Optimal. Leaf size=60 \[ -\frac{8}{b \left (e^{2 a+2 b x}+1\right )^2}+\frac{32}{3 b \left (e^{2 a+2 b x}+1\right )^3}-\frac{4}{b \left (e^{2 a+2 b x}+1\right )^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0477429, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2282, 12, 266, 43} \[ -\frac{8}{b \left (e^{2 a+2 b x}+1\right )^2}+\frac{32}{3 b \left (e^{2 a+2 b x}+1\right )^3}-\frac{4}{b \left (e^{2 a+2 b x}+1\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 12
Rule 266
Rule 43
Rubi steps
\begin{align*} \int e^{a+b x} \text{sech}^5(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{32 x^5}{\left (1+x^2\right )^5} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{32 \operatorname{Subst}\left (\int \frac{x^5}{\left (1+x^2\right )^5} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac{16 \operatorname{Subst}\left (\int \frac{x^2}{(1+x)^5} \, dx,x,e^{2 a+2 b x}\right )}{b}\\ &=\frac{16 \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 a+2 b x}\right )}{b}\\ &=-\frac{4}{b \left (1+e^{2 a+2 b x}\right )^4}+\frac{32}{3 b \left (1+e^{2 a+2 b x}\right )^3}-\frac{8}{b \left (1+e^{2 a+2 b x}\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0342389, size = 44, normalized size = 0.73 \[ -\frac{4 \left (4 e^{2 (a+b x)}+6 e^{4 (a+b x)}+1\right )}{3 b \left (e^{2 (a+b x)}+1\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.035, size = 61, normalized size = 1. \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{4\, \left ( \cosh \left ( bx+a \right ) \right ) ^{4}}}+{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{4\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}}+ \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (bx+a\right ) \right ) ^{2}}{3}} \right ) \tanh \left ( bx+a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.04132, size = 232, normalized size = 3.87 \begin{align*} -\frac{8 \, e^{\left (4 \, b x + 4 \, a\right )}}{b{\left (e^{\left (8 \, b x + 8 \, a\right )} + 4 \, e^{\left (6 \, b x + 6 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} + 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} - \frac{16 \, e^{\left (2 \, b x + 2 \, a\right )}}{3 \, b{\left (e^{\left (8 \, b x + 8 \, a\right )} + 4 \, e^{\left (6 \, b x + 6 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} + 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} - \frac{4}{3 \, b{\left (e^{\left (8 \, b x + 8 \, a\right )} + 4 \, e^{\left (6 \, b x + 6 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} + 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.78063, size = 635, normalized size = 10.58 \begin{align*} -\frac{4 \,{\left (7 \, \cosh \left (b x + a\right )^{2} + 10 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 7 \, \sinh \left (b x + a\right )^{2} + 4\right )}}{3 \,{\left (b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} + 4 \, b \cosh \left (b x + a\right )^{4} +{\left (15 \, b \cosh \left (b x + a\right )^{2} + 4 \, b\right )} \sinh \left (b x + a\right )^{4} + 4 \,{\left (5 \, b \cosh \left (b x + a\right )^{3} + 4 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 7 \, b \cosh \left (b x + a\right )^{2} +{\left (15 \, b \cosh \left (b x + a\right )^{4} + 24 \, b \cosh \left (b x + a\right )^{2} + 7 \, b\right )} \sinh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b \cosh \left (b x + a\right )^{5} + 8 \, b \cosh \left (b x + a\right )^{3} + 5 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 4 \, b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21968, size = 57, normalized size = 0.95 \begin{align*} -\frac{4 \,{\left (6 \, e^{\left (4 \, b x + 4 \, a\right )} + 4 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}}{3 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]