Integrand size = 25, antiderivative size = 250 \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{5/2}} \, dx=\frac {e^{-4 c (a+b x)} \text {csch}(a c+b c x)}{128 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {5 e^{-2 c (a+b x)} \text {csch}(a c+b c x)}{64 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {5 e^{2 c (a+b x)} \text {csch}(a c+b c x)}{32 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {5 e^{4 c (a+b x)} \text {csch}(a c+b c x)}{128 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {e^{6 c (a+b x)} \text {csch}(a c+b c x)}{192 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {5 x \text {csch}(a c+b c x)}{16 \sqrt {\text {csch}^2(a c+b c x)}} \]
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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} \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{5/2}} \, dx=\frac {e^{-4 c (a+b x)} \text {csch}(a c+b c x)}{128 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {5 e^{-2 c (a+b x)} \text {csch}(a c+b c x)}{64 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {5 e^{2 c (a+b x)} \text {csch}(a c+b c x)}{32 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {5 e^{4 c (a+b x)} \text {csch}(a c+b c x)}{128 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {e^{6 c (a+b x)} \text {csch}(a c+b c x)}{192 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {5 x \text {csch}(a c+b c x)}{16 \sqrt {\text {csch}^2(a c+b c x)}} \]
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Rule 12
Rule 45
Rule 272
Rule 2320
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \frac {\text {csch}(a c+b c x) \int e^{c (a+b x)} \sinh ^5(a c+b c x) \, dx}{\sqrt {\text {csch}^2(a c+b c x)}} \\ & = \frac {\text {csch}(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 {csch}^2(a c+b c x)}} \\ & = \frac {\text {csch}(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 {csch}^2(a c+b c x)}} \\ & = \frac {\text {csch}(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 {csch}^2(a c+b c x)}} \\ & = \frac {\text {csch}(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 {csch}^2(a c+b c x)}} \\ & = \frac {e^{-4 c (a+b x)} \text {csch}(a c+b c x)}{128 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {5 e^{-2 c (a+b x)} \text {csch}(a c+b c x)}{64 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {5 e^{2 c (a+b x)} \text {csch}(a c+b c x)}{32 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {5 e^{4 c (a+b x)} \text {csch}(a c+b c x)}{128 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {e^{6 c (a+b x)} \text {csch}(a c+b c x)}{192 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {5 x \text {csch}(a c+b c x)}{16 \sqrt {\text {csch}^2(a c+b c x)}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.44 \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{5/2}} \, dx=\frac {\left (\frac {1}{128} e^{-4 c (a+b x)}-\frac {5}{64} e^{-2 c (a+b x)}+\frac {5}{32} e^{2 c (a+b x)}-\frac {5}{128} e^{4 c (a+b x)}+\frac {1}{192} e^{6 c (a+b x)}-\frac {5 b c x}{16}\right ) \text {csch}^5(c (a+b x))}{b c \text {csch}^2(c (a+b x))^{5/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.73 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.35
method | result | size |
default | \(\frac {\operatorname {csgn}\left (\operatorname {csch}\left (c \left (b x +a \right )\right )\right ) \left (\left (\frac {\sinh \left (b c x +a c \right )^{5}}{6}-\frac {5 \sinh \left (b c x +a c \right )^{3}}{24}+\frac {5 \sinh \left (b c x +a c \right )}{16}\right ) \cosh \left (b c x +a c \right )-\frac {5 b c x}{16}-\frac {5 a c}{16}+\frac {\sinh \left (b c x +a c \right )^{6}}{6}\right )}{c b}\) | \(88\) |
risch | \(-\frac {5 x \,{\mathrm e}^{c \left (b x +a \right )}}{16 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}+\frac {{\mathrm e}^{7 c \left (b x +a \right )}}{192 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}-\frac {5 \,{\mathrm e}^{5 c \left (b x +a \right )}}{128 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}+\frac {5 \,{\mathrm e}^{3 c \left (b x +a \right )}}{32 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}-\frac {5 \,{\mathrm e}^{-c \left (b x +a \right )}}{64 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}+\frac {{\mathrm e}^{-3 c \left (b x +a \right )}}{128 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}\) | \(326\) |
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Time = 0.26 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.87 \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{5/2}} \, dx=\frac {5 \, \cosh \left (b c x + a c\right )^{5} + 25 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{4} - \sinh \left (b c x + a c\right )^{5} - 5 \, {\left (2 \, \cosh \left (b c x + a c\right )^{2} - 3\right )} \sinh \left (b c x + a c\right )^{3} - 45 \, \cosh \left (b c x + a c\right )^{3} + 5 \, {\left (10 \, \cosh \left (b c x + a c\right )^{3} - 27 \, \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} - 60 \, {\left (2 \, b c x - 1\right )} \cosh \left (b c x + a c\right ) - 5 \, {\left (\cosh \left (b c x + a c\right )^{4} - 24 \, b c x - 9 \, \cosh \left (b c x + a c\right )^{2} - 12\right )} \sinh \left (b c x + a c\right )}{384 \, {\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\right )}} \]
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\[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{5/2}} \, dx=e^{a c} \int \frac {e^{b c x}}{\left (\operatorname {csch}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.36 \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{5/2}} \, dx=\frac {{\left (2 \, e^{\left (10 \, b c x + 10 \, a c\right )} - 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 60 \, e^{\left (6 \, b c x + 6 \, a c\right )} - 30 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 3\right )} e^{\left (-4 \, b c x - 4 \, a c\right )}}{384 \, b c} - \frac {5 \, {\left (b c x + a c\right )}}{16 \, b c} \]
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Time = 0.29 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.11 \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{5/2}} \, dx=-\frac {{\left (120 \, b c x e^{\left (-a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 3 \, {\left (30 \, e^{\left (4 \, b c x + 4 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 10 \, e^{\left (2 \, b c x + 2 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) + \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )\right )} e^{\left (-4 \, b c x - 5 \, a c\right )} - {\left (2 \, e^{\left (6 \, b c x + 20 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 15 \, e^{\left (4 \, b c x + 18 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) + 60 \, e^{\left (2 \, b c x + 16 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )\right )} e^{\left (-15 \, a c\right )}\right )} e^{\left (a c\right )}}{384 \, b c} \]
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Timed out. \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{5/2}} \, dx=\int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left (\frac {1}{{\mathrm {sinh}\left (a\,c+b\,c\,x\right )}^2}\right )}^{5/2}} \,d x \]
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