Integrand size = 25, antiderivative size = 162 \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{3/2}} \, dx=\frac {e^{-2 c (a+b x)} \text {csch}(a c+b c x)}{16 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {3 e^{2 c (a+b x)} \text {csch}(a c+b c x)}{16 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {e^{4 c (a+b x)} \text {csch}(a c+b c x)}{32 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {3 x \text {csch}(a c+b c x)}{8 \sqrt {\text {csch}^2(a c+b c x)}} \]
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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} \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{3/2}} \, dx=\frac {e^{-2 c (a+b x)} \text {csch}(a c+b c x)}{16 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {3 e^{2 c (a+b x)} \text {csch}(a c+b c x)}{16 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {e^{4 c (a+b x)} \text {csch}(a c+b c x)}{32 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {3 x \text {csch}(a c+b c x)}{8 \sqrt {\text {csch}^2(a c+b c x)}} \]
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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 ^3(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 )^3}{8 x^3} \, 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 )^3}{x^3} \, dx,x,e^{c (a+b x)}\right )}{8 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)^3}{x^2} \, dx,x,e^{2 c (a+b x)}\right )}{16 b c \sqrt {\text {csch}^2(a c+b c x)}} \\ & = \frac {\text {csch}(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 {csch}^2(a c+b c x)}} \\ & = \frac {e^{-2 c (a+b x)} \text {csch}(a c+b c x)}{16 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {3 e^{2 c (a+b x)} \text {csch}(a c+b c x)}{16 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)}{32 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {3 x \text {csch}(a c+b c x)}{8 \sqrt {\text {csch}^2(a c+b c x)}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.47 \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{3/2}} \, dx=\frac {\left (e^{-2 c (a+b x)}-3 e^{2 c (a+b x)}+\frac {1}{2} e^{4 c (a+b x)}+6 b c x\right ) \text {csch}^3(c (a+b x))}{16 b c \text {csch}^2(c (a+b x))^{3/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.72 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.46
method | result | size |
default | \(\frac {\operatorname {csgn}\left (\operatorname {csch}\left (c \left (b x +a \right )\right )\right ) \left (\left (\frac {\sinh \left (b c x +a c \right )^{3}}{4}-\frac {3 \sinh \left (b c x +a c \right )}{8}\right ) \cosh \left (b c x +a c \right )+\frac {3 b c x}{8}+\frac {3 a c}{8}+\frac {\sinh \left (b c x +a c \right )^{4}}{4}\right )}{c b}\) | \(75\) |
risch | \(\frac {3 x \,{\mathrm e}^{c \left (b x +a \right )}}{8 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}+\frac {{\mathrm e}^{5 c \left (b x +a \right )}}{32 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}-\frac {3 \,{\mathrm e}^{3 c \left (b x +a \right )}}{16 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}+\frac {{\mathrm e}^{-c \left (b x +a \right )}}{16 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}\) | \(216\) |
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Time = 0.25 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.78 \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{3/2}} \, dx=\frac {3 \, \cosh \left (b c x + a c\right )^{3} + 9 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} - \sinh \left (b c x + a c\right )^{3} + 6 \, {\left (2 \, b c x - 1\right )} \cosh \left (b c x + a c\right ) - 3 \, {\left (4 \, b c x + \cosh \left (b c x + a c\right )^{2} + 2\right )} \sinh \left (b c x + a c\right )}{32 \, {\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\right )}} \]
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\[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{3/2}} \, dx=e^{a c} \int \frac {e^{b c x}}{\left (\operatorname {csch}^{2}{\left (a c + b c x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.38 \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{3/2}} \, dx=\frac {{\left (e^{\left (6 \, b c x + 6 \, a c\right )} - 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 2\right )} e^{\left (-2 \, b c x - 2 \, a c\right )}}{32 \, b c} + \frac {3 \, {\left (b c x + a c\right )}}{8 \, b c} \]
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Time = 0.30 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.26 \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{3/2}} \, dx=\frac {{\left (12 \, b c x e^{\left (-a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 2 \, {\left (3 \, e^{\left (2 \, b c x + 2 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )\right )} e^{\left (-2 \, b c x - 3 \, a c\right )} + {\left (e^{\left (4 \, b c x + 9 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 6 \, e^{\left (2 \, b c x + 7 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )\right )} e^{\left (-6 \, a c\right )}\right )} e^{\left (a c\right )}}{32 \, b c} \]
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Timed out. \[ \int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left (\frac {1}{{\mathrm {sinh}\left (a\,c+b\,c\,x\right )}^2}\right )}^{3/2}} \,d x \]
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