4.1 Problem number 64

$$\int \frac{a+b\text{ArcTan}(cx)}{x^2(d+icdx{)}^3}dx$$

Optimal antiderivative $$\frac{bc}{8d^3{(\mathrm{I}-cx)}^2}-\frac{9\mathrm{I}bc}{8d^3(\mathrm{I}-cx)}+\frac{9\mathrm{I}bc\arctan \left(cx\right)}{8d^3}+\frac{-a-b\arctan \left(cx\right)}{d^{3}x}+\frac{\mathrm{I}c(a+b\arctan \left(cx\right))}{2d^3{(\mathrm{I}-cx)}^2}+\frac{2c(a+b\arctan \left(cx\right))}{d^3(\mathrm{I}-cx)}-\frac{3\mathrm{I}ac\ln \left(x\right)}{d^3}+\frac{bc\ln \left(x\right)}{d^3}-\frac{3\mathrm{I}c(a+b\arctan \left(cx\right))\ln \left(\frac{2}{1+\mathrm{I}cx}\right)}{d^3}-\frac{bc\ln (c^2{x}^2+1)}{2d^3}+\frac{3bc\mathrm{polylog}(2,-\mathrm{I}cx)}{2d^3}-\frac{3bc\mathrm{polylog}(2,\mathrm{I}cx)}{2d^3}+\frac{3bc\mathrm{polylog}(2,1-\frac{2}{1+\mathrm{I}cx})}{2d^3}$$

command

integrate((a+b*arctan(c*x))/x^2/(d+I*c*d*x)^3,x, algorithm="maxima")

Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output

$$\text{Exception raised: RuntimeError}$$

Maxima 5.44 via sagemath 9.3 output

$$-\frac{17ib{c}^3{x}^3\arctan (1,cx)+(b(34\arctan (1,cx)-18i)+48a)c^2{x}^2+(b(-17i\arctan (1,cx)-20)-72ia)cx+(12b{c}^3{x}^3-24ib{c}^2{x}^2-12bcx)\arctan {\left(cx\right)}^2+(3b{c}^3{x}^3-6ib{c}^2{x}^2-3bcx)\log {(c^2{x}^2+1)}^2+(-12ib{c}^3{x}^3-24b{c}^2{x}^2+12ibcx)\arctan \left(cx\right)\log (\frac{1}{4}{c}^2{x}^2+\frac{1}{4})+(48ib{c}^3{x}^3+96b{c}^2{x}^2-48ibcx)\arctan \left(cx\right)\log \left(cx\right)+((48a-ib)c^3{x}^3+(-96ia+46b)c^2{x}^2-(48a+71ib)cx-16b)\arctan \left(cx\right)+(24b{c}^3{x}^3-48ib{c}^2{x}^2-24bcx){\mathrm{L}\mathrm{i}}_2(icx+1)-(24b{c}^3{x}^3-48ib{c}^2{x}^2-24bcx){\mathrm{L}\mathrm{i}}_2(\frac{1}{2}icx+\frac{1}{2})-(24b{c}^3{x}^3-48ib{c}^2{x}^2-24bcx){\mathrm{L}\mathrm{i}}_2(-icx+1)-(4((3i\pi -2)b+6ia)c^3{x}^3+((24\pi +16i)b+48a)c^2{x}^2+4((-3i\pi +2)b-6ia)cx+(6b{c}^3{x}^3-12ib{c}^2{x}^2-6bcx)\log (\frac{1}{4}{c}^2{x}^2+\frac{1}{4}))\log (c^2{x}^2+1)+((48ia-16b)c^3{x}^3+32(3a+ib)c^2{x}^2+(-48ia+16b)cx)\log \left(x\right)-16a}{16(c^2{d}^3{x}^3-2ic{d}^3{x}^2-d^{3}x)}$$