Optimal. Leaf size=191 \[ -\frac{64 \cosh (a c+b c x) \sqrt{\text{sech}^2(a c+b c x)}}{3 b c \left (e^{2 c (a+b x)}+1\right )^3}+\frac{48 \cosh (a c+b c x) \sqrt{\text{sech}^2(a c+b c x)}}{b c \left (e^{2 c (a+b x)}+1\right )^4}-\frac{192 \cosh (a c+b c x) \sqrt{\text{sech}^2(a c+b c x)}}{5 b c \left (e^{2 c (a+b x)}+1\right )^5}+\frac{32 \cosh (a c+b c x) \sqrt{\text{sech}^2(a c+b c x)}}{3 b c \left (e^{2 c (a+b x)}+1\right )^6} \]
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Rubi [A] time = 0.282013, antiderivative size = 191, 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} \[ -\frac{64 \cosh (a c+b c x) \sqrt{\text{sech}^2(a c+b c x)}}{3 b c \left (e^{2 c (a+b x)}+1\right )^3}+\frac{48 \cosh (a c+b c x) \sqrt{\text{sech}^2(a c+b c x)}}{b c \left (e^{2 c (a+b x)}+1\right )^4}-\frac{192 \cosh (a c+b c x) \sqrt{\text{sech}^2(a c+b c x)}}{5 b c \left (e^{2 c (a+b x)}+1\right )^5}+\frac{32 \cosh (a c+b c x) \sqrt{\text{sech}^2(a c+b c x)}}{3 b c \left (e^{2 c (a+b x)}+1\right )^6} \]
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)^{7/2} \, 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}^7(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{128 x^7}{\left (1+x^2\right )^7} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac{\left (128 \cosh (a c+b c x) \sqrt{\text{sech}^2(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 \cosh (a c+b c x) \sqrt{\text{sech}^2(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 \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)^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 \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 )^6}-\frac{192 \cosh (a c+b c x) \sqrt{\text{sech}^2(a c+b c x)}}{5 b c \left (1+e^{2 c (a+b x)}\right )^5}+\frac{48 \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{64 \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}\\ \end{align*}
Mathematica [A] time = 0.087682, size = 84, normalized size = 0.44 \[ -\frac{16 \left (6 e^{2 c (a+b x)}+15 e^{4 c (a+b x)}+20 e^{6 c (a+b x)}+1\right ) \cosh (c (a+b x)) \sqrt{\text{sech}^2(c (a+b x))}}{15 b c \left (e^{2 c (a+b x)}+1\right )^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.228, size = 91, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 320\,{{\rm e}^{6\,c \left ( bx+a \right ) }}+240\,{{\rm e}^{4\,c \left ( bx+a \right ) }}+96\,{{\rm e}^{2\,c \left ( bx+a \right ) }}+16 \right ){{\rm e}^{-c \left ( bx+a \right ) }}}{15\, \left ( 1+{{\rm e}^{2\,c \left ( bx+a \right ) }} \right ) ^{5}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 [B] time = 1.13235, size = 521, normalized size = 2.73 \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 = 3.88298, size = 1516, normalized size = 7.94 \begin{align*} -\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 )}} \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 = 1.13784, size = 86, normalized size = 0.45 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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