Integrand size = 25, antiderivative size = 147 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{5/2} \, dx=-\frac {4 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4}+\frac {32 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac {8 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2} \]
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Time = 0.12 (sec) , antiderivative size = 147, 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} \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{5/2} \, dx=-\frac {8 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{b c \left (1-e^{2 c (a+b x)}\right )^2}+\frac {32 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac {4 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{b c \left (1-e^{2 c (a+b x)}\right )^4} \]
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Rule 12
Rule 45
Rule 272
Rule 2320
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \int e^{c (a+b x)} \text {csch}^5(a c+b c x) \, dx \\ & = \frac {\left (\sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \text {Subst}\left (\int \frac {32 x^5}{\left (-1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c} \\ & = \frac {\left (32 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \text {Subst}\left (\int \frac {x^5}{\left (-1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c} \\ & = \frac {\left (16 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \text {Subst}\left (\int \frac {x^2}{(-1+x)^5} \, dx,x,e^{2 c (a+b x)}\right )}{b c} \\ & = \frac {\left (16 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \text {Subst}\left (\int \left (\frac {1}{(-1+x)^5}+\frac {2}{(-1+x)^4}+\frac {1}{(-1+x)^3}\right ) \, dx,x,e^{2 c (a+b x)}\right )}{b c} \\ & = -\frac {4 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4}+\frac {32 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac {8 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.49 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{5/2} \, dx=-\frac {4 \left (1-4 e^{2 c (a+b x)}+6 e^{4 c (a+b x)}\right ) \sqrt {\text {csch}^2(c (a+b x))} \sinh (c (a+b x))}{3 b c \left (-1+e^{2 c (a+b x)}\right )^4} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.44
method | result | size |
default | \(-\frac {\operatorname {csgn}\left (\operatorname {csch}\left (c \left (b x +a \right )\right )\right ) \left (\frac {\coth \left (c \left (b x +a \right )\right )^{4}}{4}+\frac {\coth \left (c \left (b x +a \right )\right )^{3}}{3}-\frac {\coth \left (c \left (b x +a \right )\right )^{2}}{2}-\coth \left (c \left (b x +a \right )\right )\right )}{c b}\) | \(65\) |
risch | \(-\frac {4 \left (6 \,{\mathrm e}^{4 c \left (b x +a \right )}-4 \,{\mathrm e}^{2 c \left (b x +a \right )}+1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, {\mathrm e}^{-c \left (b x +a \right )}}{3 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{3}}\) | \(80\) |
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Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (130) = 260\).
Time = 0.25 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.14 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{5/2} \, dx=-\frac {4 \, {\left (7 \, \cosh \left (b c x + a c\right )^{2} + 10 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + 7 \, \sinh \left (b c x + a c\right )^{2} - 4\right )}}{3 \, {\left (b c \cosh \left (b c x + a c\right )^{6} + 6 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{5} + b c \sinh \left (b c x + a c\right )^{6} - 4 \, b c \cosh \left (b c x + a c\right )^{4} + {\left (15 \, b c \cosh \left (b c x + a c\right )^{2} - 4 \, b c\right )} \sinh \left (b c x + a c\right )^{4} + 7 \, b c \cosh \left (b c x + a c\right )^{2} + 4 \, {\left (5 \, b c \cosh \left (b c x + a c\right )^{3} - 4 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{3} + {\left (15 \, b c \cosh \left (b c x + a c\right )^{4} - 24 \, b c \cosh \left (b c x + a c\right )^{2} + 7 \, b c\right )} \sinh \left (b c x + a c\right )^{2} - 4 \, b c + 2 \, {\left (3 \, b c \cosh \left (b c x + a c\right )^{5} - 8 \, b c \cosh \left (b c x + a c\right )^{3} + 5 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )\right )}} \]
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\[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{5/2} \, dx=e^{a c} \int \left (\operatorname {csch}^{2}{\left (a c + b c x \right )}\right )^{\frac {5}{2}} e^{b c x}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.42 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{5/2} \, dx=-\frac {8 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} - 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} + \frac {16 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{3 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} - 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {4}{3 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} - 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.52 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{5/2} \, dx=-\frac {4 \, {\left (6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}}{3 \, b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}^{4} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )} \]
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Time = 2.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.62 \[ \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{5/2} \, dx=-\frac {2\,{\mathrm {e}}^{-a\,c-b\,c\,x}\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left (6\,{\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}-4\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}{3\,b\,c\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^3} \]
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