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 = 0.166844, antiderivative size = 141, 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} \[ -\frac{8 \cosh (a c+b c x) \sqrt{\text{sech}^2(a c+b c x)}}{b c \left (e^{2 c (a+b x)}+1\right )^2}+\frac{32 \cosh (a c+b c x) \sqrt{\text{sech}^2(a c+b c x)}}{3 b c \left (e^{2 c (a+b x)}+1\right )^3}-\frac{4 \cosh (a c+b c x) \sqrt{\text{sech}^2(a c+b c x)}}{b c \left (e^{2 c (a+b x)}+1\right )^4} \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2282
Rule 12
Rule 266
Rule 43
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*}
Mathematica [A] time = 0.0657853, 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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Maple [A] time = 0.174, size = 80, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 24\,{{\rm e}^{4\,c \left ( bx+a \right ) }}+16\,{{\rm e}^{2\,c \left ( bx+a \right ) }}+4 \right ){{\rm e}^{-c \left ( bx+a \right ) }}}{3\, \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{3}cb}\sqrt{{\frac{{{\rm e}^{2\,c \left ( bx+a \right ) }}}{ \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1506, size = 282, normalized size = 2. \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.12381, size = 797, normalized size = 5.65 \begin{align*} -\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 )}} \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 = 1.12626, size = 69, normalized size = 0.49 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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