Integrand size = 19, antiderivative size = 66 \[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=-\frac {4 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {2 x \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3400, 3377, 2718} \[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {2 x \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}}{f}-\frac {4 \sqrt {a+i a \sinh (e+f x)}}{f^2} \]
[In]
[Out]
Rule 2718
Rule 3377
Rule 3400
Rubi steps \begin{align*} \text {integral}& = \left (\text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int x \sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx \\ & = \frac {2 x \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f}-\frac {\left (2 \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \cosh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{f} \\ & = -\frac {4 \sqrt {a+i a \sinh (e+f x)}}{f^2}+\frac {2 x \sqrt {a+i a \sinh (e+f x)} \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{f} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.32 \[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {2 \left ((-2+i f x) \cosh \left (\frac {1}{2} (e+f x)\right )+(-2 i+f x) \sinh \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a+i a \sinh (e+f x)}}{f^2 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )} \]
[In]
[Out]
Time = 0.48 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.59
method | result | size |
risch | \(\frac {i \sqrt {2}\, \sqrt {a \left (i {\mathrm e}^{2 f x +2 e}-i+2 \,{\mathrm e}^{f x +e}\right ) {\mathrm e}^{-f x -e}}\, \left (i x f +f x \,{\mathrm e}^{f x +e}+2 i-2 \,{\mathrm e}^{f x +e}\right ) \left ({\mathrm e}^{f x +e}-i\right )}{\left (i {\mathrm e}^{2 f x +2 e}-i+2 \,{\mathrm e}^{f x +e}\right ) f^{2}}\) | \(105\) |
[In]
[Out]
Exception generated. \[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\int x \sqrt {i a \left (\sinh {\left (e + f x \right )} - i\right )}\, dx \]
[In]
[Out]
\[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\int { \sqrt {i \, a \sinh \left (f x + e\right ) + a} x \,d x } \]
[In]
[Out]
\[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\int { \sqrt {i \, a \sinh \left (f x + e\right ) + a} x \,d x } \]
[In]
[Out]
Time = 1.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.21 \[ \int x \sqrt {a+i a \sinh (e+f x)} \, dx=\frac {\sqrt {2}\,\left ({\mathrm {e}}^{e+f\,x}+1{}\mathrm {i}\right )\,\left (f\,x\,{\mathrm {e}}^{e+f\,x}+f\,x\,1{}\mathrm {i}-2\,{\mathrm {e}}^{e+f\,x}+2{}\mathrm {i}\right )\,\sqrt {a\,{\mathrm {e}}^{-e-f\,x}\,{\left ({\mathrm {e}}^{e+f\,x}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}}{f^2\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )} \]
[In]
[Out]