Integrand size = 18, antiderivative size = 98 \[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=-\frac {12 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{25 b^2 \sqrt {i \sinh (a+b x)}}-\frac {4 \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b} \]
[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, 2719} \[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=-\frac {4 \sinh ^{\frac {3}{2}}(a+b x) \cosh (a+b x)}{25 b^2}-\frac {12 i \sqrt {\sinh (a+b x)} E\left (\left .\frac {1}{2} \left (i a+i b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2 \sqrt {i \sinh (a+b x)}}+\frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b} \]
[In]
[Out]
Rule 2715
Rule 2719
Rule 2721
Rule 5480
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \int \sinh ^{\frac {5}{2}}(a+b x) \, dx}{5 b} \\ & = -\frac {4 \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {6 \int \sqrt {\sinh (a+b x)} \, dx}{25 b} \\ & = -\frac {4 \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b}+\frac {\left (6 \sqrt {\sinh (a+b x)}\right ) \int \sqrt {i \sinh (a+b x)} \, dx}{25 b \sqrt {i \sinh (a+b x)}} \\ & = -\frac {12 i E\left (\left .\frac {1}{2} \left (i a-\frac {\pi }{2}+i b x\right )\right |2\right ) \sqrt {\sinh (a+b x)}}{25 b^2 \sqrt {i \sinh (a+b x)}}-\frac {4 \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sinh ^{\frac {5}{2}}(a+b x)}{5 b} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.95 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.46 \[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\frac {e^{-3 (a+b x)} \left (\left (-1+e^{2 (a+b x)}\right ) \left (2+5 b x+e^{2 (a+b x)} (24-10 b x)+e^{4 (a+b x)} (-2+5 b x)\right )+48 e^{2 (a+b x)} \sqrt {1-e^{2 (a+b x)}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},e^{2 (a+b x)}\right )\right )}{50 \sqrt {2} b^2 \sqrt {-e^{-a-b x}+e^{a+b x}}} \]
[In]
[Out]
\[\int x \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{\frac {3}{2}}d x\]
[In]
[Out]
Exception generated. \[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\int x \sinh ^{\frac {3}{2}}{\left (a + b x \right )} \cosh {\left (a + b x \right )}\, dx \]
[In]
[Out]
\[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\int { x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int x \cosh (a+b x) \sinh ^{\frac {3}{2}}(a+b x) \, dx=\int x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^{3/2} \,d x \]
[In]
[Out]