Integrand size = 25, antiderivative size = 250 \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{5/2}} \, dx=-\frac {e^{-4 c (a+b x)} \text {sech}(a c+b c x)}{128 b c \sqrt {\text {sech}^2(a c+b c x)}}-\frac {5 e^{-2 c (a+b x)} \text {sech}(a c+b c x)}{64 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 e^{2 c (a+b x)} \text {sech}(a c+b c x)}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 e^{4 c (a+b x)} \text {sech}(a c+b c x)}{128 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {e^{6 c (a+b x)} \text {sech}(a c+b c x)}{192 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 x \text {sech}(a c+b c x)}{16 \sqrt {\text {sech}^2(a c+b c x)}} \]
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Time = 0.14 (sec) , antiderivative size = 250, 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} \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{5/2}} \, dx=-\frac {e^{-4 c (a+b x)} \text {sech}(a c+b c x)}{128 b c \sqrt {\text {sech}^2(a c+b c x)}}-\frac {5 e^{-2 c (a+b x)} \text {sech}(a c+b c x)}{64 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 e^{2 c (a+b x)} \text {sech}(a c+b c x)}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 e^{4 c (a+b x)} \text {sech}(a c+b c x)}{128 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {e^{6 c (a+b x)} \text {sech}(a c+b c x)}{192 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 x \text {sech}(a c+b c x)}{16 \sqrt {\text {sech}^2(a c+b c x)}} \]
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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 ^5(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 )^5}{32 x^5} \, 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 )^5}{x^5} \, dx,x,e^{c (a+b x)}\right )}{32 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)^5}{x^3} \, dx,x,e^{2 c (a+b x)}\right )}{64 b c \sqrt {\text {sech}^2(a c+b c x)}} \\ & = \frac {\text {sech}(a c+b c x) \text {Subst}\left (\int \left (10+\frac {1}{x^3}+\frac {5}{x^2}+\frac {10}{x}+5 x+x^2\right ) \, dx,x,e^{2 c (a+b x)}\right )}{64 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)}{128 b c \sqrt {\text {sech}^2(a c+b c x)}}-\frac {5 e^{-2 c (a+b x)} \text {sech}(a c+b c x)}{64 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 e^{2 c (a+b x)} \text {sech}(a c+b c x)}{32 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 e^{4 c (a+b x)} \text {sech}(a c+b c x)}{128 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {e^{6 c (a+b x)} \text {sech}(a c+b c x)}{192 b c \sqrt {\text {sech}^2(a c+b c x)}}+\frac {5 x \text {sech}(a c+b c x)}{16 \sqrt {\text {sech}^2(a c+b c x)}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.44 \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{5/2}} \, dx=\frac {\left (-\frac {1}{128} e^{-4 c (a+b x)}-\frac {5}{64} e^{-2 c (a+b x)}+\frac {5}{32} e^{2 c (a+b x)}+\frac {5}{128} e^{4 c (a+b x)}+\frac {1}{192} e^{6 c (a+b x)}+\frac {5 b c x}{16}\right ) \text {sech}^5(c (a+b x))}{b c \text {sech}^2(c (a+b x))^{5/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.67 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.35
method | result | size |
default | \(\frac {\operatorname {csgn}\left (\operatorname {sech}\left (c \left (b x +a \right )\right )\right ) \left (\frac {\cosh \left (b c x +a c \right )^{6}}{6}+\left (\frac {\cosh \left (b c x +a c \right )^{5}}{6}+\frac {5 \cosh \left (b c x +a c \right )^{3}}{24}+\frac {5 \cosh \left (b c x +a c \right )}{16}\right ) \sinh \left (b c x +a c \right )+\frac {5 b c x}{16}+\frac {5 a c}{16}\right )}{c b}\) | \(88\) |
risch | \(\frac {5 x \,{\mathrm e}^{c \left (b x +a \right )}}{16 \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}+\frac {{\mathrm e}^{7 c \left (b x +a \right )}}{192 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}+\frac {5 \,{\mathrm e}^{5 c \left (b x +a \right )}}{128 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}+\frac {5 \,{\mathrm e}^{3 c \left (b x +a \right )}}{32 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}-\frac {5 \,{\mathrm e}^{-c \left (b x +a \right )}}{64 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}-\frac {{\mathrm e}^{-3 c \left (b x +a \right )}}{128 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}}\) | \(326\) |
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Time = 0.26 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.87 \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{5/2}} \, dx=-\frac {\cosh \left (b c x + a c\right )^{5} + 5 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{4} - 5 \, \sinh \left (b c x + a c\right )^{5} - 5 \, {\left (10 \, \cosh \left (b c x + a c\right )^{2} + 9\right )} \sinh \left (b c x + a c\right )^{3} + 15 \, \cosh \left (b c x + a c\right )^{3} + 5 \, {\left (2 \, \cosh \left (b c x + a c\right )^{3} + 9 \, \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} - 60 \, {\left (2 \, b c x + 1\right )} \cosh \left (b c x + a c\right ) - 5 \, {\left (5 \, \cosh \left (b c x + a c\right )^{4} - 24 \, b c x + 27 \, \cosh \left (b c x + a c\right )^{2} + 12\right )} \sinh \left (b c x + a c\right )}{384 \, {\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (238) = 476\).
Time = 86.60 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.12 \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{5/2}} \, dx=\begin {cases} x & \text {for}\: b = 0 \wedge c = 0 \\\frac {x e^{a c}}{\left (\operatorname {sech}^{2}{\left (a c \right )}\right )^{\frac {5}{2}}} & \text {for}\: b = 0 \\x & \text {for}\: c = 0 \\- \frac {5 x e^{a c} e^{b c x} \tanh ^{5}{\left (a c + b c x \right )}}{16 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {5 x e^{a c} e^{b c x} \tanh ^{4}{\left (a c + b c x \right )}}{16 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {5 x e^{a c} e^{b c x} \tanh ^{3}{\left (a c + b c x \right )}}{8 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {5 x e^{a c} e^{b c x} \tanh ^{2}{\left (a c + b c x \right )}}{8 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {5 x e^{a c} e^{b c x} \tanh {\left (a c + b c x \right )}}{16 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {5 x e^{a c} e^{b c x}}{16 \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {8 e^{a c} e^{b c x} \tanh ^{5}{\left (a c + b c x \right )}}{15 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {53 e^{a c} e^{b c x} \tanh ^{4}{\left (a c + b c x \right )}}{240 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {331 e^{a c} e^{b c x} \tanh ^{3}{\left (a c + b c x \right )}}{240 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {131 e^{a c} e^{b c x} \tanh ^{2}{\left (a c + b c x \right )}}{240 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} + \frac {253 e^{a c} e^{b c x} \tanh {\left (a c + b c x \right )}}{240 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} - \frac {11 e^{a c} e^{b c x}}{30 b c \left (\operatorname {sech}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.45 \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{5/2}} \, dx=\frac {5 \, {\left (b c x + a c\right )}}{16 \, b c} + \frac {e^{\left (6 \, b c x + 6 \, a c\right )}}{192 \, b c} + \frac {5 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{128 \, b c} + \frac {5 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{32 \, b c} - \frac {5 \, e^{\left (-2 \, b c x - 2 \, a c\right )}}{64 \, b c} - \frac {e^{\left (-4 \, b c x - 4 \, a c\right )}}{128 \, b c} \]
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Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.44 \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{5/2}} \, dx=\frac {{\left (120 \, b c x e^{\left (-a c\right )} - 3 \, {\left (30 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 10 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )} e^{\left (-4 \, b c x - 5 \, a c\right )} + {\left (2 \, e^{\left (6 \, b c x + 20 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 18 \, a c\right )} + 60 \, e^{\left (2 \, b c x + 16 \, a c\right )}\right )} e^{\left (-15 \, a c\right )}\right )} e^{\left (a c\right )}}{384 \, b c} \]
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Timed out. \[ \int \frac {e^{c (a+b x)}}{\text {sech}^2(a c+b c x)^{5/2}} \, dx=\int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left (\frac {1}{{\mathrm {cosh}\left (a\,c+b\,c\,x\right )}^2}\right )}^{5/2}} \,d x \]
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