Optimal. Leaf size=199 \[ \frac {64 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac {48 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{b c \left (1-e^{2 c (a+b x)}\right )^4}+\frac {192 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{5 b c \left (1-e^{2 c (a+b x)}\right )^5}-\frac {32 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{3 b c \left (1-e^{2 c (a+b x)}\right )^6} \]
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Rubi [A] time = 0.30, antiderivative size = 199, 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} \[ \frac {64 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac {48 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{b c \left (1-e^{2 c (a+b x)}\right )^4}+\frac {192 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{5 b c \left (1-e^{2 c (a+b x)}\right )^5}-\frac {32 \sinh (a c+b c x) \sqrt {\text {csch}^2(a c+b c x)}}{3 b c \left (1-e^{2 c (a+b x)}\right )^6} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 266
Rule 2282
Rule 6720
Rubi steps
\begin {align*} \int e^{c (a+b x)} \text {csch}^2(a c+b c x)^{7/2} \, dx &=\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}^7(a c+b c x) \, dx\\ &=\frac {\left (\sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \operatorname {Subst}\left (\int \frac {128 x^7}{\left (-1+x^2\right )^7} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac {\left (128 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \operatorname {Subst}\left (\int \frac {x^7}{\left (-1+x^2\right )^7} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac {\left (64 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \operatorname {Subst}\left (\int \frac {x^3}{(-1+x)^7} \, dx,x,e^{2 c (a+b x)}\right )}{b c}\\ &=\frac {\left (64 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \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}\\ &=-\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 )^6}+\frac {192 \sqrt {\text {csch}^2(a c+b c x)} \sinh (a c+b c x)}{5 b c \left (1-e^{2 c (a+b x)}\right )^5}-\frac {48 \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 {64 \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}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 84, normalized size = 0.42 \[ -\frac {16 \left (6 e^{2 c (a+b x)}-15 e^{4 c (a+b x)}+20 e^{6 c (a+b x)}-1\right ) \sinh (c (a+b x)) \sqrt {\text {csch}^2(c (a+b x))}}{15 b c \left (e^{2 c (a+b x)}-1\right )^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 592, normalized size = 2.97 \[ -\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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 90, normalized size = 0.45 \[ -\frac {16 \, {\left (20 \, e^{\left (6 \, b c x + 6 \, a c\right )} - 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}}{15 \, b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}^{6} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 91, normalized size = 0.46 \[ -\frac {16 \left (20 \,{\mathrm e}^{6 c \left (b x +a \right )}-15 \,{\mathrm e}^{4 c \left (b x +a \right )}+6 \,{\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}\, {\mathrm e}^{-c \left (b x +a \right )}}{15 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 386, normalized size = 1.94 \[ -\frac {64 \, e^{\left (6 \, b c x + 6 \, a c\right )}}{3 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} - 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} - 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} + \frac {16 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} - 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} - 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {32 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{5 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} - 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} - 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} + \frac {16}{15 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} - 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} - 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.60, size = 413, normalized size = 2.08 \[ \frac {32\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left ({\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}-2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}{3\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}-{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^3}+\frac {24\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left ({\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}-2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}{b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}-{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^4}+\frac {96\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left ({\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}-2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}{5\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}-{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^5}+\frac {16\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left ({\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}-2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}{3\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}-{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}-1\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{a c} \int \left (\operatorname {csch}^{2}{\left (a c + b c x \right )}\right )^{\frac {7}{2}} e^{b c x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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