\(\int \frac {\arctan (a+b x)}{c+\frac {d}{x^3}} \, dx\) [57]

Optimal result

Integrand size = 16, antiderivative size = 933 \[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x^3}} \, dx=-\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} \operatorname {PolyLog}\left (2,\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} \operatorname {PolyLog}\left (2,\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} \operatorname {PolyLog}\left (2,\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} \operatorname {PolyLog}\left (2,\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} \operatorname {PolyLog}\left (2,\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} \operatorname {PolyLog}\left (2,\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}} \]

[Out]

-1/2*(1+I*a+I*b*x)*ln(1+I*a+I*b*x)/b/c-1/2*(1-I*a-I*b*x)*ln(-I*(I+a+b*x))/b/c-1/6*I*d^(1/3)*ln(1-I*a-I*b*x)*ln
(-b*(d^(1/3)+c^(1/3)*x)/((I+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)+1/6*I*d^(1/3)*ln(1+I*a+I*b*x)*ln(b*(d^(1/3)+c^(1/3)
*x)/((I-a)*c^(1/3)+b*d^(1/3)))/c^(4/3)-1/6*(-1)^(1/6)*d^(1/3)*ln(1+I*a+I*b*x)*ln(-b*(d^(1/3)-(-1)^(1/3)*c^(1/3
)*x)/((-1)^(1/3)*(I-a)*c^(1/3)-b*d^(1/3)))/c^(4/3)+1/6*(-1)^(1/6)*d^(1/3)*ln(1-I*a-I*b*x)*ln(b*(d^(1/3)-(-1)^(
1/3)*c^(1/3)*x)/((-1)^(1/3)*(I+a)*c^(1/3)+b*d^(1/3)))/c^(4/3)-1/6*(-1)^(5/6)*d^(1/3)*ln(1+I*a+I*b*x)*ln(b*(d^(
1/3)+(-1)^(2/3)*c^(1/3)*x)/((-1)^(2/3)*(I-a)*c^(1/3)+b*d^(1/3)))/c^(4/3)+1/6*(-1)^(5/6)*d^(1/3)*ln(1-I*a-I*b*x
)*ln(b*(d^(1/3)+(-1)^(2/3)*c^(1/3)*x)/((-1)^(1/6)*(1-I*a)*c^(1/3)+b*d^(1/3)))/c^(4/3)-1/6*(-1)^(1/6)*d^(1/3)*p
olylog(2,(-1)^(1/3)*c^(1/3)*(I-a-b*x)/((-1)^(1/3)*(I-a)*c^(1/3)-b*d^(1/3)))/c^(4/3)-1/6*(-1)^(5/6)*d^(1/3)*pol
ylog(2,(-1)^(1/6)*c^(1/3)*(I-a-b*x)/((-1)^(1/6)*(I-a)*c^(1/3)-I*b*d^(1/3)))/c^(4/3)+1/6*I*d^(1/3)*polylog(2,c^
(1/3)*(I-a-b*x)/((I-a)*c^(1/3)+b*d^(1/3)))/c^(4/3)-1/6*I*d^(1/3)*polylog(2,c^(1/3)*(I+a+b*x)/((I+a)*c^(1/3)-b*
d^(1/3)))/c^(4/3)+1/6*(-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)))/c^(4/3)+1/6*(-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)))/c^(4/3)

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5159, 2456, 2436, 2332, 2441, 2440, 2438} \[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x^3}} \, dx=-\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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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}} \]

[In]

Int[ArcTan[a + b*x]/(c + d/x^3),x]

[Out]

-1/2*((1 + I*a + I*b*x)*Log[1 + I*a + I*b*x])/(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 2332

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

Rule 2436

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 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*} \text {integral}& = \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 {\text {Subst}(\int \log (x) \, dx,x,1-i a-i b x)}{2 b c}-\frac {\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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} \operatorname {PolyLog}\left (2,\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} \operatorname {PolyLog}\left (2,\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} \operatorname {PolyLog}\left (2,\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} \operatorname {PolyLog}\left (2,\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} \operatorname {PolyLog}\left (2,\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} \operatorname {PolyLog}\left (2,\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 [A] (verified)

Time = 0.93 (sec) , antiderivative size = 896, normalized size of antiderivative = 0.96 \[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x^3}} \, dx=\frac {i \left (3 i \sqrt [3]{c} \log (1+i a+i b x)-3 a \sqrt [3]{c} \log (1+i a+i b x)-3 b \sqrt [3]{c} x \log (1+i a+i b x)+3 i \sqrt [3]{c} \log (-i (i+a+b x))+3 a \sqrt [3]{c} \log (-i (i+a+b x))+3 b \sqrt [3]{c} x \log (-i (i+a+b x))+b \sqrt [3]{d} \log (1+i a+i b x) \log \left (\frac {b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{-\left ((-i+a) \sqrt [3]{c}\right )+b \sqrt [3]{d}}\right )-b \sqrt [3]{d} \log (-i (i+a+b x)) \log \left (\frac {b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{-\left ((i+a) \sqrt [3]{c}\right )+b \sqrt [3]{d}}\right )+(-1)^{2/3} b \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 )-(-1)^{2/3} b \sqrt [3]{d} \log (-i (i+a+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 )+\sqrt [3]{-1} b \sqrt [3]{d} \log (-i (i+a+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 )-\sqrt [3]{-1} b \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 )+b \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (-i+a+b x)}{(-i+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )-\sqrt [3]{-1} b \sqrt [3]{d} \operatorname {PolyLog}\left (2,\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 )+(-1)^{2/3} b \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{c} (-i+a+b x)}{\sqrt [3]{-1} (-i+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )-b \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (i+a+b x)}{(i+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )+\sqrt [3]{-1} b \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{c} (i+a+b x)}{(-1)^{2/3} (i+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )-(-1)^{2/3} b \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{c} (i+a+b x)}{\sqrt [3]{-1} (i+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )\right )}{6 b c^{4/3}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.09 (sec) , antiderivative size = 511, normalized size of antiderivative = 0.55

method	result	size
risch	\(\frac {i \ln \left (-b x i-i a +1\right ) x}{2 c}+\frac {i \ln \left (-b x i-i a +1\right ) a}{2 b c}-\frac {\ln \left (-b x i-i a +1\right )}{2 b c}+\frac {1}{b c}+\frac {i b^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}+\left (3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a c -3 c \right ) \textit {\_Z}^{2}+\left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a c -3 a^{2} c +3 c \right ) \textit {\_Z} -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a^{3} c +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) b^{3} d +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a c +3 a^{2} c -c \right )}{\sum }\frac {\ln \left (-b x i-i a +1\right ) \ln \left (\frac {b x i+i a +\textit {\_R1} -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {b x i+i a +\textit {\_R1} -1}{\textit {\_R1}}\right )}{2 \textit {\_R1} a i+\textit {\_R1}^{2}-a^{2}-2 i a -2 \textit {\_R1} +1}\right )}{6 c^{2}}-\frac {i \ln \left (b x i+i a +1\right ) x}{2 c}-\frac {i \ln \left (b x i+i a +1\right ) a}{2 b c}-\frac {\ln \left (b x i+i a +1\right )}{2 b c}-\frac {i b^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}+\left (-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a c -3 c \right ) \textit {\_Z}^{2}+\left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a c -3 a^{2} c +3 c \right ) \textit {\_Z} +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a^{3} c -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) b^{3} d -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a c +3 a^{2} c -c \right )}{\sum }\frac {\ln \left (b x i+i a +1\right ) \ln \left (\frac {-b x i-i a +\textit {\_R1} -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-b x i-i a +\textit {\_R1} -1}{\textit {\_R1}}\right )}{-2 \textit {\_R1} a i+\textit {\_R1}^{2}-a^{2}+2 i a -2 \textit {\_R1} +1}\right )}{6 c^{2}}\)	\(511\)
derivativedivides	\(\frac {\frac {\arctan \left (b x +a \right ) \left (b x +a \right )}{c}+\frac {\arctan \left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-3 a c \,\textit {\_Z}^{2}+3 a^{2} c \textit {\_Z} -a^{3} c +b^{3} d \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right ) d \,b^{3}}{3 c^{2}}-\frac {d \,b^{3} \left (\arctan \left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-3 a c \,\textit {\_Z}^{2}+3 a^{2} c \textit {\_Z} -a^{3} c +b^{3} d \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right )-3 c \left (-\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (a^{3} c +3 i a^{2} c -b^{3} d -3 a c -i c \right ) \textit {\_Z}^{6}+\left (3 a^{3} c +3 i a^{2} c -3 b^{3} d +3 a c +3 i c \right ) \textit {\_Z}^{4}+\left (3 a^{3} c -3 i a^{2} c -3 b^{3} d +3 a c -3 i c \right ) \textit {\_Z}^{2}-3 i a^{2} c +a^{3} c -b^{3} d +i c -3 a c \right )}{\sum }\frac {i \arctan \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )}{a^{3} c \,\textit {\_R1}^{4}+3 i a^{2} c \,\textit {\_R1}^{4}-b^{3} d \,\textit {\_R1}^{4}-3 a c \,\textit {\_R1}^{4}-i c \,\textit {\_R1}^{4}+2 a^{3} c \,\textit {\_R1}^{2}+2 i a^{2} c \,\textit {\_R1}^{2}-2 b^{3} d \,\textit {\_R1}^{2}+2 a c \,\textit {\_R1}^{2}+2 i c \,\textit {\_R1}^{2}+a^{3} c -i a^{2} c -b^{3} d +a c -i c}\right )}{3}-\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (a^{3} c +3 i a^{2} c -b^{3} d -3 a c -i c \right ) \textit {\_Z}^{6}+\left (3 a^{3} c +3 i a^{2} c -3 b^{3} d +3 a c +3 i c \right ) \textit {\_Z}^{4}+\left (3 a^{3} c -3 i a^{2} c -3 b^{3} d +3 a c -3 i c \right ) \textit {\_Z}^{2}-3 i a^{2} c +a^{3} c -b^{3} d +i c -3 a c \right )}{\sum }\frac {\textit {\_R1}^{2} \left (i \arctan \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )\right )}{a^{3} c \,\textit {\_R1}^{4}+3 i a^{2} c \,\textit {\_R1}^{4}-b^{3} d \,\textit {\_R1}^{4}-3 a c \,\textit {\_R1}^{4}-i c \,\textit {\_R1}^{4}+2 a^{3} c \,\textit {\_R1}^{2}+2 i a^{2} c \,\textit {\_R1}^{2}-2 b^{3} d \,\textit {\_R1}^{2}+2 a c \,\textit {\_R1}^{2}+2 i c \,\textit {\_R1}^{2}+a^{3} c -i a^{2} c -b^{3} d +a c -i c}\right )}{3}\right )\right )}{3 c^{2}}-\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2 c}}{b}\)	\(819\)
default	\(\frac {\frac {\arctan \left (b x +a \right ) \left (b x +a \right )}{c}+\frac {\arctan \left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-3 a c \,\textit {\_Z}^{2}+3 a^{2} c \textit {\_Z} -a^{3} c +b^{3} d \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right ) d \,b^{3}}{3 c^{2}}-\frac {d \,b^{3} \left (\arctan \left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-3 a c \,\textit {\_Z}^{2}+3 a^{2} c \textit {\_Z} -a^{3} c +b^{3} d \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right )-3 c \left (-\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (a^{3} c +3 i a^{2} c -b^{3} d -3 a c -i c \right ) \textit {\_Z}^{6}+\left (3 a^{3} c +3 i a^{2} c -3 b^{3} d +3 a c +3 i c \right ) \textit {\_Z}^{4}+\left (3 a^{3} c -3 i a^{2} c -3 b^{3} d +3 a c -3 i c \right ) \textit {\_Z}^{2}-3 i a^{2} c +a^{3} c -b^{3} d +i c -3 a c \right )}{\sum }\frac {i \arctan \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )}{a^{3} c \,\textit {\_R1}^{4}+3 i a^{2} c \,\textit {\_R1}^{4}-b^{3} d \,\textit {\_R1}^{4}-3 a c \,\textit {\_R1}^{4}-i c \,\textit {\_R1}^{4}+2 a^{3} c \,\textit {\_R1}^{2}+2 i a^{2} c \,\textit {\_R1}^{2}-2 b^{3} d \,\textit {\_R1}^{2}+2 a c \,\textit {\_R1}^{2}+2 i c \,\textit {\_R1}^{2}+a^{3} c -i a^{2} c -b^{3} d +a c -i c}\right )}{3}-\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (a^{3} c +3 i a^{2} c -b^{3} d -3 a c -i c \right ) \textit {\_Z}^{6}+\left (3 a^{3} c +3 i a^{2} c -3 b^{3} d +3 a c +3 i c \right ) \textit {\_Z}^{4}+\left (3 a^{3} c -3 i a^{2} c -3 b^{3} d +3 a c -3 i c \right ) \textit {\_Z}^{2}-3 i a^{2} c +a^{3} c -b^{3} d +i c -3 a c \right )}{\sum }\frac {\textit {\_R1}^{2} \left (i \arctan \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )\right )}{a^{3} c \,\textit {\_R1}^{4}+3 i a^{2} c \,\textit {\_R1}^{4}-b^{3} d \,\textit {\_R1}^{4}-3 a c \,\textit {\_R1}^{4}-i c \,\textit {\_R1}^{4}+2 a^{3} c \,\textit {\_R1}^{2}+2 i a^{2} c \,\textit {\_R1}^{2}-2 b^{3} d \,\textit {\_R1}^{2}+2 a c \,\textit {\_R1}^{2}+2 i c \,\textit {\_R1}^{2}+a^{3} c -i a^{2} c -b^{3} d +a c -i c}\right )}{3}\right )\right )}{3 c^{2}}-\frac {\ln \left (1+\left (b x +a \right )^{2}\right )}{2 c}}{b}\)	\(819\)

[In]

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

[Out]

1/2*I/c*ln(1-I*a-I*b*x)*x+1/2*I/b/c*ln(1-I*a-I*b*x)*a-1/2/b/c*ln(1-I*a-I*b*x)+1/b/c+1/6*I*b^2*d/c^2*sum(1/(1+2
*I*a*_R1-2*I*a+_R1^2-a^2-2*_R1)*(ln(1-I*a-I*b*x)*ln((_R1+I*b*x+I*a-1)/_R1)+dilog((_R1+I*b*x+I*a-1)/_R1)),_R1=R
ootOf(c*_Z^3+(3*RootOf(_Z^2+1,index=1)*a*c-3*c)*_Z^2+(-6*RootOf(_Z^2+1,index=1)*a*c-3*a^2*c+3*c)*_Z-RootOf(_Z^
2+1,index=1)*a^3*c+RootOf(_Z^2+1,index=1)*b^3*d+3*RootOf(_Z^2+1,index=1)*a*c+3*a^2*c-c))-1/2*I/c*ln(1+I*a+I*b*
x)*x-1/2*I/b/c*ln(1+I*a+I*b*x)*a-1/2/b/c*ln(1+I*a+I*b*x)-1/6*I*b^2*d/c^2*sum(1/(1-2*I*a*_R1+2*I*a+_R1^2-a^2-2*
_R1)*(ln(1+I*a+I*b*x)*ln((_R1-I*b*x-I*a-1)/_R1)+dilog((_R1-I*b*x-I*a-1)/_R1)),_R1=RootOf(c*_Z^3+(-3*RootOf(_Z^
2+1,index=1)*a*c-3*c)*_Z^2+(6*RootOf(_Z^2+1,index=1)*a*c-3*a^2*c+3*c)*_Z+RootOf(_Z^2+1,index=1)*a^3*c-RootOf(_
Z^2+1,index=1)*b^3*d-3*RootOf(_Z^2+1,index=1)*a*c+3*a^2*c-c))

Fricas [F]

\[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x^3}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{c + \frac {d}{x^{3}}} \,d x } \]

[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)]

Timed out. \[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x^3}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x^3}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{c + \frac {d}{x^{3}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x^3}} \, dx=\int { \frac {\arctan \left (b x + a\right )}{c + \frac {d}{x^{3}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a+b x)}{c+\frac {d}{x^3}} \, dx=\int \frac {\mathrm {atan}\left (a+b\,x\right )}{c+\frac {d}{x^3}} \,d x \]

[In]

int(atan(a + b*x)/(c + d/x^3),x)

[Out]

int(atan(a + b*x)/(c + d/x^3), x)