Question Number 223923 by Mathspace last updated on 09/Aug/25
$${find}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left(\mathrm{1}+{x}\right)}{\mathrm{1}+{x}^{\mathrm{3}} }{dx} \\ $$
Answered by Tawa11 last updated on 14/Aug/25
$$\:\mathrm{Finally}:\: \\ $$$$\:\:\:\:\int_{\:\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{ln}\left(\mathrm{1}\:\:+\:\:\mathrm{x}\right)}{\mathrm{1}\:\:+\:\:\mathrm{x}^{\mathrm{3}} }\:\mathrm{dx}\:\:\:=\:\:\:\frac{\mathrm{2}}{\:\sqrt{\mathrm{3}}}\:\mathrm{Ti}_{\mathrm{2}} \left(\sqrt{\mathrm{3}}\right)\:\:−\:\:\:\frac{\mathrm{1}}{\mathrm{36}}\:\psi_{\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{6}}\right)\:\:+\:\:\:\frac{\mathrm{1}}{\mathrm{36}}\:\psi_{\mathrm{1}} \left(\frac{\mathrm{2}}{\mathrm{3}}\right) \\ $$