right )}{b c} \]
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\[ \int e^{c (a+b x)} \sqrt {\text {sech}^2(a c+b c x)} \, dx=e^{a c} \int \sqrt {\operatorname {sech}^{2}{\left (a c + b c x \right )}} e^{b c x}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.48 \[ \int e^{c (a+b x)} \sqrt {\text {sech}^2(a c+b c x)} \, dx=\frac {\log \left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}{b c} \]
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Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.45 \[ \int e^{c (a+b x)} \sqrt {\text {sech}^2(a c+b c x)} \, dx=\frac {\log \left (e^{\left (2 \, b c x\right )} + e^{\left (-2 \, a c\right )}\right )}{b c} \]
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Timed out. \[ \int e^{c (a+b x)} \sqrt {\text {sech}^2(a c+b c x)} \, dx=\int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,\sqrt {\frac {1}{{\mathrm {cosh}\left (a\,c+b\,c\,x\right )}^2}} \,d x \]
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