Integrand size = 18, antiderivative size = 98 \[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=-\frac {4 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right ),2\right ) \sqrt {i \sinh (a+b x)}}{9 b^2 \sqrt {\sinh (a+b x)}}-\frac {4 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{9 b^2}+\frac {2 x \sinh ^{\frac {3}{2}}(a+b x)}{3 b} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5480, 2715, 2721, 2720} \[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=-\frac {4 \sqrt {\sinh (a+b x)} \cosh (a+b x)}{9 b^2}-\frac {4 i \sqrt {i \sinh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right ),2\right )}{9 b^2 \sqrt {\sinh (a+b x)}}+\frac {2 x \sinh ^{\frac {3}{2}}(a+b x)}{3 b} \]
[In]
[Out]
Rule 2715
Rule 2720
Rule 2721
Rule 5480
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \sinh ^{\frac {3}{2}}(a+b x)}{3 b}-\frac {2 \int \sinh ^{\frac {3}{2}}(a+b x) \, dx}{3 b} \\ & = -\frac {4 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{9 b^2}+\frac {2 x \sinh ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {2 \int \frac {1}{\sqrt {\sinh (a+b x)}} \, dx}{9 b} \\ & = -\frac {4 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{9 b^2}+\frac {2 x \sinh ^{\frac {3}{2}}(a+b x)}{3 b}+\frac {\left (2 \sqrt {i \sinh (a+b x)}\right ) \int \frac {1}{\sqrt {i \sinh (a+b x)}} \, dx}{9 b \sqrt {\sinh (a+b x)}} \\ & = -\frac {4 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right ),2\right ) \sqrt {i \sinh (a+b x)}}{9 b^2 \sqrt {\sinh (a+b x)}}-\frac {4 \cosh (a+b x) \sqrt {\sinh (a+b x)}}{9 b^2}+\frac {2 x \sinh ^{\frac {3}{2}}(a+b x)}{3 b} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79 \[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=\frac {2 \left (2 i \operatorname {EllipticF}\left (\frac {1}{4} (-2 i a+\pi -2 i b x),2\right ) \sqrt {i \sinh (a+b x)}+3 b x \sinh ^2(a+b x)-\sinh (2 (a+b x))\right )}{9 b^2 \sqrt {\sinh (a+b x)}} \]
[In]
[Out]
\[\int x \cosh \left (b x +a \right ) \sqrt {\sinh \left (b x +a \right )}d x\]
[In]
[Out]
Exception generated. \[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=\int x \sqrt {\sinh {\left (a + b x \right )}} \cosh {\left (a + b x \right )}\, dx \]
[In]
[Out]
\[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=\int { x \cosh \left (b x + a\right ) \sqrt {\sinh \left (b x + a\right )} \,d x } \]
[In]
[Out]
\[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=\int { x \cosh \left (b x + a\right ) \sqrt {\sinh \left (b x + a\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int x \cosh (a+b x) \sqrt {\sinh (a+b x)} \, dx=\int x\,\mathrm {cosh}\left (a+b\,x\right )\,\sqrt {\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
[In]
[Out]