Integrand size = 10, antiderivative size = 15 \[ \int \frac {1}{\sqrt {a \cosh ^4(x)}} \, dx=\frac {\cosh (x) \sinh (x)}{\sqrt {a \cosh ^4(x)}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 3852, 8} \[ \int \frac {1}{\sqrt {a \cosh ^4(x)}} \, dx=\frac {\sinh (x) \cosh (x)}{\sqrt {a \cosh ^4(x)}} \]
[In]
[Out]
Rule 8
Rule 3286
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \frac {\cosh ^2(x) \int \text {sech}^2(x) \, dx}{\sqrt {a \cosh ^4(x)}} \\ & = \frac {\left (i \cosh ^2(x)\right ) \text {Subst}(\int 1 \, dx,x,-i \tanh (x))}{\sqrt {a \cosh ^4(x)}} \\ & = \frac {\cosh (x) \sinh (x)}{\sqrt {a \cosh ^4(x)}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a \cosh ^4(x)}} \, dx=\frac {\cosh (x) \sinh (x)}{\sqrt {a \cosh ^4(x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(28\) vs. \(2(13)=26\).
Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.93
method | result | size |
risch | \(-\frac {2 \,{\mathrm e}^{-2 x} \left (1+{\mathrm e}^{2 x}\right )}{\sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}\) | \(29\) |
default | \(\frac {\sqrt {8}\, \sqrt {2}\, \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (1+\cosh \left (2 x \right )\right )}\, \sqrt {a \sinh \left (2 x \right )^{2}}}{4 a \sinh \left (2 x \right ) \sqrt {\left (1+\cosh \left (2 x \right )\right )^{2} a}}\) | \(56\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (13) = 26\).
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 7.73 \[ \int \frac {1}{\sqrt {a \cosh ^4(x)}} \, dx=-\frac {2 \, \sqrt {a e^{\left (8 \, x\right )} + 4 \, a e^{\left (6 \, x\right )} + 6 \, a e^{\left (4 \, x\right )} + 4 \, a e^{\left (2 \, x\right )} + a}}{a \cosh \left (x\right )^{2} + {\left (a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} + {\left (a \cosh \left (x\right )^{2} + a\right )} e^{\left (4 \, x\right )} + 2 \, {\left (a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + 2 \, {\left (a \cosh \left (x\right ) e^{\left (4 \, x\right )} + 2 \, a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {a \cosh ^4(x)}} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {a \cosh ^4(x)}} \, dx=\frac {2}{\sqrt {a} e^{\left (-2 \, x\right )} + \sqrt {a}} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\sqrt {a \cosh ^4(x)}} \, dx=-\frac {2}{\sqrt {a} {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.60 \[ \int \frac {1}{\sqrt {a \cosh ^4(x)}} \, dx=-\frac {{\mathrm {e}}^{-x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^3} \]
[In]
[Out]