Integrand size = 25, antiderivative size = 74 \[ \int \frac {e^{c (a+b x)}}{\sqrt {\text {sech}^2(a c+b c x)}} \, dx=\frac {e^{2 c (a+b x)} \text {sech}(a c+b c x)}{4 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {x \text {sech}(a c+b c x)}{2 \sqrt {\text {sech}^2(a c+b c x)}} \]
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Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6852, 2320, 12, 14} \[ \int \frac {e^{c (a+b x)}}{\sqrt {\text {sech}^2(a c+b c x)}} \, dx=\frac {e^{2 c (a+b x)} \text {sech}(a c+b c x)}{4 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {x \text {sech}(a c+b c x)}{2 \sqrt {\text {sech}^2(a c+b c x)}} \]
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Rule 12
Rule 14
Rule 2320
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \frac {\text {sech}(a c+b c x) \int e^{c (a+b x)} \cosh (a c+b c x) \, dx}{\sqrt {\text {sech}^2(a c+b c x)}} \\ & = \frac {\text {sech}(a c+b c x) \text {Subst}\left (\int \frac {1+x^2}{2 x} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\text {sech}^2(a c+b c x)}} \\ & = \frac {\text {sech}(a c+b c x) \text {Subst}\left (\int \frac {1+x^2}{x} \, dx,x,e^{c (a+b x)}\right )}{2 b c \sqrt {\text {sech}^2(a c+b c x)}} \\ & = \frac {\text {sech}(a c+b c x) \text {Subst}\left (\int \left (\frac {1}{x}+x\right ) \, dx,x,e^{c (a+b x)}\right )}{2 b c \sqrt {\text {sech}^2(a c+b c x)}} \\ & = \frac {e^{2 c (a+b x)} \text {sech}(a c+b c x)}{4 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {x \text {sech}(a c+b c x)}{2 \sqrt {\text {sech}^2(a c+b c x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.65 \[ \int \frac {e^{c (a+b x)}}{\sqrt {\text {sech}^2(a c+b c x)}} \, dx=\frac {\left (e^{2 c (a+b x)}+2 b c x\right ) \text {sech}(c (a+b x))}{4 b c \sqrt {\text {sech}^2(c (a+b x))}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.43 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.81
method | result | size |
default | \(\frac {\operatorname {csgn}\left (\operatorname {sech}\left (c \left (b x +a \right )\right )\right ) \left (\frac {\cosh \left (b c x +a c \right )^{2}}{2}+\frac {\sinh \left (b c x +a c \right ) \cosh \left (b c x +a c \right )}{2}+\frac {b c x}{2}+\frac {a c}{2}\right )}{c b}\) | \(60\) |
risch | \(\frac {x \,{\mathrm e}^{c \left (b x +a \right )}}{2 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}+\frac {{\mathrm e}^{3 c \left (b x +a \right )}}{4 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}\) | \(106\) |
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Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.89 \[ \int \frac {e^{c (a+b x)}}{\sqrt {\text {sech}^2(a c+b c x)}} \, dx=\frac {{\left (2 \, b c x + 1\right )} \cosh \left (b c x + a c\right ) - {\left (2 \, b c x - 1\right )} \sinh \left (b c x + a c\right )}{4 \, {\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\right )}} \]
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\[ \int \frac {e^{c (a+b x)}}{\sqrt {\text {sech}^2(a c+b c x)}} \, dx=e^{a c} \int \frac {e^{b c x}}{\sqrt {\operatorname {sech}^{2}{\left (a c + b c x \right )}}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.39 \[ \int \frac {e^{c (a+b x)}}{\sqrt {\text {sech}^2(a c+b c x)}} \, dx=\frac {1}{2} \, x + \frac {a}{2 \, b} + \frac {e^{\left (2 \, b c x + 2 \, a c\right )}}{4 \, b c} \]
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.45 \[ \int \frac {e^{c (a+b x)}}{\sqrt {\text {sech}^2(a c+b c x)}} \, dx=\frac {{\left (2 \, b c x e^{\left (-a c\right )} + e^{\left (2 \, b c x + a c\right )}\right )} e^{\left (a c\right )}}{4 \, b c} \]
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Timed out. \[ \int \frac {e^{c (a+b x)}}{\sqrt {\text {sech}^2(a c+b c x)}} \, dx=\int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{\sqrt {\frac {1}{{\mathrm {cosh}\left (a\,c+b\,c\,x\right )}^2}}} \,d x \]
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