Integrand size = 25, antiderivative size = 56 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{3/2} \, dx=\frac {2 e^{4 c (a+b x)} \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c \left (1+e^{2 c (a+b x)}\right )^2} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6852, 2320, 12, 270} \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{3/2} \, dx=\frac {2 e^{4 c (a+b x)} \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}}{b c \left (e^{2 c (a+b x)}+1\right )^2} \]
[In]
[Out]
Rule 12
Rule 270
Rule 2320
Rule 6852
Rubi steps \begin{align*} \text {integral}& = \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}^3(a c+b c x) \, dx \\ & = \frac {\left (\cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \text {Subst}\left (\int \frac {8 x^3}{\left (1+x^2\right )^3} \, dx,x,e^{c (a+b x)}\right )}{b c} \\ & = \frac {\left (8 \cosh (a c+b c x) \sqrt {\text {sech}^2(a c+b c x)}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+x^2\right )^3} \, dx,x,e^{c (a+b x)}\right )}{b c} \\ & = \frac {2 e^{4 c (a+b x)} \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*}
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.79 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{3/2} \, dx=\frac {e^{3 c (a+b x)} \sqrt {\text {sech}^2(c (a+b x))}}{b c+b c e^{2 c (a+b x)}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.36 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {\operatorname {csgn}\left (\operatorname {sech}\left (c \left (b x +a \right )\right )\right ) \left (\frac {\tanh \left (c \left (b x +a \right )\right )^{2}}{2}+\tanh \left (c \left (b x +a \right )\right )\right )}{c b}\) | \(38\) |
risch | \(-\frac {2 \left (2 \,{\mathrm e}^{2 c \left (b x +a \right )}+1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )^{2}}}\, {\mathrm e}^{-c \left (b x +a \right )}}{b c \left (1+{\mathrm e}^{2 c \left (b x +a \right )}\right )}\) | \(69\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (52) = 104\).
Time = 0.25 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.14 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{3/2} \, dx=-\frac {2 \, {\left (3 \, \cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )}}{b c \cosh \left (b c x + a c\right )^{3} + 3 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} + b c \sinh \left (b c x + a c\right )^{3} + 3 \, b c \cosh \left (b c x + a c\right ) + {\left (3 \, b c \cosh \left (b c x + a c\right )^{2} + b c\right )} \sinh \left (b c x + a c\right )} \]
[In]
[Out]
Timed out. \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{3/2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.50 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{3/2} \, dx=-\frac {4 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{b c {\left (e^{\left (4 \, b c x + 4 \, a c\right )} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac {2}{b c {\left (e^{\left (4 \, b c x + 4 \, a c\right )} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.68 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{3/2} \, dx=-\frac {2 \, {\left (2 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}}{b c {\left (e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}^{2}} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.39 \[ \int e^{c (a+b x)} \text {sech}^2(a c+b c x)^{3/2} \, dx=-\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}\,\left (2\,{\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{a\,c+b\,c\,x}}{2}+\frac {{\mathrm {e}}^{-a\,c-b\,c\,x}}{2}\right )}^2}}}{b\,c\,\left ({\mathrm {e}}^{2\,a\,c+2\,b\,c\,x}+1\right )} \]
[In]
[Out]