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

Optimal. Leaf size=863 \[ -\frac{i \log (i a+i b x+1) \log \left (\frac{b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\sqrt [3]{d} (i-a)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{i \log (-i a-i b x+1) \log \left (\frac{b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{b \sqrt [3]{c}-(a+i) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [6]{-1} \log (i a+i b x+1) \log \left (\frac{b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (i-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [6]{-1} \log (-i a-i b x+1) \log \left (\frac{b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} \sqrt [3]{d} (a+i)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{5/6} \log (i a+i b x+1) \log \left (\frac{b \left ((-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}\right )}{(-1)^{2/3} \sqrt [3]{d} (i-a)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{5/6} \log (-i a-i b x+1) \log \left (\frac{b \left ((-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}\right )}{\sqrt [6]{-1} \sqrt [3]{d} (1-i a)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{i \text{PolyLog}\left (2,\frac{\sqrt [3]{d} (-a-b x+i)}{\sqrt [3]{d} (i-a)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{5/6} \text{PolyLog}\left (2,-\frac{\sqrt [6]{-1} \sqrt [3]{d} (-a-b x+i)}{i b \sqrt [3]{c}-\sqrt [6]{-1} (i-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [6]{-1} \text{PolyLog}\left (2,-\frac{\sqrt [3]{-1} \sqrt [3]{d} (-a-b x+i)}{b \sqrt [3]{c}-\sqrt [3]{-1} (i-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{i \text{PolyLog}\left (2,-\frac{\sqrt [3]{d} (a+b x+i)}{b \sqrt [3]{c}-(a+i) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [6]{-1} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{d} (a+b x+i)}{\sqrt [3]{-1} \sqrt [3]{d} (a+i)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{5/6} \text{PolyLog}\left (2,-\frac{(-1)^{2/3} \sqrt [3]{d} (a+b x+i)}{b \sqrt [3]{c}-(-1)^{2/3} (a+i) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.20694, antiderivative size = 863, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5051, 2409, 2394, 2393, 2391} \[ -\frac{i \log (i a+i b x+1) \log \left (\frac{b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\sqrt [3]{d} (i-a)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{i \log (-i a-i b x+1) \log \left (\frac{b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{b \sqrt [3]{c}-(a+i) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [6]{-1} \log (i a+i b x+1) \log \left (\frac{b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (i-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [6]{-1} \log (-i a-i b x+1) \log \left (\frac{b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} \sqrt [3]{d} (a+i)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{5/6} \log (i a+i b x+1) \log \left (\frac{b \left ((-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}\right )}{(-1)^{2/3} \sqrt [3]{d} (i-a)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{5/6} \log (-i a-i b x+1) \log \left (\frac{b \left ((-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}\right )}{\sqrt [6]{-1} \sqrt [3]{d} (1-i a)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{i \text{PolyLog}\left (2,\frac{\sqrt [3]{d} (-a-b x+i)}{\sqrt [3]{d} (i-a)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{5/6} \text{PolyLog}\left (2,-\frac{\sqrt [6]{-1} \sqrt [3]{d} (-a-b x+i)}{i b \sqrt [3]{c}-\sqrt [6]{-1} (i-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [6]{-1} \text{PolyLog}\left (2,-\frac{\sqrt [3]{-1} \sqrt [3]{d} (-a-b x+i)}{b \sqrt [3]{c}-\sqrt [3]{-1} (i-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{i \text{PolyLog}\left (2,-\frac{\sqrt [3]{d} (a+b x+i)}{b \sqrt [3]{c}-(a+i) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [6]{-1} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{d} (a+b x+i)}{\sqrt [3]{-1} \sqrt [3]{d} (a+i)+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{5/6} \text{PolyLog}\left (2,-\frac{(-1)^{2/3} \sqrt [3]{d} (a+b x+i)}{b \sqrt [3]{c}-(-1)^{2/3} (a+i) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a + b*x]/(c + d*x^3),x]

[Out]

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

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

Mathematica [A]  time = 0.804645, size = 701, normalized size = 0.81 \[ \frac{-i \text{PolyLog}\left (2,\frac{\sqrt [3]{d} (a+b x-i)}{-b \sqrt [3]{c}+(a-i) \sqrt [3]{d}}\right )+(-1)^{5/6} \text{PolyLog}\left (2,\frac{\sqrt [6]{-1} \sqrt [3]{d} (a+b x-i)}{\sqrt [6]{-1} (a-i) \sqrt [3]{d}+i b \sqrt [3]{c}}\right )+\sqrt [6]{-1} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{d} (a+b x-i)}{b \sqrt [3]{c}+\sqrt [3]{-1} (a-i) \sqrt [3]{d}}\right )+i \text{PolyLog}\left (2,\frac{\sqrt [3]{d} (a+b x+i)}{-b \sqrt [3]{c}+(a+i) \sqrt [3]{d}}\right )-\sqrt [6]{-1} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{d} (a+b x+i)}{b \sqrt [3]{c}+\sqrt [3]{-1} (a+i) \sqrt [3]{d}}\right )-(-1)^{5/6} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{d} (a+b x+i)}{-b \sqrt [3]{c}+(-1)^{2/3} (a+i) \sqrt [3]{d}}\right )-i \log (i a+i b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(a-i) \sqrt [3]{d}}\right )+i \log (-i (a+b x+i)) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(a+i) \sqrt [3]{d}}\right )+\sqrt [6]{-1} \log (i a+i b x+1) \log \left (\frac{b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [3]{-1} (a-i) \sqrt [3]{d}}\right )-\sqrt [6]{-1} \log (-i (a+b x+i)) \log \left (\frac{b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [3]{-1} (a+i) \sqrt [3]{d}}\right )-(-1)^{5/6} \log (-i (a+b x+i)) \log \left (\frac{b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [6]{-1} (1-i a) \sqrt [3]{d}}\right )+(-1)^{5/6} \log (i a+i b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (a-i) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.636, size = 631, normalized size = 0.7 \begin{align*}{\frac{2\,{b}^{2}}{3}\sum _{{\it \_R1}={\it RootOf} \left ( \left ( 3\,i{a}^{2}d+{a}^{3}d-c{b}^{3}-id-3\,ad \right ){{\it \_Z}}^{6}+ \left ( 3\,i{a}^{2}d+3\,{a}^{3}d-3\,c{b}^{3}+3\,id+3\,ad \right ){{\it \_Z}}^{4}+ \left ( -3\,i{a}^{2}d+3\,{a}^{3}d-3\,c{b}^{3}-3\,id+3\,ad \right ){{\it \_Z}}^{2}-3\,i{a}^{2}d+{a}^{3}d-c{b}^{3}+id-3\,ad \right ) }{\frac{1}{{a}^{3}d{{\it \_R1}}^{4}+3\,i{a}^{2}d{{\it \_R1}}^{4}-{b}^{3}c{{\it \_R1}}^{4}-3\,ad{{\it \_R1}}^{4}-id{{\it \_R1}}^{4}+2\,{a}^{3}d{{\it \_R1}}^{2}+2\,i{a}^{2}d{{\it \_R1}}^{2}-2\,{b}^{3}c{{\it \_R1}}^{2}+2\,{{\it \_R1}}^{2}ad+2\,id{{\it \_R1}}^{2}+{a}^{3}d-i{a}^{2}d-c{b}^{3}+ad-id} \left ( i\arctan \left ( bx+a \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{(1+i \left ( bx+a \right ) ){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{(1+i \left ( bx+a \right ) ){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) } \right ) \right ) }}+{\frac{2\,{b}^{2}}{3}\sum _{{\it \_R1}={\it RootOf} \left ( \left ( 3\,i{a}^{2}d+{a}^{3}d-c{b}^{3}-id-3\,ad \right ){{\it \_Z}}^{6}+ \left ( 3\,i{a}^{2}d+3\,{a}^{3}d-3\,c{b}^{3}+3\,id+3\,ad \right ){{\it \_Z}}^{4}+ \left ( -3\,i{a}^{2}d+3\,{a}^{3}d-3\,c{b}^{3}-3\,id+3\,ad \right ){{\it \_Z}}^{2}-3\,i{a}^{2}d+{a}^{3}d-c{b}^{3}+id-3\,ad \right ) }{\frac{{{\it \_R1}}^{2}}{{a}^{3}d{{\it \_R1}}^{4}+3\,i{a}^{2}d{{\it \_R1}}^{4}-{b}^{3}c{{\it \_R1}}^{4}-3\,ad{{\it \_R1}}^{4}-id{{\it \_R1}}^{4}+2\,{a}^{3}d{{\it \_R1}}^{2}+2\,i{a}^{2}d{{\it \_R1}}^{2}-2\,{b}^{3}c{{\it \_R1}}^{2}+2\,{{\it \_R1}}^{2}ad+2\,id{{\it \_R1}}^{2}+{a}^{3}d-i{a}^{2}d-c{b}^{3}+ad-id} \left ( i\arctan \left ( bx+a \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{(1+i \left ( bx+a \right ) ){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{(1+i \left ( bx+a \right ) ){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) } \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b^2*sum(1/(a^3*d*_R1^4+3*I*a^2*d*_R1^4-b^3*c*_R1^4-3*a*d*_R1^4-I*d*_R1^4+2*a^3*d*_R1^2+2*I*a^2*d*_R1^2-2*b
^3*c*_R1^2+2*_R1^2*a*d+2*I*d*_R1^2+a^3*d-I*a^2*d-c*b^3+a*d-I*d)*(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*d+a^3*d-c*b^3-I*d-3
*a*d)*_Z^6+(3*I*a^2*d+3*a^3*d-3*c*b^3+3*I*d+3*a*d)*_Z^4+(-3*I*a^2*d+3*a^3*d-3*c*b^3-3*I*d+3*a*d)*_Z^2-3*I*a^2*
d+a^3*d-c*b^3+I*d-3*a*d))+2/3*b^2*sum(_R1^2/(a^3*d*_R1^4+3*I*a^2*d*_R1^4-b^3*c*_R1^4-3*a*d*_R1^4-I*d*_R1^4+2*a
^3*d*_R1^2+2*I*a^2*d*_R1^2-2*b^3*c*_R1^2+2*_R1^2*a*d+2*I*d*_R1^2+a^3*d-I*a^2*d-c*b^3+a*d-I*d)*(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=RootO
f((3*I*a^2*d+a^3*d-c*b^3-I*d-3*a*d)*_Z^6+(3*I*a^2*d+3*a^3*d-3*c*b^3+3*I*d+3*a*d)*_Z^4+(-3*I*a^2*d+3*a^3*d-3*c*
b^3-3*I*d+3*a*d)*_Z^2-3*I*a^2*d+a^3*d-c*b^3+I*d-3*a*d))

________________________________________________________________________________________

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)/(d*x^3+c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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