Optimal. Leaf size=199 \[ \frac{64 \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3 \sqrt{\sinh ^2(a c+b c x)}}-\frac{48 \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4 \sqrt{\sinh ^2(a c+b c x)}}+\frac{192 \sinh (a c+b c x)}{5 b c \left (1-e^{2 c (a+b x)}\right )^5 \sqrt{\sinh ^2(a c+b c x)}}-\frac{32 \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^6 \sqrt{\sinh ^2(a c+b c x)}} \]
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Rubi [A] time = 0.279457, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6720, 2282, 12, 266, 43} \[ \frac{64 \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3 \sqrt{\sinh ^2(a c+b c x)}}-\frac{48 \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4 \sqrt{\sinh ^2(a c+b c x)}}+\frac{192 \sinh (a c+b c x)}{5 b c \left (1-e^{2 c (a+b x)}\right )^5 \sqrt{\sinh ^2(a c+b c x)}}-\frac{32 \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^6 \sqrt{\sinh ^2(a c+b c x)}} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2282
Rule 12
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{c (a+b x)}}{\sinh ^2(a c+b c x)^{7/2}} \, dx &=\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) \operatorname{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)) \operatorname{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)) \operatorname{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)) \operatorname{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*}
Mathematica [A] time = 0.078297, size = 84, normalized size = 0.42 \[ -\frac{16 \left (6 e^{2 c (a+b x)}-15 e^{4 c (a+b x)}+20 e^{6 c (a+b x)}-1\right ) \sinh (c (a+b x))}{15 b c \left (e^{2 c (a+b x)}-1\right )^6 \sqrt{\sinh ^2(c (a+b x))}} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{{{\rm e}^{c \left ( bx+a \right ) }} \left ( \left ( \sinh \left ( bcx+ac \right ) \right ) ^{2} \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.7505, size = 521, normalized size = 2.62 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98596, size = 1512, normalized size = 7.6 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33243, size = 217, normalized size = 1.09 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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