Integrand size = 25, antiderivative size = 162 \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{3/2}} \, dx=-\frac {e^{-2 c (a+b x)} \text {sech}(a c+b c x)}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {3 e^{2 c (a+b x)} \text {sech}(a c+b c x)}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {e^{4 c (a+b x)} \text {sech}(a c+b c x)}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {3 x \text {sech}(a c+b c x)}{8 \sqrt {\text {sech}^2(a c+b c x)}} \]
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Time = 0.11 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6852, 2320, 12, 272, 45} \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{3/2}} \, dx=-\frac {e^{-2 c (a+b x)} \text {sech}(a c+b c x)}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {3 e^{2 c (a+b x)} \text {sech}(a c+b c x)}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {e^{4 c (a+b x)} \text {sech}(a c+b c x)}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {3 x \text {sech}(a c+b c x)}{8 \sqrt {\text {sech}^2(a c+b c x)}} \]
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Rule 12
Rule 45
Rule 272
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 ^3(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 {\left (1+x^2\right )^3}{8 x^3} \, 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 {\left (1+x^2\right )^3}{x^3} \, dx,x,e^{c (a+b x)}\right )}{8 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)^3}{x^2} \, dx,x,e^{2 c (a+b x)}\right )}{16 b c \sqrt {\text {sech}^2(a c+b c x)}} \\ & = \frac {\text {sech}(a c+b c x) \text {Subst}\left (\int \left (3+\frac {1}{x^2}+\frac {3}{x}+x\right ) \, dx,x,e^{2 c (a+b x)}\right )}{16 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)}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {3 e^{2 c (a+b x)} \text {sech}(a c+b c x)}{16 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {e^{4 c (a+b x)} \text {sech}(a c+b c x)}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {3 x \text {sech}(a c+b c x)}{8 \sqrt {\text {sech}^2(a c+b c x)}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.50 \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{3/2}} \, dx=\frac {\left (-\frac {1}{16} e^{-2 c (a+b x)}+\frac {3}{16} e^{2 c (a+b x)}+\frac {1}{32} e^{4 c (a+b x)}+\frac {3 b c x}{8}\right ) \text {sech}^3(c (a+b x))}{b c \text {sech}^2(c (a+b x))^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.64 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.46
method | result | size |
default | \(\frac {\operatorname {csgn}\left (\operatorname {sech}\left (c \left (b x +a \right )\right )\right ) \left (\frac {\cosh \left (b c x +a c \right )^{4}}{4}+\left (\frac {\cosh \left (b c x +a c \right )^{3}}{4}+\frac {3 \cosh \left (b c x +a c \right )}{8}\right ) \sinh \left (b c x +a c \right )+\frac {3 b c x}{8}+\frac {3 a c}{8}\right )}{c b}\) | \(75\) |
risch | \(\frac {3 x \,{\mathrm e}^{c \left (b x +a \right )}}{8 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}+\frac {{\mathrm e}^{5 c \left (b x +a \right )}}{32 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}+\frac {3 \,{\mathrm e}^{3 c \left (b x +a \right )}}{16 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}-\frac {{\mathrm e}^{-c \left (b x +a \right )}}{16 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}\) | \(216\) |
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Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.78 \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{3/2}} \, dx=-\frac {\cosh \left (b c x + a c\right )^{3} + 3 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} - 3 \, \sinh \left (b c x + a c\right )^{3} - 6 \, {\left (2 \, b c x + 1\right )} \cosh \left (b c x + a c\right ) + 3 \, {\left (4 \, b c x - 3 \, \cosh \left (b c x + a c\right )^{2} - 2\right )} \sinh \left (b c x + a c\right )}{32 \, {\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\right )}} \]
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\[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{3/2}} \, dx=e^{a c} \int \frac {e^{b c x}}{\left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.46 \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{3/2}} \, dx=\frac {3 \, {\left (b c x + a c\right )}}{8 \, b c} + \frac {e^{\left (4 \, b c x + 4 \, a c\right )}}{32 \, b c} + \frac {3 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{16 \, b c} - \frac {e^{\left (-2 \, b c x - 2 \, a c\right )}}{16 \, b c} \]
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Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.51 \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{3/2}} \, dx=\frac {{\left (12 \, b c x e^{\left (-a c\right )} - 2 \, {\left (3 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )} e^{\left (-2 \, b c x - 3 \, a c\right )} + {\left (e^{\left (4 \, b c x + 9 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 7 \, a c\right )}\right )} e^{\left (-6 \, a c\right )}\right )} e^{\left (a c\right )}}{32 \, b c} \]
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Timed out. \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left (\frac {1}{{\mathrm {cosh}\left (a\,c+b\,c\,x\right )}^2}\right )}^{3/2}} \,d x \]
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