Optimal. Leaf size=141 \[ -\frac {8 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c \left (e^{2 c (a+b x)}+1\right )^2}+\frac {32 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{3 b c \left (e^{2 c (a+b x)}+1\right )^3}-\frac {4 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c \left (e^{2 c (a+b x)}+1\right )^4} \]
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Rubi [A] time = 0.17, antiderivative size = 141, 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} \[ -\frac {8 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c \left (e^{2 c (a+b x)}+1\right )^2}+\frac {32 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{3 b c \left (e^{2 c (a+b x)}+1\right )^3}-\frac {4 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c \left (e^{2 c (a+b x)}+1\right )^4} \]
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 {sech}^2(a c+b c x)^{5/2} \, dx &=\left (\cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \int e^{c (a+b x)} \text {sech}^5(a c+b c x) \, dx\\ &=\frac {\left (\cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \operatorname {Subst}\left (\int \frac {32 x^5}{\left (1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac {\left (32 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \operatorname {Subst}\left (\int \frac {x^5}{\left (1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac {\left (16 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \operatorname {Subst}\left (\int \frac {x^2}{(1+x)^5} \, dx,x,e^{2 c (a+b x)}\right )}{b c}\\ &=\frac {\left (16 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \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}\\ &=-\frac {4 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c \left (1+e^{2 c (a+b x)}\right )^4}+\frac {32 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{3 b c \left (1+e^{2 c (a+b x)}\right )^3}-\frac {8 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c \left (1+e^{2 c (a+b x)}\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 72, normalized size = 0.51 \[ -\frac {4 \left (4 e^{2 c (a+b x)}+6 e^{4 c (a+b x)}+1\right ) \cosh (c (a+b x)) \sqrt {\text {sech}^2(c (a+b x))}}{3 b c \left (e^{2 c (a+b x)}+1\right )^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 315, normalized size = 2.23 \[ -\frac {4 \, {\left (7 \, \cosh \left (b c x + a c\right )^{2} + 10 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + 7 \, \sinh \left (b c x + a c\right )^{2} + 4\right )}}{3 \, {\left (b c \cosh \left (b c x + a c\right )^{6} + 6 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{5} + b c \sinh \left (b c x + a c\right )^{6} + 4 \, b c \cosh \left (b c x + a c\right )^{4} + {\left (15 \, b c \cosh \left (b c x + a c\right )^{2} + 4 \, b c\right )} \sinh \left (b c x + a c\right )^{4} + 7 \, b c \cosh \left (b c x + a c\right )^{2} + 4 \, {\left (5 \, b c \cosh \left (b c x + a c\right )^{3} + 4 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{3} + {\left (15 \, b c \cosh \left (b c x + a c\right )^{4} + 24 \, b c \cosh \left (b c x + a c\right )^{2} + 7 \, b c\right )} \sinh \left (b c x + a c\right )^{2} + 4 \, b c + 2 \, {\left (3 \, b c \cosh \left (b c x + a c\right )^{5} + 8 \, b c \cosh \left (b c x + a c\right )^{3} + 5 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 51, normalized size = 0.36 \[ -\frac {4 \, {\left (6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}}{3 \, b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.68, size = 80, normalized size = 0.57 \[ -\frac {4 \left (6 \,{\mathrm e}^{4 c \left (b x +a \right )}+4 \,{\mathrm e}^{2 c \left (b x +a \right )}+1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, {\mathrm e}^{-c \left (b x +a \right )}}{3 b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 209, normalized size = 1.48 \[ -\frac {8 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} + 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {16 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{3 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} + 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {4}{3 \, b c {\left (e^{\left (8 \, b c x + 8 \, a c\right )} + 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 91, normalized size = 0.65 \[ -\frac {2\,{\mathrm {e}}^{-a\,c-b\,c\,x}\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}\,\left (4\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+6\,{\mathrm {e}}^{4\,a\,c+4\,b\,c\,x}+1\right )}{3\,b\,c\,{\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )}^3} \]
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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