Optimal. Leaf size=147 \[ -\frac {8 \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2 \sqrt {\sinh ^2(a c+b c x)}}+\frac {32 \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3 \sqrt {\sinh ^2(a c+b c x)}}-\frac {4 \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4 \sqrt {\sinh ^2(a c+b c x)}} \]
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Rubi [A] time = 0.21, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6720, 2282, 12, 266, 43} \[ -\frac {8 \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2 \sqrt {\sinh ^2(a c+b c x)}}+\frac {32 \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3 \sqrt {\sinh ^2(a c+b c x)}}-\frac {4 \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4 \sqrt {\sinh ^2(a c+b c x)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 266
Rule 2282
Rule 6720
Rubi steps
\begin {align*} \int \frac {e^{c (a+b x)}}{\sinh ^2(a c+b c x)^{5/2}} \, dx &=\frac {\sinh (a c+b c x) \int e^{c (a+b x)} \text {csch}^5(a c+b c x) \, dx}{\sqrt {\sinh ^2(a c+b c x)}}\\ &=\frac {\sinh (a c+b c x) \operatorname {Subst}\left (\int \frac {32 x^5}{\left (-1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\sinh ^2(a c+b c x)}}\\ &=\frac {(32 \sinh (a c+b c x)) \operatorname {Subst}\left (\int \frac {x^5}{\left (-1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c \sqrt {\sinh ^2(a c+b c x)}}\\ &=\frac {(16 \sinh (a c+b c x)) \operatorname {Subst}\left (\int \frac {x^2}{(-1+x)^5} \, dx,x,e^{2 c (a+b x)}\right )}{b c \sqrt {\sinh ^2(a c+b c x)}}\\ &=\frac {(16 \sinh (a c+b c x)) \operatorname {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 \sqrt {\sinh ^2(a c+b c x)}}\\ &=-\frac {4 \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 {32 \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 {8 \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2 \sqrt {\sinh ^2(a c+b c x)}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 72, normalized size = 0.49 \[ -\frac {4 \left (-4 e^{2 c (a+b x)}+6 e^{4 c (a+b x)}+1\right ) \sinh (c (a+b x))}{3 b c \left (e^{2 c (a+b x)}-1\right )^4 \sqrt {\sinh ^2(c (a+b x))}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 315, normalized size = 2.14 \[ -\frac {4 \, {\left (7 \, \cosh \left (b c x + a c\right )^{2} + 10 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + 7 \, \sinh \left (b c x + a c\right )^{2} - 4\right )}}{3 \, {\left (b c \cosh \left (b c x + a c\right )^{6} + 6 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{5} + b c \sinh \left (b c x + a c\right )^{6} - 4 \, b c \cosh \left (b c x + a c\right )^{4} + {\left (15 \, b c \cosh \left (b c x + a c\right )^{2} - 4 \, b c\right )} \sinh \left (b c x + a c\right )^{4} + 7 \, b c \cosh \left (b c x + a c\right )^{2} + 4 \, {\left (5 \, b c \cosh \left (b c x + a c\right )^{3} - 4 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{3} + {\left (15 \, b c \cosh \left (b c x + a c\right )^{4} - 24 \, b c \cosh \left (b c x + a c\right )^{2} + 7 \, b c\right )} \sinh \left (b c x + a c\right )^{2} - 4 \, b c + 2 \, {\left (3 \, b c \cosh \left (b c x + a c\right )^{5} - 8 \, b c \cosh \left (b c x + a c\right )^{3} + 5 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 122, normalized size = 0.83 \[ -\frac {4 \, {\left (6 \, e^{\left (4 \, b c x + 4 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) + \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )\right )}}{3 \, b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {32 \,{\mathrm e}^{c \left (b x +a \right )}}{\left (-2+2 \cosh \left (2 b c x +2 a c \right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 209, normalized size = 1.42 \[ -\frac {8 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} - 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} + \frac {16 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{3 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} - 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {4}{3 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} - 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 89, normalized size = 0.61 \[ -\frac {8\,{\mathrm {e}}^{a\,c+b\,c\,x}\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}\,\left (6\,{\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}-4\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}{3\,b\,c\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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