3.1.52 \(\int \frac {\text {ArcTan}(a+b x)}{c+d x^3} \, dx\) [52]

Optimal. Leaf size=863 \[ -\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 {PolyLog}\left (2,\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,\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 {PolyLog}\left (2,-\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}} \]

[Out]

-1/6*I*ln(1+I*a+I*b*x)*ln(b*(c^(1/3)+d^(1/3)*x)/(b*c^(1/3)+(I-a)*d^(1/3)))/c^(2/3)/d^(1/3)+1/6*I*ln(1-I*a-I*b*
x)*ln(b*(c^(1/3)+d^(1/3)*x)/(b*c^(1/3)-(I+a)*d^(1/3)))/c^(2/3)/d^(1/3)+1/6*(-1)^(1/6)*ln(1+I*a+I*b*x)*ln(b*(c^
(1/3)-(-1)^(1/3)*d^(1/3)*x)/(b*c^(1/3)-(-1)^(1/3)*(I-a)*d^(1/3)))/c^(2/3)/d^(1/3)-1/6*(-1)^(1/6)*ln(1-I*a-I*b*
x)*ln(b*(c^(1/3)-(-1)^(1/3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^(1/3)*(I+a)*d^(1/3)))/c^(2/3)/d^(1/3)+1/6*(-1)^(5/6)*ln
(1+I*a+I*b*x)*ln(b*(c^(1/3)+(-1)^(2/3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^(2/3)*(I-a)*d^(1/3)))/c^(2/3)/d^(1/3)-1/6*(-
1)^(5/6)*ln(1-I*a-I*b*x)*ln(b*(c^(1/3)+(-1)^(2/3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^(1/6)*(1-I*a)*d^(1/3)))/c^(2/3)/d
^(1/3)-1/6*I*polylog(2,d^(1/3)*(I-a-b*x)/(b*c^(1/3)+(I-a)*d^(1/3)))/c^(2/3)/d^(1/3)+1/6*(-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)))/c^(2/3)/d^(1/3)+1/6*(-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)))/c^(2/3)/d^(1/3)+1/6*I*polylog(2,-d^(1/3)*(
I+a+b*x)/(b*c^(1/3)-(I+a)*d^(1/3)))/c^(2/3)/d^(1/3)-1/6*(-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)))/c^(2/3)/d^(1/3)-1/6*(-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)))/c^(2/3)/d^(1/3)

________________________________________________________________________________________

Rubi [A]
time = 1.13, antiderivative size = 863, normalized size of antiderivative = 1.00, 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 = {5159, 2456, 2441, 2440, 2438} \begin {gather*} -\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 {Li}_2\left (\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 {Li}_2\left (-\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 {Li}_2\left (-\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 {Li}_2\left (-\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 {Li}_2\left (\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 {Li}_2\left (-\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTan[a + b*x]/(c + d*x^3),x]

[Out]

((-1/6*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))])/(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 2438

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]

Rule 2440

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 2441

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 2456

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 5159

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]

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 \text {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 \text {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} \text {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} \text {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} \text {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} \text {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*}

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Mathematica [A]
time = 0.64, size = 701, normalized size = 0.81 \begin {gather*} \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 )+i \log (-i (i+a+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 )+\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 )-\sqrt [6]{-1} \log (-i (i+a+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 )-(-1)^{5/6} \log (-i (i+a+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 )+(-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 )-i \text {PolyLog}\left (2,\frac {\sqrt [3]{d} (-i+a+b x)}{-b \sqrt [3]{c}+(-i+a) \sqrt [3]{d}}\right )+(-1)^{5/6} \text {PolyLog}\left (2,\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 )+\sqrt [6]{-1} \text {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{d} (-i+a+b x)}{b \sqrt [3]{c}+\sqrt [3]{-1} (-i+a) \sqrt [3]{d}}\right )+i \text {PolyLog}\left (2,\frac {\sqrt [3]{d} (i+a+b x)}{-b \sqrt [3]{c}+(i+a) \sqrt [3]{d}}\right )-\sqrt [6]{-1} \text {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{d} (i+a+b x)}{b \sqrt [3]{c}+\sqrt [3]{-1} (i+a) \sqrt [3]{d}}\right )-(-1)^{5/6} \text {PolyLog}\left (2,\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 {gather*}

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))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.43, size = 787, normalized size = 0.91 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {atan}\left (a+b\,x\right )}{d\,x^3+c} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(a + b*x)/(c + d*x^3),x)

[Out]

int(atan(a + b*x)/(c + d*x^3), x)

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