Optimal. Leaf size=191 \[ -\frac {64 \cosh (a c+b c x)}{3 b c \left (e^{2 c (a+b x)}+1\right )^3 \sqrt {\cosh ^2(a c+b c x)}}+\frac {48 \cosh (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^4 \sqrt {\cosh ^2(a c+b c x)}}-\frac {192 \cosh (a c+b c x)}{5 b c \left (e^{2 c (a+b x)}+1\right )^5 \sqrt {\cosh ^2(a c+b c x)}}+\frac {32 \cosh (a c+b c x)}{3 b c \left (e^{2 c (a+b x)}+1\right )^6 \sqrt {\cosh ^2(a c+b c x)}} \]
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Rubi [A] time = 0.26, antiderivative size = 191, 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} \[ -\frac {64 \cosh (a c+b c x)}{3 b c \left (e^{2 c (a+b x)}+1\right )^3 \sqrt {\cosh ^2(a c+b c x)}}+\frac {48 \cosh (a c+b c x)}{b c \left (e^{2 c (a+b x)}+1\right )^4 \sqrt {\cosh ^2(a c+b c x)}}-\frac {192 \cosh (a c+b c x)}{5 b c \left (e^{2 c (a+b x)}+1\right )^5 \sqrt {\cosh ^2(a c+b c x)}}+\frac {32 \cosh (a c+b c x)}{3 b c \left (e^{2 c (a+b x)}+1\right )^6 \sqrt {\cosh ^2(a c+b c x)}} \]
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)}}{\cosh ^2(a c+b c x)^{7/2}} \, dx &=\frac {\cosh (a c+b c x) \int e^{c (a+b x)} \text {sech}^7(a c+b c x) \, dx}{\sqrt {\cosh ^2(a c+b c x)}}\\ &=\frac {\cosh (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 {\cosh ^2(a c+b c x)}}\\ &=\frac {(128 \cosh (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 {\cosh ^2(a c+b c x)}}\\ &=\frac {(64 \cosh (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 {\cosh ^2(a c+b c x)}}\\ &=\frac {(64 \cosh (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 {\cosh ^2(a c+b c x)}}\\ &=\frac {32 \cosh (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^6 \sqrt {\cosh ^2(a c+b c x)}}-\frac {192 \cosh (a c+b c x)}{5 b c \left (1+e^{2 c (a+b x)}\right )^5 \sqrt {\cosh ^2(a c+b c x)}}+\frac {48 \cosh (a c+b c x)}{b c \left (1+e^{2 c (a+b x)}\right )^4 \sqrt {\cosh ^2(a c+b c x)}}-\frac {64 \cosh (a c+b c x)}{3 b c \left (1+e^{2 c (a+b x)}\right )^3 \sqrt {\cosh ^2(a c+b c x)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 84, normalized size = 0.44 \[ -\frac {16 \left (6 e^{2 c (a+b x)}+15 e^{4 c (a+b x)}+20 e^{6 c (a+b x)}+1\right ) \cosh (c (a+b x))}{15 b c \left (e^{2 c (a+b x)}+1\right )^6 \sqrt {\cosh ^2(c (a+b x))}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 589, normalized size = 3.08 \[ -\frac {16 \, {\left (21 \, \cosh \left (b c x + a c\right )^{3} + 63 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} + 19 \, \sinh \left (b c x + a c\right )^{3} + 3 \, {\left (19 \, \cosh \left (b c x + a c\right )^{2} + 3\right )} \sinh \left (b c x + a c\right ) + 21 \, \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} + 21 \, 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} + {\left (84 \, b c \cosh \left (b c x + a c\right )^{6} + 210 \, b c \cosh \left (b c x + a c\right )^{4} + 150 \, b c \cosh \left (b c x + a c\right )^{2} + 19 \, b c\right )} \sinh \left (b c x + a c\right )^{3} + 21 \, 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} + 21 \, 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} + 19 \, b c \cosh \left (b c x + a c\right )^{2} + 3 \, 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 = 0.12, size = 64, normalized size = 0.34 \[ -\frac {16 \, {\left (20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}}{15 \, b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{6}} \]
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 \[ \int \frac {128 \,{\mathrm e}^{c \left (b x +a \right )}}{\left (2 \cosh \left (2 b c x +2 a c \right )+2\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 386, normalized size = 2.02 \[ -\frac {64 \, e^{\left (6 \, b c x + 6 \, a c\right )}}{3 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} + 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {16 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} + 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {32 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{5 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} + 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {16}{15 \, b c {\left (e^{\left (12 \, b c x + 12 \, a c\right )} + 6 \, e^{\left (10 \, b c x + 10 \, a c\right )} + 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 20 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 15 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 345, normalized size = 1.81 \[ \frac {96\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}{b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}+{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^4}-\frac {128\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}{3\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}+{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^3}-\frac {384\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}{5\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}+{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^5}+\frac {64\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}\,\sqrt {{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}{3\,b\,c\,\left ({\mathrm {e}}^{a\,c+b\,c\,x}+{\mathrm {e}}^{3\,a\,c+3\,b\,c\,x}\right )\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^6} \]
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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