How to find the laplace transform of $\sin(t^2 + \ln(t))?$

Question

How to find the Laplace transform of $$f(t) = \sin(t^2 + \ln(t))$$

I need to calculate it and then compare to some python functions and libraries, I can't really calculate it.

Answer 1

Only with help of Mathematica:

$$\mathcal{L}_t\left[\sin \left(t^2+\ln (t)\right)\right](s)=\\\frac{1}{48} e^{-\frac{\pi }{4}} \Im\left(6 \sqrt{2} \Gamma \left(\frac{1}{2}+\frac{i}{2}\right) \left((2+2 i) \, _2F_3\left(\frac{1}{4}+\frac{i}{4},\frac{3}{4}+\frac{i}{4};\frac{1}{4},\frac{1}{2},\frac{3}{4};-\frac{s^4}{64}\right)-s^2 \, _2F_3\left(\frac{3}{4}+\frac{i}{4},\frac{5}{4}+\frac{i}{4};\frac{3}{4},\frac{5}{4},\frac{3}{2};-\frac{s^4}{64}\right)\right)+s \Gamma \left(\frac{i}{2}\right) \left(12 \, _2F_3\left(\frac{1}{2}+\frac{i}{4},1+\frac{i}{4};\frac{1}{2},\frac{3}{4},\frac{5}{4};-\frac{s^4}{64}\right)-(1-2 i) s^2 \, _2F_3\left(1+\frac{i}{4},\frac{3}{2}+\frac{i}{4};\frac{5}{4},\frac{3}{2},\frac{7}{4};-\frac{s^4}{64}\right)\right)\right)=\\\frac{1}{2} e^{-\pi /4} \Im\left(\sqrt[4]{-1} \Gamma \left(\frac{1}{2}+\frac{i}{2}\right) \, _1F_1\left(\frac{1}{2}+\frac{i}{2};\frac{1}{2};\frac{i s^2}{4}\right)-i s \Gamma \left(1+\frac{i}{2}\right) \, _1F_1\left(1+\frac{i}{2};\frac{3}{2};\frac{i s^2}{4}\right)\right)$$

MMA code:

LaplaceTransform[Sin[ t^2 + Log[t]], t, s] == Im[6*Sqrt[2]* Gamma[1/2 + I/2]*((2 + 2*I)* HypergeometricPFQ[{1/4 + I/4, 3/4 + I/4}, {1/4, 1/2, 3/4}, -1/64*s^4] - s^2*HypergeometricPFQ[{3/4 + I/4, 5/4 + I/4}, {3/4, 5/4, 3/2}, -1/64*s^4]) + s*Gamma[I/ 2]*(12*HypergeometricPFQ[{1/2 + I/4, 1 + I/4}, {1/2, 3/4, 5/4}, -1/64*s^4] - (1 - 2*I)*s^2* HypergeometricPFQ[{1 + I/4, 3/2 + I/4}, {5/4, 3/2, 7/4}, -1/64*s^4])]/(48*E^(Pi/4)) = Im[1/2 (-1)^(1/4) (-I)^(-(I/2)) Gamma[1/2 + I/2] Hypergeometric1F1[1/2 + I/2, 1/2, (I s^2)/4] - 1/2 I (-I)^(-(I/2)) s Gamma[1 + I/2] Hypergeometric1F1[1 + I/2, 3/2, (I s^2)/4]]