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

Optimal. Leaf size=780 \[ -\frac{\text{PolyLog}\left (2,\frac{\sqrt [3]{d} (-a-b x+1)}{(1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{2/3} \text{PolyLog}\left (2,-\frac{\sqrt [3]{-1} \sqrt [3]{d} (-a-b x+1)}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [3]{-1} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{d} (-a-b x+1)}{(-1)^{2/3} (1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\text{PolyLog}\left (2,-\frac{\sqrt [3]{d} (a+b x+1)}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{2/3} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{d} (a+b x+1)}{\sqrt [3]{-1} (a+1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [3]{-1} \text{PolyLog}\left (2,-\frac{(-1)^{2/3} \sqrt [3]{d} (a+b x+1)}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\log (-a-b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{(1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\log (a+b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{2/3} \log (-a-b x+1) \log \left (\frac{b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{2/3} \log (a+b x+1) \log \left (\frac{b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} (a+1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [3]{-1} \log (-a-b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{(-1)^{2/3} (1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [3]{-1} \log (a+b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.39528, antiderivative size = 780, 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 = {6115, 2409, 2394, 2393, 2391} \[ -\frac{\text{PolyLog}\left (2,\frac{\sqrt [3]{d} (-a-b x+1)}{(1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{2/3} \text{PolyLog}\left (2,-\frac{\sqrt [3]{-1} \sqrt [3]{d} (-a-b x+1)}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [3]{-1} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{d} (-a-b x+1)}{(-1)^{2/3} (1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\text{PolyLog}\left (2,-\frac{\sqrt [3]{d} (a+b x+1)}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{2/3} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{d} (a+b x+1)}{\sqrt [3]{-1} (a+1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [3]{-1} \text{PolyLog}\left (2,-\frac{(-1)^{2/3} \sqrt [3]{d} (a+b x+1)}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\log (-a-b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{(1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\log (a+b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{2/3} \log (-a-b x+1) \log \left (\frac{b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{2/3} \log (a+b x+1) \log \left (\frac{b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} (a+1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [3]{-1} \log (-a-b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{(-1)^{2/3} (1-a) \sqrt [3]{d}+b \sqrt [3]{c}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [3]{-1} \log (a+b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a + b*x]/(c + d*x^3),x]

[Out]

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

Rule 6115

Int[ArcTanh[(c_) + (d_.)*(x_)]/((e_) + (f_.)*(x_)^(n_.)), x_Symbol] :> Dist[1/2, Int[Log[1 + c + d*x]/(e + f*x
^n), x], x] - Dist[1/2, Int[Log[1 - c - d*x]/(e + f*x^n), x], x] /; FreeQ[{c, d, e, f}, 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{\tanh ^{-1}(a+b x)}{c+d x^3} \, dx &=-\left (\frac{1}{2} \int \frac{\log (1-a-b x)}{c+d x^3} \, dx\right )+\frac{1}{2} \int \frac{\log (1+a+b x)}{c+d x^3} \, dx\\ &=-\left (\frac{1}{2} \int \left (-\frac{\log (1-a-b x)}{3 c^{2/3} \left (-\sqrt [3]{c}-\sqrt [3]{d} x\right )}-\frac{\log (1-a-b x)}{3 c^{2/3} \left (-\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d} x\right )}-\frac{\log (1-a-b x)}{3 c^{2/3} \left (-\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d} x\right )}\right ) \, dx\right )+\frac{1}{2} \int \left (-\frac{\log (1+a+b x)}{3 c^{2/3} \left (-\sqrt [3]{c}-\sqrt [3]{d} x\right )}-\frac{\log (1+a+b x)}{3 c^{2/3} \left (-\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d} x\right )}-\frac{\log (1+a+b x)}{3 c^{2/3} \left (-\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d} x\right )}\right ) \, dx\\ &=\frac{\int \frac{\log (1-a-b x)}{-\sqrt [3]{c}-\sqrt [3]{d} x} \, dx}{6 c^{2/3}}+\frac{\int \frac{\log (1-a-b x)}{-\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d} x} \, dx}{6 c^{2/3}}+\frac{\int \frac{\log (1-a-b x)}{-\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d} x} \, dx}{6 c^{2/3}}-\frac{\int \frac{\log (1+a+b x)}{-\sqrt [3]{c}-\sqrt [3]{d} x} \, dx}{6 c^{2/3}}-\frac{\int \frac{\log (1+a+b x)}{-\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d} x} \, dx}{6 c^{2/3}}-\frac{\int \frac{\log (1+a+b x)}{-\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d} x} \, dx}{6 c^{2/3}}\\ &=-\frac{\log (1-a-b x) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\log (1+a+b x) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{2/3} \log (1-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} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{2/3} \log (1+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} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [3]{-1} \log (1-a-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} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [3]{-1} \log (1+a+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} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{b \int \frac{\log \left (-\frac{b \left (-\sqrt [3]{c}-\sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{1-a-b x} \, dx}{6 c^{2/3} \sqrt [3]{d}}-\frac{b \int \frac{\log \left (\frac{b \left (-\sqrt [3]{c}-\sqrt [3]{d} x\right )}{-b \sqrt [3]{c}+(1+a) \sqrt [3]{d}}\right )}{1+a+b x} \, dx}{6 c^{2/3} \sqrt [3]{d}}+\frac{\left (\sqrt [3]{-1} b\right ) \int \frac{\log \left (-\frac{b \left (-\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}}\right )}{1-a-b x} \, dx}{6 c^{2/3} \sqrt [3]{d}}+\frac{\left (\sqrt [3]{-1} b\right ) \int \frac{\log \left (\frac{b \left (-\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d} x\right )}{-b \sqrt [3]{c}+(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )}{1+a+b x} \, dx}{6 c^{2/3} \sqrt [3]{d}}-\frac{\left ((-1)^{2/3} b\right ) \int \frac{\log \left (-\frac{b \left (-\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{1-a-b x} \, dx}{6 c^{2/3} \sqrt [3]{d}}-\frac{\left ((-1)^{2/3} b\right ) \int \frac{\log \left (\frac{b \left (-\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d} x\right )}{-b \sqrt [3]{c}-\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )}{1+a+b x} \, dx}{6 c^{2/3} \sqrt [3]{d}}\\ &=-\frac{\log (1-a-b x) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\log (1+a+b x) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{2/3} \log (1-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} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{2/3} \log (1+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} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [3]{-1} \log (1-a-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} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [3]{-1} \log (1+a+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} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [3]{d} x}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{x} \, dx,x,1-a-b x\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [3]{d} x}{-b \sqrt [3]{c}+(1+a) \sqrt [3]{d}}\right )}{x} \, dx,x,1+a+b x\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [3]{-1} \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{(-1)^{2/3} \sqrt [3]{d} x}{b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}}\right )}{x} \, dx,x,1-a-b x\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [3]{-1} \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{(-1)^{2/3} \sqrt [3]{d} x}{-b \sqrt [3]{c}+(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )}{x} \, dx,x,1+a+b x\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{2/3} \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{-1} \sqrt [3]{d} x}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{x} \, dx,x,1-a-b x\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{2/3} \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{-1} \sqrt [3]{d} x}{-b \sqrt [3]{c}-\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )}{x} \, dx,x,1+a+b x\right )}{6 c^{2/3} \sqrt [3]{d}}\\ &=-\frac{\log (1-a-b x) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\log (1+a+b x) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{2/3} \log (1-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} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{2/3} \log (1+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} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [3]{-1} \log (1-a-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} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [3]{-1} \log (1+a+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} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\text{Li}_2\left (\frac{\sqrt [3]{d} (1-a-b x)}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{(-1)^{2/3} \text{Li}_2\left (-\frac{\sqrt [3]{-1} \sqrt [3]{d} (1-a-b x)}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\sqrt [3]{-1} \text{Li}_2\left (\frac{(-1)^{2/3} \sqrt [3]{d} (1-a-b x)}{b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{\text{Li}_2\left (-\frac{\sqrt [3]{d} (1+a+b x)}{b \sqrt [3]{c}-(1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac{(-1)^{2/3} \text{Li}_2\left (\frac{\sqrt [3]{-1} \sqrt [3]{d} (1+a+b x)}{b \sqrt [3]{c}+\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac{\sqrt [3]{-1} \text{Li}_2\left (-\frac{(-1)^{2/3} \sqrt [3]{d} (1+a+b x)}{b \sqrt [3]{c}-(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}\\ \end{align*}

Mathematica [A]  time = 0.727825, size = 623, normalized size = 0.8 \[ \frac{-\text{PolyLog}\left (2,-\frac{\sqrt [3]{d} (a+b x-1)}{b \sqrt [3]{c}-(a-1) \sqrt [3]{d}}\right )-(-1)^{2/3} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{d} (a+b x-1)}{\sqrt [3]{-1} (a-1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )+\sqrt [3]{-1} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{d} (a+b x-1)}{(-1)^{2/3} (a-1) \sqrt [3]{d}-b \sqrt [3]{c}}\right )+\text{PolyLog}\left (2,-\frac{\sqrt [3]{d} (a+b x+1)}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )+(-1)^{2/3} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{d} (a+b x+1)}{\sqrt [3]{-1} (a+1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )-\sqrt [3]{-1} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{d} (a+b x+1)}{(-1)^{2/3} (a+1) \sqrt [3]{d}-b \sqrt [3]{c}}\right )-\log (-a-b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(a-1) \sqrt [3]{d}}\right )+\log (a+b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )-(-1)^{2/3} \log (-a-b x+1) \log \left (\frac{b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} (a-1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )+(-1)^{2/3} \log (a+b x+1) \log \left (\frac{b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} (a+1) \sqrt [3]{d}+b \sqrt [3]{c}}\right )+\sqrt [3]{-1} \log (-a-b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (a-1) \sqrt [3]{d}}\right )-\sqrt [3]{-1} \log (a+b x+1) \log \left (\frac{b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.493, size = 587, normalized size = 0.8 \begin{align*} -{\frac{2\,{b}^{2}}{3}\sum _{{\it \_R1}={\it RootOf} \left ( \left ({a}^{3}d-c{b}^{3}-3\,{a}^{2}d+3\,ad-d \right ){{\it \_Z}}^{6}+ \left ( 3\,{a}^{3}d-3\,c{b}^{3}-3\,{a}^{2}d-3\,ad+3\,d \right ){{\it \_Z}}^{4}+ \left ( 3\,{a}^{3}d-3\,c{b}^{3}+3\,{a}^{2}d-3\,ad-3\,d \right ){{\it \_Z}}^{2}+{a}^{3}d-c{b}^{3}+3\,{a}^{2}d+3\,ad+d \right ) }{\frac{1}{{{\it \_R1}}^{4}{a}^{3}d-{{\it \_R1}}^{4}{b}^{3}c-3\,{{\it \_R1}}^{4}{a}^{2}d+3\,{{\it \_R1}}^{4}ad+2\,{{\it \_R1}}^{2}{a}^{3}d-2\,{{\it \_R1}}^{2}{b}^{3}c-{{\it \_R1}}^{4}d-2\,{{\it \_R1}}^{2}{a}^{2}d-2\,{{\it \_R1}}^{2}ad+{a}^{3}d-c{b}^{3}+2\,{{\it \_R1}}^{2}d+{a}^{2}d-ad-d} \left ({\it Artanh} \left ( bx+a \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{(bx+a+1){\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}}} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{(bx+a+1){\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}}} \right ) } \right ) \right ) }}-{\frac{2\,{b}^{2}}{3}\sum _{{\it \_R1}={\it RootOf} \left ( \left ({a}^{3}d-c{b}^{3}-3\,{a}^{2}d+3\,ad-d \right ){{\it \_Z}}^{6}+ \left ( 3\,{a}^{3}d-3\,c{b}^{3}-3\,{a}^{2}d-3\,ad+3\,d \right ){{\it \_Z}}^{4}+ \left ( 3\,{a}^{3}d-3\,c{b}^{3}+3\,{a}^{2}d-3\,ad-3\,d \right ){{\it \_Z}}^{2}+{a}^{3}d-c{b}^{3}+3\,{a}^{2}d+3\,ad+d \right ) }{\frac{{{\it \_R1}}^{2}}{{{\it \_R1}}^{4}{a}^{3}d-{{\it \_R1}}^{4}{b}^{3}c-3\,{{\it \_R1}}^{4}{a}^{2}d+3\,{{\it \_R1}}^{4}ad+2\,{{\it \_R1}}^{2}{a}^{3}d-2\,{{\it \_R1}}^{2}{b}^{3}c-{{\it \_R1}}^{4}d-2\,{{\it \_R1}}^{2}{a}^{2}d-2\,{{\it \_R1}}^{2}ad+{a}^{3}d-c{b}^{3}+2\,{{\it \_R1}}^{2}d+{a}^{2}d-ad-d} \left ({\it Artanh} \left ( bx+a \right ) \ln \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{(bx+a+1){\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}}} \right ) } \right ) +{\it dilog} \left ({\frac{1}{{\it \_R1}} \left ({\it \_R1}-{(bx+a+1){\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(arctanh(b*x+a)/(d*x^3+c),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{artanh}\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(arctanh(b*x+a)/(d*x^3+c),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\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(arctanh(b*x+a)/(d*x^3+c),x, algorithm="giac")

[Out]

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