Integrand size = 11, antiderivative size = 30 \[ \int \cosh (x) \cosh (2 x) \cosh (3 x) \, dx=\frac {x}{4}+\frac {1}{8} \sinh (2 x)+\frac {1}{16} \sinh (4 x)+\frac {1}{24} \sinh (6 x) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4440, 2717} \[ \int \cosh (x) \cosh (2 x) \cosh (3 x) \, dx=\frac {x}{4}+\frac {1}{8} \sinh (2 x)+\frac {1}{16} \sinh (4 x)+\frac {1}{24} \sinh (6 x) \]
[In]
[Out]
Rule 2717
Rule 4440
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{4}+\frac {1}{4} \cosh (2 x)+\frac {1}{4} \cosh (4 x)+\frac {1}{4} \cosh (6 x)\right ) \, dx \\ & = \frac {x}{4}+\frac {1}{4} \int \cosh (2 x) \, dx+\frac {1}{4} \int \cosh (4 x) \, dx+\frac {1}{4} \int \cosh (6 x) \, dx \\ & = \frac {x}{4}+\frac {1}{8} \sinh (2 x)+\frac {1}{16} \sinh (4 x)+\frac {1}{24} \sinh (6 x) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \cosh (x) \cosh (2 x) \cosh (3 x) \, dx=\frac {x}{4}+\frac {1}{8} \sinh (2 x)+\frac {1}{16} \sinh (4 x)+\frac {1}{24} \sinh (6 x) \]
[In]
[Out]
Time = 2.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {x}{4}+\frac {\sinh \left (2 x \right )}{8}+\frac {\sinh \left (4 x \right )}{16}+\frac {\sinh \left (6 x \right )}{24}\) | \(23\) |
risch | \(\frac {x}{4}+\frac {{\mathrm e}^{6 x}}{48}+\frac {{\mathrm e}^{4 x}}{32}+\frac {{\mathrm e}^{2 x}}{16}-\frac {{\mathrm e}^{-2 x}}{16}-\frac {{\mathrm e}^{-4 x}}{32}-\frac {{\mathrm e}^{-6 x}}{48}\) | \(41\) |
parallelrisch | \(\frac {\left (-24+24 \cosh \left (x \right )\right ) \ln \left (1-\coth \left (x \right )+\operatorname {csch}\left (x \right )\right )+\left (-24 \cosh \left (x \right )+24\right ) \ln \left (\coth \left (x \right )-\operatorname {csch}\left (x \right )+1\right )+48 x \cosh \left (x \right )-48 x -4 \sinh \left (6 x \right )+2 \sinh \left (7 x \right )+6 \sinh \left (x \right )-12 \sinh \left (2 x \right )+9 \sinh \left (3 x \right )-6 \sinh \left (4 x \right )+5 \sinh \left (5 x \right )}{96 \cosh \left (x \right )-96}\) | \(91\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \cosh (x) \cosh (2 x) \cosh (3 x) \, dx=\frac {1}{4} \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \frac {1}{12} \, {\left (10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + \frac {1}{4} \, {\left (\cosh \left (x\right )^{5} + \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + \frac {1}{4} \, x \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (22) = 44\).
Time = 0.94 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.87 \[ \int \cosh (x) \cosh (2 x) \cosh (3 x) \, dx=\frac {x \sinh {\left (x \right )} \sinh {\left (2 x \right )} \cosh {\left (3 x \right )}}{4} - \frac {x \sinh {\left (x \right )} \sinh {\left (3 x \right )} \cosh {\left (2 x \right )}}{4} - \frac {x \sinh {\left (2 x \right )} \sinh {\left (3 x \right )} \cosh {\left (x \right )}}{4} + \frac {x \cosh {\left (x \right )} \cosh {\left (2 x \right )} \cosh {\left (3 x \right )}}{4} - \frac {3 \sinh {\left (x \right )} \sinh {\left (2 x \right )} \sinh {\left (3 x \right )}}{8} + \frac {\sinh {\left (x \right )} \cosh {\left (2 x \right )} \cosh {\left (3 x \right )}}{3} + \frac {5 \sinh {\left (2 x \right )} \cosh {\left (x \right )} \cosh {\left (3 x \right )}}{24} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \cosh (x) \cosh (2 x) \cosh (3 x) \, dx=\frac {1}{96} \, {\left (3 \, e^{\left (-2 \, x\right )} + 6 \, e^{\left (-4 \, x\right )} + 2\right )} e^{\left (6 \, x\right )} + \frac {1}{4} \, x - \frac {1}{16} \, e^{\left (-2 \, x\right )} - \frac {1}{32} \, e^{\left (-4 \, x\right )} - \frac {1}{48} \, e^{\left (-6 \, x\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (22) = 44\).
Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \cosh (x) \cosh (2 x) \cosh (3 x) \, dx=-\frac {1}{96} \, {\left (22 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-6 \, x\right )} + \frac {1}{4} \, x + \frac {1}{48} \, e^{\left (6 \, x\right )} + \frac {1}{32} \, e^{\left (4 \, x\right )} + \frac {1}{16} \, e^{\left (2 \, x\right )} \]
[In]
[Out]
Time = 0.50 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \cosh (x) \cosh (2 x) \cosh (3 x) \, dx=\frac {x}{4}-\frac {{\mathrm {e}}^{-2\,x}}{16}+\frac {{\mathrm {e}}^{2\,x}}{16}-\frac {{\mathrm {e}}^{-4\,x}}{32}+\frac {{\mathrm {e}}^{4\,x}}{32}-\frac {{\mathrm {e}}^{-6\,x}}{48}+\frac {{\mathrm {e}}^{6\,x}}{48} \]
[In]
[Out]