Integrand size = 18, antiderivative size = 37 \[ \int \frac {x \sinh (a+b x)}{\sqrt {\cosh (a+b x)}} \, dx=\frac {2 x \sqrt {\cosh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5481, 2719} \[ \int \frac {x \sinh (a+b x)}{\sqrt {\cosh (a+b x)}} \, dx=\frac {2 x \sqrt {\cosh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \]
[In]
[Out]
Rule 2719
Rule 5481
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \sqrt {\cosh (a+b x)}}{b}-\frac {2 \int \sqrt {\cosh (a+b x)} \, dx}{b} \\ & = \frac {2 x \sqrt {\cosh (a+b x)}}{b}+\frac {4 i E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.95 \[ \int \frac {x \sinh (a+b x)}{\sqrt {\cosh (a+b x)}} \, dx=\frac {(\cosh (a+b x)-\sinh (a+b x)) \left (4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\cosh (2 (a+b x))-\sinh (2 (a+b x))\right ) \sqrt {1+\cosh (2 (a+b x))+\sinh (2 (a+b x))}+(-2+b x) (1+\cosh (2 (a+b x))+\sinh (2 (a+b x)))\right )}{b^2 \sqrt {\cosh (a+b x)}} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (61 ) = 122\).
Time = 0.26 (sec) , antiderivative size = 250, normalized size of antiderivative = 6.76
method | result | size |
risch | \(\frac {\left (b x -2\right ) \left (1+{\mathrm e}^{2 b x +2 a}\right ) \sqrt {2}\, {\mathrm e}^{-b x -a}}{b^{2} \sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}}-\frac {2 \left (-\frac {2 \left (1+{\mathrm e}^{2 b x +2 a}\right )}{\sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}}+\frac {i \sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{b x +a}-i\right )}\, \sqrt {i {\mathrm e}^{b x +a}}\, \left (-2 i \operatorname {EllipticE}\left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticF}\left (\sqrt {-i \left ({\mathrm e}^{b x +a}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 b x +3 a}+{\mathrm e}^{b x +a}}}\right ) \sqrt {2}\, \sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{b x +a}}\, {\mathrm e}^{-b x -a}}{b^{2} \sqrt {\left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}}\) | \(250\) |
[In]
[Out]
Exception generated. \[ \int \frac {x \sinh (a+b x)}{\sqrt {\cosh (a+b x)}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {x \sinh (a+b x)}{\sqrt {\cosh (a+b x)}} \, dx=\int \frac {x \sinh {\left (a + b x \right )}}{\sqrt {\cosh {\left (a + b x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {x \sinh (a+b x)}{\sqrt {\cosh (a+b x)}} \, dx=\int { \frac {x \sinh \left (b x + a\right )}{\sqrt {\cosh \left (b x + a\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {x \sinh (a+b x)}{\sqrt {\cosh (a+b x)}} \, dx=\int { \frac {x \sinh \left (b x + a\right )}{\sqrt {\cosh \left (b x + a\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x \sinh (a+b x)}{\sqrt {\cosh (a+b x)}} \, dx=\int \frac {x\,\mathrm {sinh}\left (a+b\,x\right )}{\sqrt {\mathrm {cosh}\left (a+b\,x\right )}} \,d x \]
[In]
[Out]