Integrand size = 25, antiderivative size = 199 \[ \int \frac {e^{c (a+b x)}}{\sinh ^2(a c+b c x)^{7/2}} \, dx=-\frac {32 \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^6 \sqrt {\sinh ^2(a c+b c x)}}+\frac {192 \sinh (a c+b c x)}{5 b c \left (1-e^{2 c (a+b x)}\right )^5 \sqrt {\sinh ^2(a c+b c x)}}-\frac {48 \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4 \sqrt {\sinh ^2(a c+b c x)}}+\frac {64 \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3 \sqrt {\sinh ^2(a c+b c x)}} \]
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Time = 0.19 (sec) , antiderivative size = 199, 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} \[ \int \frac {e^{c (a+b x)}}{\sinh ^2(a c+b c x)^{7/2}} \, dx=\frac {64 \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3 \sqrt {\sinh ^2(a c+b c x)}}-\frac {48 \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4 \sqrt {\sinh ^2(a c+b c x)}}+\frac {192 \sinh (a c+b c x)}{5 b c \left (1-e^{2 c (a+b x)}\right )^5 \sqrt {\sinh ^2(a c+b c x)}}-\frac {32 \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^6 \sqrt {\sinh ^2(a c+b c x)}} \]
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Rule 12
Rule 45
Rule 272
Rule 2320
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \frac {\sinh (a c+b c x) \int e^{c (a+b x)} \text {csch}^7(a c+b c x) \, dx}{\sqrt {\sinh ^2(a c+b c x)}} \\ & = \frac {\sinh (a c+b c x) \text {Subst}\left (\int \frac {128 x^7}{\left (-1+x^2\right )^7} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\sinh ^2(a c+b c x)}} \\ & = \frac {(128 \sinh (a c+b c x)) \text {Subst}\left (\int \frac {x^7}{\left (-1+x^2\right )^7} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\sinh ^2(a c+b c x)}} \\ & = \frac {(64 \sinh (a c+b c x)) \text {Subst}\left (\int \frac {x^3}{(-1+x)^7} \, dx,x,e^{2 c (a+b x)}\right )}{b c \sqrt {\sinh ^2(a c+b c x)}} \\ & = \frac {(64 \sinh (a c+b c x)) \text {Subst}\left (\int \left (\frac {1}{(-1+x)^7}+\frac {3}{(-1+x)^6}+\frac {3}{(-1+x)^5}+\frac {1}{(-1+x)^4}\right ) \, dx,x,e^{2 c (a+b x)}\right )}{b c \sqrt {\sinh ^2(a c+b c x)}} \\ & = -\frac {32 \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^6 \sqrt {\sinh ^2(a c+b c x)}}+\frac {192 \sinh (a c+b c x)}{5 b c \left (1-e^{2 c (a+b x)}\right )^5 \sqrt {\sinh ^2(a c+b c x)}}-\frac {48 \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4 \sqrt {\sinh ^2(a c+b c x)}}+\frac {64 \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3 \sqrt {\sinh ^2(a c+b c x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.42 \[ \int \frac {e^{c (a+b x)}}{\sinh ^2(a c+b c x)^{7/2}} \, dx=-\frac {16 \left (-1+6 e^{2 c (a+b x)}-15 e^{4 c (a+b x)}+20 e^{6 c (a+b x)}\right ) \sinh (c (a+b x))}{15 b c \left (-1+e^{2 c (a+b x)}\right )^6 \sqrt {\sinh ^2(c (a+b x))}} \]
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Time = 0.87 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.46
method | result | size |
risch | \(-\frac {16 \left (20 \,{\mathrm e}^{6 c \left (b x +a \right )}-15 \,{\mathrm e}^{4 c \left (b x +a \right )}+6 \,{\mathrm e}^{2 c \left (b x +a \right )}-1\right ) {\mathrm e}^{-c \left (b x +a \right )}}{15 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{5} \sqrt {\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2} {\mathrm e}^{-2 c \left (b x +a \right )}}}\) | \(91\) |
default | \(\frac {\operatorname {csgn}\left (\sinh \left (c \left (b x +a \right )\right )\right ) \left (-\frac {\coth \left (b c x +a c \right )^{6}}{6}-\frac {\coth \left (b c x +a c \right )^{5}}{5}+\frac {\coth \left (b c x +a c \right )^{4}}{2}+\frac {2 \coth \left (b c x +a c \right )^{3}}{3}-\frac {\coth \left (b c x +a c \right )^{2}}{2}-\coth \left (b c x +a c \right )\right )}{c b}\) | \(94\) |
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Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (173) = 346\).
Time = 0.29 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.97 \[ \int \frac {e^{c (a+b x)}}{\sinh ^2(a c+b c x)^{7/2}} \, dx=-\frac {16 \, {\left (19 \, \cosh \left (b c x + a c\right )^{3} + 57 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} + 21 \, \sinh \left (b c x + a c\right )^{3} + 21 \, {\left (3 \, \cosh \left (b c x + a c\right )^{2} - 1\right )} \sinh \left (b c x + a c\right ) - 9 \, \cosh \left (b c x + a c\right )\right )}}{15 \, {\left (b c \cosh \left (b c x + a c\right )^{9} + 9 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{8} + b c \sinh \left (b c x + a c\right )^{9} - 6 \, b c \cosh \left (b c x + a c\right )^{7} + 6 \, {\left (6 \, b c \cosh \left (b c x + a c\right )^{2} - b c\right )} \sinh \left (b c x + a c\right )^{7} + 15 \, b c \cosh \left (b c x + a c\right )^{5} + 42 \, {\left (2 \, b c \cosh \left (b c x + a c\right )^{3} - b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{6} + 3 \, {\left (42 \, b c \cosh \left (b c x + a c\right )^{4} - 42 \, b c \cosh \left (b c x + a c\right )^{2} + 5 \, b c\right )} \sinh \left (b c x + a c\right )^{5} - 19 \, b c \cosh \left (b c x + a c\right )^{3} + 3 \, {\left (42 \, b c \cosh \left (b c x + a c\right )^{5} - 70 \, b c \cosh \left (b c x + a c\right )^{3} + 25 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{4} + 3 \, {\left (28 \, b c \cosh \left (b c x + a c\right )^{6} - 70 \, b c \cosh \left (b c x + a c\right )^{4} + 50 \, b c \cosh \left (b c x + a c\right )^{2} - 7 \, b c\right )} \sinh \left (b c x + a c\right )^{3} + 9 \, b c \cosh \left (b c x + a c\right ) + 3 \, {\left (12 \, b c \cosh \left (b c x + a c\right )^{7} - 42 \, b c \cosh \left (b c x + a c\right )^{5} + 50 \, b c \cosh \left (b c x + a c\right )^{3} - 19 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} + 3 \, {\left (3 \, b c \cosh \left (b c x + a c\right )^{8} - 14 \, b c \cosh \left (b c x + a c\right )^{6} + 25 \, b c \cosh \left (b c x + a c\right )^{4} - 21 \, b c \cosh \left (b c x + a c\right )^{2} + 7 \, b c\right )} \sinh \left (b c x + a c\right )\right )}} \]
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Timed out. \[ \int \frac {e^{c (a+b x)}}{\sinh ^2(a c+b c x)^{7/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (173) = 346\).
Time = 0.32 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.94 \[ \int \frac {e^{c (a+b x)}}{\sinh ^2(a c+b c x)^{7/2}} \, dx=-\frac {64 \, e^{\left (6 \, b c x + 6 \, a c\right )}}{3 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} - 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} - 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} + \frac {16 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} - 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} - 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {32 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{5 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} - 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} - 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} + \frac {16}{15 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} - 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} - 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.81 \[ \int \frac {e^{c (a+b x)}}{\sinh ^2(a c+b c x)^{7/2}} \, dx=-\frac {16 \, {\left (20 \, e^{\left (6 \, b c x + 6 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )\right )}}{15 \, b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}^{6}} \]
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Time = 1.35 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.77 \[ \int \frac {e^{c (a+b x)}}{\sinh ^2(a c+b c x)^{7/2}} \, dx=\frac {128\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}{3\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}-{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^3}+\frac {96\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}{b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}-{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^4}+\frac {384\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}{5\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}-{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^5}+\frac {64\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}{3\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}-{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^6} \]
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