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

Optimal. Leaf size=933 \[ -\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 {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} \text {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} \text {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} \text {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} \text {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} \text {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]
time = 1.26, antiderivative size = 933, normalized size of antiderivative = 1.00, 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 = {5159, 2456, 2436, 2332, 2441, 2440, 2438} \begin {gather*} -\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 {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (\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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {\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} \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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 4.14, size = 933, normalized size = 1.00 \begin {gather*} \frac {6 \left ((a+b x) \text {ArcTan}(a+b x)+\log \left (\frac {1}{\sqrt {1+(a+b x)^2}}\right )\right )-b^3 d \text {RootSum}\left [i c-3 a c-3 i a^2 c+a^3 c-b^3 d-3 i c \text {$\#$1}+3 a c \text {$\#$1}-3 i a^2 c \text {$\#$1}+3 a^3 c \text {$\#$1}-3 b^3 d \text {$\#$1}+3 i c \text {$\#$1}^2+3 a c \text {$\#$1}^2+3 i a^2 c \text {$\#$1}^2+3 a^3 c \text {$\#$1}^2-3 b^3 d \text {$\#$1}^2-i c \text {$\#$1}^3-3 a c \text {$\#$1}^3+3 i a^2 c \text {$\#$1}^3+a^3 c \text {$\#$1}^3-b^3 d \text {$\#$1}^3\&,\frac {-\pi \text {ArcTan}(a+b x)-2 \text {ArcTan}(a+b x)^2+2 i \text {ArcTan}(a+b x) \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right )+i \pi \log \left (1+e^{-2 i \text {ArcTan}(a+b x)}\right )+2 i \text {ArcTan}(a+b x) \log \left (1-e^{2 i \text {ArcTan}(a+b x)-2 \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right )}\right )-2 \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right ) \log \left (1-e^{2 i \text {ArcTan}(a+b x)-2 \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right )}\right )-i \pi \log \left (\frac {1}{\sqrt {1+(a+b x)^2}}\right )+2 \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right ) \log \left (\sin \left (\text {ArcTan}(a+b x)+i \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right )\right )\right )+\text {PolyLog}\left (2,e^{2 i \text {ArcTan}(a+b x)-2 \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right )}\right )-2 \text {ArcTan}(a+b x)^2 \text {$\#$1}+\pi \text {ArcTan}(a+b x) \text {$\#$1}^2-2 i \text {ArcTan}(a+b x) \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right ) \text {$\#$1}^2-i \pi \log \left (1+e^{-2 i \text {ArcTan}(a+b x)}\right ) \text {$\#$1}^2-2 i \text {ArcTan}(a+b x) \log \left (1-e^{2 i \text {ArcTan}(a+b x)-2 \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right )}\right ) \text {$\#$1}^2+2 \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right ) \log \left (1-e^{2 i \text {ArcTan}(a+b x)-2 \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right )}\right ) \text {$\#$1}^2+i \pi \log \left (\frac {1}{\sqrt {1+(a+b x)^2}}\right ) \text {$\#$1}^2-2 \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right ) \log \left (\sin \left (\text {ArcTan}(a+b x)+i \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right )\right )\right ) \text {$\#$1}^2-\text {PolyLog}\left (2,e^{2 i \text {ArcTan}(a+b x)-2 \tanh ^{-1}\left (\frac {-1+\text {$\#$1}}{1+\text {$\#$1}}\right )}\right ) \text {$\#$1}^2+2 e^{\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{1+\text {$\#$1}}\right )} \text {ArcTan}(a+b x)^2 \sqrt {\frac {\text {$\#$1}}{(1+\text {$\#$1})^2}}+4 e^{\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{1+\text {$\#$1}}\right )} \text {ArcTan}(a+b x)^2 \text {$\#$1} \sqrt {\frac {\text {$\#$1}}{(1+\text {$\#$1})^2}}+2 e^{\tanh ^{-1}\left (\frac {1-\text {$\#$1}}{1+\text {$\#$1}}\right )} \text {ArcTan}(a+b x)^2 \text {$\#$1}^2 \sqrt {\frac {\text {$\#$1}}{(1+\text {$\#$1})^2}}}{-a c-2 i a^2 c+a^3 c-b^3 d+2 a c \text {$\#$1}+2 a^3 c \text {$\#$1}-2 b^3 d \text {$\#$1}-a c \text {$\#$1}^2+2 i a^2 c \text {$\#$1}^2+a^3 c \text {$\#$1}^2-b^3 d \text {$\#$1}^2}\&\right ]}{6 b c} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.40, size = 673, normalized size = 0.72 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)/(c+d/x^3),x, algorithm="maxima")

[Out]

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

[Out]

integral(x^3*arctan(b*x + a)/(c*x^3 + d), 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)/(c+d/x**3),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)/(c+d/x^3),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 )}{c+\frac {d}{x^3}} \,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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