45.1 Problem number 145

\[ \int x^{-1+2 n} \sin \left (a+b x^n\right ) \, dx \]

Optimal antiderivative \[ -\frac {x^{n} \cos \left (a +b \,x^{n}\right )}{b n}+\frac {\sin \left (a +b \,x^{n}\right )}{b^{2} n} \]

command

integrate(x**(-1+2*n)*sin(a+b*x**n),x)

Sympy 1.10.1 under Python 3.10.4 output

\[ \begin {cases} \log {\left (x \right )} \sin {\left (a \right )} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x^{2 n} \sin {\left (a \right )}}{2 n} & \text {for}\: b = 0 \\\log {\left (x \right )} \sin {\left (a + b \right )} & \text {for}\: n = 0 \\- \frac {x^{n} \cos {\left (a + b x^{n} \right )}}{b n} + \frac {\sin {\left (a + b x^{n} \right )}}{b^{2} n} & \text {otherwise} \end {cases} \]

Sympy 1.8 under Python 3.8.8 output

\[ \text {Timed out} \]________________________________________________________________________________________