3.57 \(\int \frac{\tan ^{-1}(a+b x)}{c+\frac{d}{x^3}} \, dx\)

Optimal. Leaf size=933 \[ \text{result too large to display} \]

[Out]

-((1 + I*a + I*b*x)*Log[1 + I*a + I*b*x])/(2*b*c) - ((1 - I*a - I*b*x)*Log[(-I)*(I + a + b*x)])/(2*b*c) - ((I/
6)*d^(1/3)*Log[1 - I*a - I*b*x]*Log[-((b*(d^(1/3) + c^(1/3)*x))/((I + a)*c^(1/3) - b*d^(1/3)))])/c^(4/3) + ((I
/6)*d^(1/3)*Log[1 + I*a + I*b*x]*Log[(b*(d^(1/3) + c^(1/3)*x))/((I - a)*c^(1/3) + b*d^(1/3))])/c^(4/3) - ((-1)
^(1/6)*d^(1/3)*Log[1 + I*a + I*b*x]*Log[-((b*(d^(1/3) - (-1)^(1/3)*c^(1/3)*x))/((-1)^(1/3)*(I - a)*c^(1/3) - b
*d^(1/3)))])/(6*c^(4/3)) + ((-1)^(1/6)*d^(1/3)*Log[1 - I*a - I*b*x]*Log[(b*(d^(1/3) - (-1)^(1/3)*c^(1/3)*x))/(
(-1)^(1/3)*(I + a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) - ((-1)^(5/6)*d^(1/3)*Log[1 + I*a + I*b*x]*Log[(b*(d^(1/
3) + (-1)^(2/3)*c^(1/3)*x))/((-1)^(2/3)*(I - a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) + ((-1)^(5/6)*d^(1/3)*Log[1
- I*a - I*b*x]*Log[(b*(d^(1/3) + (-1)^(2/3)*c^(1/3)*x))/((-1)^(1/6)*(1 - I*a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/
3)) - ((-1)^(1/6)*d^(1/3)*PolyLog[2, ((-1)^(1/3)*c^(1/3)*(I - a - b*x))/((-1)^(1/3)*(I - a)*c^(1/3) - b*d^(1/3
))])/(6*c^(4/3)) - ((-1)^(5/6)*d^(1/3)*PolyLog[2, ((-1)^(1/6)*c^(1/3)*(I - a - b*x))/((-1)^(1/6)*(I - a)*c^(1/
3) - I*b*d^(1/3))])/(6*c^(4/3)) + ((I/6)*d^(1/3)*PolyLog[2, (c^(1/3)*(I - a - b*x))/((I - a)*c^(1/3) + b*d^(1/
3))])/c^(4/3) - ((I/6)*d^(1/3)*PolyLog[2, (c^(1/3)*(I + a + b*x))/((I + a)*c^(1/3) - b*d^(1/3))])/c^(4/3) + ((
-1)^(5/6)*d^(1/3)*PolyLog[2, ((-1)^(2/3)*c^(1/3)*(I + a + b*x))/((-1)^(2/3)*(I + a)*c^(1/3) - b*d^(1/3))])/(6*
c^(4/3)) + ((-1)^(1/6)*d^(1/3)*PolyLog[2, ((-1)^(1/3)*c^(1/3)*(I + a + b*x))/((-1)^(1/3)*(I + a)*c^(1/3) + b*d
^(1/3))])/(6*c^(4/3))

________________________________________________________________________________________

Rubi [A]  time = 1.36743, antiderivative size = 933, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5051, 2409, 2389, 2295, 2394, 2393, 2391} \[ -\frac{(i a+i b x+1) \log (i a+i b x+1)}{2 b c}+\frac{i \sqrt [3]{d} \log \left (\frac{b \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{\sqrt [3]{c} (i-a)+b \sqrt [3]{d}}\right ) \log (i a+i b x+1)}{6 c^{4/3}}-\frac{\sqrt [6]{-1} \sqrt [3]{d} \log \left (-\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (i-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right ) \log (i a+i b x+1)}{6 c^{4/3}}-\frac{(-1)^{5/6} \sqrt [3]{d} \log \left (\frac{b \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{(-1)^{2/3} \sqrt [3]{c} (i-a)+b \sqrt [3]{d}}\right ) \log (i a+i b x+1)}{6 c^{4/3}}-\frac{(-i a-i b x+1) \log (-i (a+b x+i))}{2 b c}-\frac{i \sqrt [3]{d} \log (-i a-i b x+1) \log \left (-\frac{b \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{(a+i) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [6]{-1} \sqrt [3]{d} \log (-i a-i b x+1) \log \left (\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} \sqrt [3]{c} (a+i)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{(-1)^{5/6} \sqrt [3]{d} \log (-i a-i b x+1) \log \left (\frac{b \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{\sqrt [6]{-1} \sqrt [3]{c} (1-i a)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [6]{-1} \sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{c} (-a-b x+i)}{\sqrt [3]{-1} (i-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{5/6} \sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [6]{-1} \sqrt [3]{c} (-a-b x+i)}{\sqrt [6]{-1} (i-a) \sqrt [3]{c}-i b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{i \sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [3]{c} (-a-b x+i)}{\sqrt [3]{c} (i-a)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{i \sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [3]{c} (a+b x+i)}{(a+i) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{(-1)^{5/6} \sqrt [3]{d} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{c} (a+b x+i)}{(-1)^{2/3} (a+i) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [6]{-1} \sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{c} (a+b x+i)}{\sqrt [3]{-1} \sqrt [3]{c} (a+i)+b \sqrt [3]{d}}\right )}{6 c^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a + b*x]/(c + d/x^3),x]

[Out]

-((1 + I*a + I*b*x)*Log[1 + I*a + I*b*x])/(2*b*c) - ((1 - I*a - I*b*x)*Log[(-I)*(I + a + b*x)])/(2*b*c) - ((I/
6)*d^(1/3)*Log[1 - I*a - I*b*x]*Log[-((b*(d^(1/3) + c^(1/3)*x))/((I + a)*c^(1/3) - b*d^(1/3)))])/c^(4/3) + ((I
/6)*d^(1/3)*Log[1 + I*a + I*b*x]*Log[(b*(d^(1/3) + c^(1/3)*x))/((I - a)*c^(1/3) + b*d^(1/3))])/c^(4/3) - ((-1)
^(1/6)*d^(1/3)*Log[1 + I*a + I*b*x]*Log[-((b*(d^(1/3) - (-1)^(1/3)*c^(1/3)*x))/((-1)^(1/3)*(I - a)*c^(1/3) - b
*d^(1/3)))])/(6*c^(4/3)) + ((-1)^(1/6)*d^(1/3)*Log[1 - I*a - I*b*x]*Log[(b*(d^(1/3) - (-1)^(1/3)*c^(1/3)*x))/(
(-1)^(1/3)*(I + a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) - ((-1)^(5/6)*d^(1/3)*Log[1 + I*a + I*b*x]*Log[(b*(d^(1/
3) + (-1)^(2/3)*c^(1/3)*x))/((-1)^(2/3)*(I - a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/3)) + ((-1)^(5/6)*d^(1/3)*Log[1
- I*a - I*b*x]*Log[(b*(d^(1/3) + (-1)^(2/3)*c^(1/3)*x))/((-1)^(1/6)*(1 - I*a)*c^(1/3) + b*d^(1/3))])/(6*c^(4/
3)) - ((-1)^(1/6)*d^(1/3)*PolyLog[2, ((-1)^(1/3)*c^(1/3)*(I - a - b*x))/((-1)^(1/3)*(I - a)*c^(1/3) - b*d^(1/3
))])/(6*c^(4/3)) - ((-1)^(5/6)*d^(1/3)*PolyLog[2, ((-1)^(1/6)*c^(1/3)*(I - a - b*x))/((-1)^(1/6)*(I - a)*c^(1/
3) - I*b*d^(1/3))])/(6*c^(4/3)) + ((I/6)*d^(1/3)*PolyLog[2, (c^(1/3)*(I - a - b*x))/((I - a)*c^(1/3) + b*d^(1/
3))])/c^(4/3) - ((I/6)*d^(1/3)*PolyLog[2, (c^(1/3)*(I + a + b*x))/((I + a)*c^(1/3) - b*d^(1/3))])/c^(4/3) + ((
-1)^(5/6)*d^(1/3)*PolyLog[2, ((-1)^(2/3)*c^(1/3)*(I + a + b*x))/((-1)^(2/3)*(I + a)*c^(1/3) - b*d^(1/3))])/(6*
c^(4/3)) + ((-1)^(1/6)*d^(1/3)*PolyLog[2, ((-1)^(1/3)*c^(1/3)*(I + a + b*x))/((-1)^(1/3)*(I + a)*c^(1/3) + b*d
^(1/3))])/(6*c^(4/3))

Rule 5051

Int[ArcTan[(a_) + (b_.)*(x_)]/((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Dist[I/2, Int[Log[1 - I*a - I*b*x]/(c +
d*x^n), x], x] - Dist[I/2, Int[Log[1 + I*a + I*b*x]/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ
[n]

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a+b x)}{c+\frac{d}{x^3}} \, dx &=\frac{1}{2} i \int \frac{\log (1-i a-i b x)}{c+\frac{d}{x^3}} \, dx-\frac{1}{2} i \int \frac{\log (1+i a+i b x)}{c+\frac{d}{x^3}} \, dx\\ &=\frac{1}{2} i \int \left (\frac{\log (1-i a-i b x)}{c}-\frac{d \log (1-i a-i b x)}{c \left (d+c x^3\right )}\right ) \, dx-\frac{1}{2} i \int \left (\frac{\log (1+i a+i b x)}{c}-\frac{d \log (1+i a+i b x)}{c \left (d+c x^3\right )}\right ) \, dx\\ &=\frac{i \int \log (1-i a-i b x) \, dx}{2 c}-\frac{i \int \log (1+i a+i b x) \, dx}{2 c}-\frac{(i d) \int \frac{\log (1-i a-i b x)}{d+c x^3} \, dx}{2 c}+\frac{(i d) \int \frac{\log (1+i a+i b x)}{d+c x^3} \, dx}{2 c}\\ &=-\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1-i a-i b x)}{2 b c}-\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1+i a+i b x)}{2 b c}-\frac{(i d) \int \left (-\frac{\log (1-i a-i b x)}{3 d^{2/3} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}-\frac{\log (1-i a-i b x)}{3 d^{2/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}-\frac{\log (1-i a-i b x)}{3 d^{2/3} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}\right ) \, dx}{2 c}+\frac{(i d) \int \left (-\frac{\log (1+i a+i b x)}{3 d^{2/3} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}-\frac{\log (1+i a+i b x)}{3 d^{2/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}-\frac{\log (1+i a+i b x)}{3 d^{2/3} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}\right ) \, dx}{2 c}\\ &=-\frac{(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac{(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}+\frac{\left (i \sqrt [3]{d}\right ) \int \frac{\log (1-i a-i b x)}{-\sqrt [3]{d}-\sqrt [3]{c} x} \, dx}{6 c}+\frac{\left (i \sqrt [3]{d}\right ) \int \frac{\log (1-i a-i b x)}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x} \, dx}{6 c}+\frac{\left (i \sqrt [3]{d}\right ) \int \frac{\log (1-i a-i b x)}{-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x} \, dx}{6 c}-\frac{\left (i \sqrt [3]{d}\right ) \int \frac{\log (1+i a+i b x)}{-\sqrt [3]{d}-\sqrt [3]{c} x} \, dx}{6 c}-\frac{\left (i \sqrt [3]{d}\right ) \int \frac{\log (1+i a+i b x)}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x} \, dx}{6 c}-\frac{\left (i \sqrt [3]{d}\right ) \int \frac{\log (1+i a+i b x)}{-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x} \, dx}{6 c}\\ &=-\frac{(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac{(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}-\frac{i \sqrt [3]{d} \log (1-i a-i b x) \log \left (-\frac{b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(i+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{i \sqrt [3]{d} \log (1+i a+i b x) \log \left (\frac{b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(i-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [6]{-1} \sqrt [3]{d} \log (1+i a+i b x) \log \left (-\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (i-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [6]{-1} \sqrt [3]{d} \log (1-i a-i b x) \log \left (\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (i+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{5/6} \sqrt [3]{d} \log (1+i a+i b x) \log \left (\frac{b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (i-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{(-1)^{5/6} \sqrt [3]{d} \log (1-i a-i b x) \log \left (\frac{b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{\sqrt [6]{-1} (1-i a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\left (b \sqrt [3]{d}\right ) \int \frac{\log \left (\frac{i b \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}{(1+i a) \sqrt [3]{c}-i b \sqrt [3]{d}}\right )}{1+i a+i b x} \, dx}{6 c^{4/3}}+\frac{\left (b \sqrt [3]{d}\right ) \int \frac{\log \left (-\frac{i b \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}{(1-i a) \sqrt [3]{c}+i b \sqrt [3]{d}}\right )}{1-i a-i b x} \, dx}{6 c^{4/3}}-\frac{\left (\sqrt [3]{-1} b \sqrt [3]{d}\right ) \int \frac{\log \left (\frac{i b \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1+i a) \sqrt [3]{c}-i b \sqrt [3]{d}}\right )}{1+i a+i b x} \, dx}{6 c^{4/3}}-\frac{\left (\sqrt [3]{-1} b \sqrt [3]{d}\right ) \int \frac{\log \left (-\frac{i b \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1-i a) \sqrt [3]{c}+i b \sqrt [3]{d}}\right )}{1-i a-i b x} \, dx}{6 c^{4/3}}+\frac{\left ((-1)^{2/3} b \sqrt [3]{d}\right ) \int \frac{\log \left (\frac{i b \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}{-\sqrt [3]{-1} (1+i a) \sqrt [3]{c}-i b \sqrt [3]{d}}\right )}{1+i a+i b x} \, dx}{6 c^{4/3}}+\frac{\left ((-1)^{2/3} b \sqrt [3]{d}\right ) \int \frac{\log \left (-\frac{i b \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}{-\sqrt [3]{-1} (1-i a) \sqrt [3]{c}+i b \sqrt [3]{d}}\right )}{1-i a-i b x} \, dx}{6 c^{4/3}}\\ &=-\frac{(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac{(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}-\frac{i \sqrt [3]{d} \log (1-i a-i b x) \log \left (-\frac{b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(i+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{i \sqrt [3]{d} \log (1+i a+i b x) \log \left (\frac{b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(i-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [6]{-1} \sqrt [3]{d} \log (1+i a+i b x) \log \left (-\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (i-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [6]{-1} \sqrt [3]{d} \log (1-i a-i b x) \log \left (\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (i+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{5/6} \sqrt [3]{d} \log (1+i a+i b x) \log \left (\frac{b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (i-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{(-1)^{5/6} \sqrt [3]{d} \log (1-i a-i b x) \log \left (\frac{b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{\sqrt [6]{-1} (1-i a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\left (i \sqrt [3]{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [3]{c} x}{(1+i a) \sqrt [3]{c}-i b \sqrt [3]{d}}\right )}{x} \, dx,x,1+i a+i b x\right )}{6 c^{4/3}}+\frac{\left (i \sqrt [3]{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [3]{c} x}{(1-i a) \sqrt [3]{c}+i b \sqrt [3]{d}}\right )}{x} \, dx,x,1-i a-i b x\right )}{6 c^{4/3}}+\frac{\left (\sqrt [6]{-1} \sqrt [3]{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{-1} \sqrt [3]{c} x}{-\sqrt [3]{-1} (1+i a) \sqrt [3]{c}-i b \sqrt [3]{d}}\right )}{x} \, dx,x,1+i a+i b x\right )}{6 c^{4/3}}-\frac{\left (\sqrt [6]{-1} \sqrt [3]{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{-1} \sqrt [3]{c} x}{-\sqrt [3]{-1} (1-i a) \sqrt [3]{c}+i b \sqrt [3]{d}}\right )}{x} \, dx,x,1-i a-i b x\right )}{6 c^{4/3}}+\frac{\left ((-1)^{5/6} \sqrt [3]{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{(-1)^{2/3} \sqrt [3]{c} x}{(-1)^{2/3} (1+i a) \sqrt [3]{c}-i b \sqrt [3]{d}}\right )}{x} \, dx,x,1+i a+i b x\right )}{6 c^{4/3}}-\frac{\left ((-1)^{5/6} \sqrt [3]{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{(-1)^{2/3} \sqrt [3]{c} x}{(-1)^{2/3} (1-i a) \sqrt [3]{c}+i b \sqrt [3]{d}}\right )}{x} \, dx,x,1-i a-i b x\right )}{6 c^{4/3}}\\ &=-\frac{(1+i a+i b x) \log (1+i a+i b x)}{2 b c}-\frac{(1-i a-i b x) \log (-i (i+a+b x))}{2 b c}-\frac{i \sqrt [3]{d} \log (1-i a-i b x) \log \left (-\frac{b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(i+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{i \sqrt [3]{d} \log (1+i a+i b x) \log \left (\frac{b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(i-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [6]{-1} \sqrt [3]{d} \log (1+i a+i b x) \log \left (-\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (i-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [6]{-1} \sqrt [3]{d} \log (1-i a-i b x) \log \left (\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (i+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{5/6} \sqrt [3]{d} \log (1+i a+i b x) \log \left (\frac{b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (i-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{(-1)^{5/6} \sqrt [3]{d} \log (1-i a-i b x) \log \left (\frac{b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{\sqrt [6]{-1} (1-i a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [6]{-1} \sqrt [3]{d} \text{Li}_2\left (\frac{\sqrt [3]{-1} \sqrt [3]{c} (i-a-b x)}{\sqrt [3]{-1} (i-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{5/6} \sqrt [3]{d} \text{Li}_2\left (\frac{\sqrt [6]{-1} \sqrt [3]{c} (i-a-b x)}{\sqrt [6]{-1} (i-a) \sqrt [3]{c}-i b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{i \sqrt [3]{d} \text{Li}_2\left (\frac{\sqrt [3]{c} (i-a-b x)}{(i-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{i \sqrt [3]{d} \text{Li}_2\left (\frac{\sqrt [3]{c} (i+a+b x)}{(i+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{(-1)^{5/6} \sqrt [3]{d} \text{Li}_2\left (\frac{(-1)^{2/3} \sqrt [3]{c} (i+a+b x)}{(-1)^{2/3} (i+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [6]{-1} \sqrt [3]{d} \text{Li}_2\left (\frac{\sqrt [3]{-1} \sqrt [3]{c} (i+a+b x)}{\sqrt [3]{-1} (i+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}\\ \end{align*}

Mathematica [C]  time = 7.07409, size = 933, normalized size = 1. \[ \frac{6 \left ((a+b x) \tan ^{-1}(a+b x)+\log \left (\frac{1}{\sqrt{(a+b x)^2+1}}\right )\right )-b^3 d \text{RootSum}\left [c \text{$\#$1}^3 a^3+3 c \text{$\#$1}^2 a^3+c a^3+3 c \text{$\#$1} a^3+3 i c \text{$\#$1}^3 a^2+3 i c \text{$\#$1}^2 a^2-3 i c a^2-3 i c \text{$\#$1} a^2-3 c \text{$\#$1}^3 a+3 c \text{$\#$1}^2 a-3 c a+3 c \text{$\#$1} a-i c \text{$\#$1}^3-b^3 d \text{$\#$1}^3+3 i c \text{$\#$1}^2-3 b^3 d \text{$\#$1}^2+i c-b^3 d-3 i c \text{$\#$1}-3 b^3 d \text{$\#$1}\& ,\frac{-2 \text{$\#$1} \tan ^{-1}(a+b x)^2+2 e^{\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )} \sqrt{\frac{\text{$\#$1}}{(\text{$\#$1}+1)^2}} \tan ^{-1}(a+b x)^2+2 e^{\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )} \text{$\#$1}^2 \sqrt{\frac{\text{$\#$1}}{(\text{$\#$1}+1)^2}} \tan ^{-1}(a+b x)^2+4 e^{\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )} \text{$\#$1} \sqrt{\frac{\text{$\#$1}}{(\text{$\#$1}+1)^2}} \tan ^{-1}(a+b x)^2-2 \tan ^{-1}(a+b x)^2-2 i \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right ) \text{$\#$1}^2 \tan ^{-1}(a+b x)-2 i \log \left (1-e^{2 i \tan ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right )}\right ) \text{$\#$1}^2 \tan ^{-1}(a+b x)+\pi \text{$\#$1}^2 \tan ^{-1}(a+b x)+2 i \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right ) \tan ^{-1}(a+b x)+2 i \log \left (1-e^{2 i \tan ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right )}\right ) \tan ^{-1}(a+b x)-\pi \tan ^{-1}(a+b x)-i \pi \log \left (1+e^{-2 i \tan ^{-1}(a+b x)}\right ) \text{$\#$1}^2+2 \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right ) \log \left (1-e^{2 i \tan ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right )}\right ) \text{$\#$1}^2+i \pi \log \left (\frac{1}{\sqrt{(a+b x)^2+1}}\right ) \text{$\#$1}^2-2 \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right ) \log \left (\sin \left (\tan ^{-1}(a+b x)+i \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right )\right )\right ) \text{$\#$1}^2-\text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right )}\right ) \text{$\#$1}^2+i \pi \log \left (1+e^{-2 i \tan ^{-1}(a+b x)}\right )-2 \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right ) \log \left (1-e^{2 i \tan ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right )}\right )-i \pi \log \left (\frac{1}{\sqrt{(a+b x)^2+1}}\right )+2 \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right ) \log \left (\sin \left (\tan ^{-1}(a+b x)+i \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right )\right )\right )+\text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac{\text{$\#$1}-1}{\text{$\#$1}+1}\right )}\right )}{c \text{$\#$1}^2 a^3+c a^3+2 c \text{$\#$1} a^3+2 i c \text{$\#$1}^2 a^2-2 i c a^2-c \text{$\#$1}^2 a-c a+2 c \text{$\#$1} a-b^3 d \text{$\#$1}^2-b^3 d-2 b^3 d \text{$\#$1}}\& \right ]}{6 b c} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcTan[a + b*x]/(c + d/x^3),x]

[Out]

(6*((a + b*x)*ArcTan[a + b*x] + Log[1/Sqrt[1 + (a + b*x)^2]]) - b^3*d*RootSum[I*c - 3*a*c - (3*I)*a^2*c + a^3*
c - b^3*d - (3*I)*c*#1 + 3*a*c*#1 - (3*I)*a^2*c*#1 + 3*a^3*c*#1 - 3*b^3*d*#1 + (3*I)*c*#1^2 + 3*a*c*#1^2 + (3*
I)*a^2*c*#1^2 + 3*a^3*c*#1^2 - 3*b^3*d*#1^2 - I*c*#1^3 - 3*a*c*#1^3 + (3*I)*a^2*c*#1^3 + a^3*c*#1^3 - b^3*d*#1
^3 & , (-(Pi*ArcTan[a + b*x]) - 2*ArcTan[a + b*x]^2 + (2*I)*ArcTan[a + b*x]*ArcTanh[(-1 + #1)/(1 + #1)] + I*Pi
*Log[1 + E^((-2*I)*ArcTan[a + b*x])] + (2*I)*ArcTan[a + b*x]*Log[1 - E^((2*I)*ArcTan[a + b*x] - 2*ArcTanh[(-1
+ #1)/(1 + #1)])] - 2*ArcTanh[(-1 + #1)/(1 + #1)]*Log[1 - E^((2*I)*ArcTan[a + b*x] - 2*ArcTanh[(-1 + #1)/(1 +
#1)])] - I*Pi*Log[1/Sqrt[1 + (a + b*x)^2]] + 2*ArcTanh[(-1 + #1)/(1 + #1)]*Log[Sin[ArcTan[a + b*x] + I*ArcTanh
[(-1 + #1)/(1 + #1)]]] + PolyLog[2, E^((2*I)*ArcTan[a + b*x] - 2*ArcTanh[(-1 + #1)/(1 + #1)])] - 2*ArcTan[a +
b*x]^2*#1 + Pi*ArcTan[a + b*x]*#1^2 - (2*I)*ArcTan[a + b*x]*ArcTanh[(-1 + #1)/(1 + #1)]*#1^2 - I*Pi*Log[1 + E^
((-2*I)*ArcTan[a + b*x])]*#1^2 - (2*I)*ArcTan[a + b*x]*Log[1 - E^((2*I)*ArcTan[a + b*x] - 2*ArcTanh[(-1 + #1)/
(1 + #1)])]*#1^2 + 2*ArcTanh[(-1 + #1)/(1 + #1)]*Log[1 - E^((2*I)*ArcTan[a + b*x] - 2*ArcTanh[(-1 + #1)/(1 + #
1)])]*#1^2 + I*Pi*Log[1/Sqrt[1 + (a + b*x)^2]]*#1^2 - 2*ArcTanh[(-1 + #1)/(1 + #1)]*Log[Sin[ArcTan[a + b*x] +
I*ArcTanh[(-1 + #1)/(1 + #1)]]]*#1^2 - PolyLog[2, E^((2*I)*ArcTan[a + b*x] - 2*ArcTanh[(-1 + #1)/(1 + #1)])]*#
1^2 + 2*E^ArcTanh[(1 - #1)/(1 + #1)]*ArcTan[a + b*x]^2*Sqrt[#1/(1 + #1)^2] + 4*E^ArcTanh[(1 - #1)/(1 + #1)]*Ar
cTan[a + b*x]^2*#1*Sqrt[#1/(1 + #1)^2] + 2*E^ArcTanh[(1 - #1)/(1 + #1)]*ArcTan[a + b*x]^2*#1^2*Sqrt[#1/(1 + #1
)^2])/(-(a*c) - (2*I)*a^2*c + a^3*c - b^3*d + 2*a*c*#1 + 2*a^3*c*#1 - 2*b^3*d*#1 - a*c*#1^2 + (2*I)*a^2*c*#1^2
+ a^3*c*#1^2 - b^3*d*#1^2) & ])/(6*b*c)

________________________________________________________________________________________

Maple [C]  time = 0.658, size = 682, normalized size = 0.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(b*x+a)/(c+d/x^3),x)

[Out]

arctan(b*x+a)/c*x+1/b*arctan(b*x+a)/c*a-1/2/b/c*ln(1+(b*x+a)^2)-2/3*b^2/c*d*sum(1/(a^3*c*_R1^4+3*I*_R1^4*a^2*c
-b^3*d*_R1^4-3*a*c*_R1^4-I*c*_R1^4+2*a^3*c*_R1^2+2*I*_R1^2*a^2*c-2*b^3*d*_R1^2+2*_R1^2*a*c+2*I*c*_R1^2+a^3*c-I
*a^2*c-b^3*d+a*c-I*c)*(I*arctan(b*x+a)*ln((_R1-(1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))/_R1)+dilog((_R1-(1+I*(b*x+a)
)/(1+(b*x+a)^2)^(1/2))/_R1)),_R1=RootOf((3*I*a^2*c+a^3*c-b^3*d-I*c-3*a*c)*_Z^6+(3*I*a^2*c+3*a^3*c-3*b^3*d+3*I*
c+3*a*c)*_Z^4+(-3*I*a^2*c+3*a^3*c-3*b^3*d-3*I*c+3*a*c)*_Z^2-3*I*a^2*c+a^3*c-b^3*d+I*c-3*a*c))-2/3*b^2/c*d*sum(
_R1^2/(a^3*c*_R1^4+3*I*_R1^4*a^2*c-b^3*d*_R1^4-3*a*c*_R1^4-I*c*_R1^4+2*a^3*c*_R1^2+2*I*_R1^2*a^2*c-2*b^3*d*_R1
^2+2*_R1^2*a*c+2*I*c*_R1^2+a^3*c-I*a^2*c-b^3*d+a*c-I*c)*(I*arctan(b*x+a)*ln((_R1-(1+I*(b*x+a))/(1+(b*x+a)^2)^(
1/2))/_R1)+dilog((_R1-(1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))/_R1)),_R1=RootOf((3*I*a^2*c+a^3*c-b^3*d-I*c-3*a*c)*_Z
^6+(3*I*a^2*c+3*a^3*c-3*b^3*d+3*I*c+3*a*c)*_Z^4+(-3*I*a^2*c+3*a^3*c-3*b^3*d-3*I*c+3*a*c)*_Z^2-3*I*a^2*c+a^3*c-
b^3*d+I*c-3*a*c))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3} \arctan \left (b x + a\right )}{c x^{3} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3*arctan(b*x + a)/(c*x^3 + d), x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(b*x+a)/(c+d/x**3),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (b x + a\right )}{c + \frac{d}{x^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arctan(b*x + a)/(c + d/x^3), x)