Integrand size = 13, antiderivative size = 54 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 x}{2 a}+\frac {4 \sinh (x)}{a}-\frac {3 \cosh (x) \sinh (x)}{2 a}-\frac {\cosh ^2(x) \sinh (x)}{a+a \text {sech}(x)}+\frac {4 \sinh ^3(x)}{3 a} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3904, 3872, 2713, 2715, 8} \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 x}{2 a}+\frac {4 \sinh ^3(x)}{3 a}+\frac {4 \sinh (x)}{a}-\frac {3 \sinh (x) \cosh (x)}{2 a}-\frac {\sinh (x) \cosh ^2(x)}{a \text {sech}(x)+a} \]
[In]
[Out]
Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 3904
Rubi steps \begin{align*} \text {integral}& = -\frac {\cosh ^2(x) \sinh (x)}{a+a \text {sech}(x)}-\frac {\int \cosh ^3(x) (-4 a+3 a \text {sech}(x)) \, dx}{a^2} \\ & = -\frac {\cosh ^2(x) \sinh (x)}{a+a \text {sech}(x)}-\frac {3 \int \cosh ^2(x) \, dx}{a}+\frac {4 \int \cosh ^3(x) \, dx}{a} \\ & = -\frac {3 \cosh (x) \sinh (x)}{2 a}-\frac {\cosh ^2(x) \sinh (x)}{a+a \text {sech}(x)}+\frac {(4 i) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (x)\right )}{a}-\frac {3 \int 1 \, dx}{2 a} \\ & = -\frac {3 x}{2 a}+\frac {4 \sinh (x)}{a}-\frac {3 \cosh (x) \sinh (x)}{2 a}-\frac {\cosh ^2(x) \sinh (x)}{a+a \text {sech}(x)}+\frac {4 \sinh ^3(x)}{3 a} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\text {sech}\left (\frac {x}{2}\right ) \left (-36 x \cosh \left (\frac {x}{2}\right )+45 \sinh \left (\frac {x}{2}\right )+18 \sinh \left (\frac {3 x}{2}\right )-2 \sinh \left (\frac {5 x}{2}\right )+\sinh \left (\frac {7 x}{2}\right )\right )}{24 a} \]
[In]
[Out]
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.65
method | result | size |
parallelrisch | \(\frac {\operatorname {csch}\left (x \right ) \left (-36 x \sinh \left (x \right )-3 \cosh \left (3 x \right )+27 \cosh \left (x \right )+\cosh \left (4 x \right )+20 \cosh \left (2 x \right )-45\right )}{24 a}\) | \(35\) |
risch | \(\frac {{\mathrm e}^{4 x}-2 \,{\mathrm e}^{3 x}+18 \,{\mathrm e}^{2 x}-69-18 \,{\mathrm e}^{-x}+2 \,{\mathrm e}^{-2 x}-36 x \,{\mathrm e}^{x}+21 \,{\mathrm e}^{x}-{\mathrm e}^{-3 x}-36 x}{24 \left ({\mathrm e}^{x}+1\right ) a}\) | \(60\) |
default | \(\frac {\tanh \left (\frac {x}{2}\right )-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {5}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {5}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}}{a}\) | \(86\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (48) = 96\).
Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.85 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\cosh \left (x\right )^{4} + {\left (4 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{3} + {\left (6 \, \cosh \left (x\right )^{2} - 9 \, \cosh \left (x\right ) + 20\right )} \sinh \left (x\right )^{2} - 3 \, {\left (12 \, x - 1\right )} \cosh \left (x\right ) + 20 \, \cosh \left (x\right )^{2} + {\left (4 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} - 36 \, x + 32 \, \cosh \left (x\right ) + 39\right )} \sinh \left (x\right ) - 36 \, x - 69}{24 \, {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + a\right )}} \]
[In]
[Out]
\[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=\frac {\int \frac {\cosh ^{3}{\left (x \right )}}{\operatorname {sech}{\left (x \right )} + 1}\, dx}{a} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 \, x}{2 \, a} - \frac {21 \, e^{\left (-x\right )} - 3 \, e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}}{24 \, a} - \frac {2 \, e^{\left (-x\right )} - 18 \, e^{\left (-2 \, x\right )} - 69 \, e^{\left (-3 \, x\right )} - 1}{24 \, {\left (a e^{\left (-3 \, x\right )} + a e^{\left (-4 \, x\right )}\right )}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=-\frac {3 \, x}{2 \, a} - \frac {{\left (69 \, e^{\left (3 \, x\right )} + 18 \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 1\right )} e^{\left (-3 \, x\right )}}{24 \, a {\left (e^{x} + 1\right )}} + \frac {a^{2} e^{\left (3 \, x\right )} - 3 \, a^{2} e^{\left (2 \, x\right )} + 21 \, a^{2} e^{x}}{24 \, a^{3}} \]
[In]
[Out]
Time = 2.01 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {\cosh ^3(x)}{a+a \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {7\,{\mathrm {e}}^{-x}}{8\,a}-\frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-3\,x}}{24\,a}+\frac {{\mathrm {e}}^{3\,x}}{24\,a}-\frac {3\,x}{2\,a}-\frac {2}{a\,\left ({\mathrm {e}}^x+1\right )}+\frac {7\,{\mathrm {e}}^x}{8\,a} \]
[In]
[Out]