Integrand size = 16, antiderivative size = 66 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {4}{b \left (1-e^{2 a+2 b x}\right )^4}+\frac {32}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {8}{b \left (1-e^{2 a+2 b x}\right )^2} \]
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Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2320, 12, 272, 45} \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {8}{b \left (1-e^{2 a+2 b x}\right )^2}+\frac {32}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {4}{b \left (1-e^{2 a+2 b x}\right )^4} \]
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Rule 12
Rule 45
Rule 272
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {32 x^5}{\left (-1+x^2\right )^5} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {32 \text {Subst}\left (\int \frac {x^5}{\left (-1+x^2\right )^5} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {16 \text {Subst}\left (\int \frac {x^2}{(-1+x)^5} \, dx,x,e^{2 a+2 b x}\right )}{b} \\ & = \frac {16 \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 a+2 b x}\right )}{b} \\ & = -\frac {4}{b \left (1-e^{2 a+2 b x}\right )^4}+\frac {32}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {8}{b \left (1-e^{2 a+2 b x}\right )^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.67 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {4 \left (1-4 e^{2 (a+b x)}+6 e^{4 (a+b x)}\right )}{3 b \left (-1+e^{2 (a+b x)}\right )^4} \]
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Time = 3.42 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.53
method | result | size |
derivativedivides | \(\frac {\left (\frac {2}{3}-\frac {\operatorname {csch}\left (b x +a \right )^{2}}{3}\right ) \coth \left (b x +a \right )-\frac {1}{4 \sinh \left (b x +a \right )^{4}}}{b}\) | \(35\) |
default | \(\frac {\left (\frac {2}{3}-\frac {\operatorname {csch}\left (b x +a \right )^{2}}{3}\right ) \coth \left (b x +a \right )-\frac {1}{4 \sinh \left (b x +a \right )^{4}}}{b}\) | \(35\) |
risch | \(-\frac {4 \left (6 \,{\mathrm e}^{4 b x +4 a}-4 \,{\mathrm e}^{2 b x +2 a}+1\right )}{3 b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{4}}\) | \(43\) |
parallelrisch | \(-\frac {{\mathrm e}^{b x +a} \operatorname {sech}\left (\frac {b x}{2}+\frac {a}{2}\right )^{4} \operatorname {csch}\left (\frac {b x}{2}+\frac {a}{2}\right )^{4} \left (4 \cosh \left (b x +a \right )+4 \sinh \left (b x +a \right )-\cosh \left (3 b x +3 a \right )-\sinh \left (3 b x +3 a \right )\right )}{192 b}\) | \(73\) |
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Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (55) = 110\).
Time = 0.26 (sec) , antiderivative size = 233, normalized size of antiderivative = 3.53 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {4 \, {\left (7 \, \cosh \left (b x + a\right )^{2} + 10 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 7 \, \sinh \left (b x + a\right )^{2} - 4\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} - 4 \, b \cosh \left (b x + a\right )^{4} + {\left (15 \, b \cosh \left (b x + a\right )^{2} - 4 \, b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} - 4 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 7 \, b \cosh \left (b x + a\right )^{2} + {\left (15 \, b \cosh \left (b x + a\right )^{4} - 24 \, b \cosh \left (b x + a\right )^{2} + 7 \, b\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{5} - 8 \, b \cosh \left (b x + a\right )^{3} + 5 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 4 \, b\right )}} \]
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\[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=e^{a} \int e^{b x} \operatorname {csch}^{5}{\left (a + b x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (55) = 110\).
Time = 0.21 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.61 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {8 \, e^{\left (4 \, b x + 4 \, a\right )}}{b {\left (e^{\left (8 \, b x + 8 \, a\right )} - 4 \, e^{\left (6 \, b x + 6 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} + \frac {16 \, e^{\left (2 \, b x + 2 \, a\right )}}{3 \, b {\left (e^{\left (8 \, b x + 8 \, a\right )} - 4 \, e^{\left (6 \, b x + 6 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} - \frac {4}{3 \, b {\left (e^{\left (8 \, b x + 8 \, a\right )} - 4 \, e^{\left (6 \, b x + 6 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.64 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {4 \, {\left (6 \, e^{\left (4 \, b x + 4 \, a\right )} - 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}{3 \, b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{4}} \]
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Time = 1.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.64 \[ \int e^{a+b x} \text {csch}^5(a+b x) \, dx=-\frac {4\,\left (6\,{\mathrm {e}}^{4\,a+4\,b\,x}-4\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}{3\,b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}^4} \]
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