Integrand size = 9, antiderivative size = 31 \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}-\frac {8}{3} \sqrt {\cosh (x) \coth (x)} \text {sech}(x) \]
[Out]
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4483, 4485, 2678, 2669} \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}-\frac {8}{3} \text {sech}(x) \sqrt {\cosh (x) \coth (x)} \]
[In]
[Out]
Rule 2669
Rule 2678
Rule 4483
Rule 4485
Rubi steps \begin{align*} \text {integral}& = \frac {\left (i \sqrt {\cosh (x) \coth (x)}\right ) \int (-i \cosh (x) \coth (x))^{3/2} \, dx}{\sqrt {-i \cosh (x) \coth (x)}} \\ & = \frac {\left (i \sqrt {\cosh (x) \coth (x)}\right ) \int \cosh ^{\frac {3}{2}}(x) (-i \coth (x))^{3/2} \, dx}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}} \\ & = \frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}+\frac {\left (4 i \sqrt {\cosh (x) \coth (x)}\right ) \int \frac {(-i \coth (x))^{3/2}}{\sqrt {\cosh (x)}} \, dx}{3 \sqrt {\cosh (x)} \sqrt {-i \coth (x)}} \\ & = \frac {2}{3} \cosh (x) \sqrt {\cosh (x) \coth (x)}-\frac {8}{3} \sqrt {\cosh (x) \coth (x)} \text {sech}(x) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\frac {2}{3} \left (-4+\cosh ^2(x)\right ) \sqrt {\cosh (x) \coth (x)} \text {sech}(x) \]
[In]
[Out]
\[\int \left (\coth \left (x \right ) \cosh \left (x \right )\right )^{\frac {3}{2}}d x\]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.13 \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 7\right )} \sinh \left (x\right )^{2} - 14 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}}{3 \, \sqrt {\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \]
[In]
[Out]
Timed out. \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\text {Timed out} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (23) = 46\).
Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.52 \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\frac {\sqrt {2} e^{\left (\frac {3}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} e^{\left (-\frac {1}{2} \, x\right )}}{2 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {5 \, \sqrt {2} e^{\left (-\frac {5}{2} \, x\right )}}{2 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {2} e^{\left (-\frac {9}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {3}{2}}} \]
[In]
[Out]
\[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\int { \left (\cosh \left (x\right ) \coth \left (x\right )\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int (\cosh (x) \coth (x))^{3/2} \, dx=\int {\left (\mathrm {cosh}\left (x\right )\,\mathrm {coth}\left (x\right )\right )}^{3/2} \,d x \]
[In]
[Out]