Why aren't these two integration methods yielding the same answer? I'm trying to solve this (not homework, if it matters), and both u-substitution and integration by parts are both yielding two different answers. Where am I going wrong?
Equation: $$\int \frac{(4x^3)}{(x^4+7)}dx$$
u-substitution answer: $$=\ln\big|(x^4+7)\big|+C$$
integration by parts answer: 
$$=\int4x^3*(x^4+7)^{-1}dx$$
$$=4x^3*\ln\big|x^4+7\big|-\int 12x^2*(x^4+7)^{-1}dx$$
$$=4x^3*\ln\big|x^4+7\big|-(12x^2*ln\big|x^4+7\big|-\int 24x*(x^4+7)^{-1}dx)$$
$$=4x^3*\ln\big|x^4+7\big|-(12x^2*ln\big|x^4+7\big|-24x*ln\big|x^4+7\big|-\int 24(x^4+7)^{-1}dx)$$
$$= 4x^3*\ln\big|x^4+7\big|-(12x^2*\ln\big|x^4+7\big|-(24x\ln\big|x^4+7\big|-24\ln\big|x^4+7\big|))$$
$$=(4x^3-12x^2+24x-24)(\ln\big|x^4+7\big|)$$

I don't understand it. For $u = x^4+7, du = 4x^3dx$ so
$$
\int \frac{4x^3}{x^4+7} dx = \int du/u = \ln |x^4+7| + C.
$$
Show work for your by parts results and it will become clear where the error is...
