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

Optimal. Leaf size=832 \[ \frac{(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac{\sqrt [3]{d} \log \left (\frac{b \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{\sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right ) \log (-a-b x+1)}{6 c^{4/3}}+\frac{(-1)^{2/3} \sqrt [3]{d} \log \left (-\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right ) \log (-a-b x+1)}{6 c^{4/3}}-\frac{\sqrt [3]{-1} \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} (1-a)+b \sqrt [3]{d}}\right ) \log (-a-b x+1)}{6 c^{4/3}}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c}-\frac{\sqrt [3]{d} \log (a+b x+1) \log \left (-\frac{b \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{(a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{d} \log (a+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+1)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{d} \log (a+b x+1) \log \left (-\frac{b \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{(-1)^{2/3} (a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{(-1)^{2/3} \sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{c} (-a-b x+1)}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [3]{c} (-a-b x+1)}{\sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{d} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{c} (-a-b x+1)}{(-1)^{2/3} \sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [3]{c} (a+b x+1)}{(a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{d} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{c} (a+b x+1)}{(-1)^{2/3} (a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{c} (a+b x+1)}{\sqrt [3]{-1} \sqrt [3]{c} (a+1)+b \sqrt [3]{d}}\right )}{6 c^{4/3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.42979, antiderivative size = 832, normalized size of antiderivative = 1., 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 = {6115, 2409, 2389, 2295, 2394, 2393, 2391} \[ \frac{(-a-b x+1) \log (-a-b x+1)}{2 b c}+\frac{\sqrt [3]{d} \log \left (\frac{b \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{\sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right ) \log (-a-b x+1)}{6 c^{4/3}}+\frac{(-1)^{2/3} \sqrt [3]{d} \log \left (-\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right ) \log (-a-b x+1)}{6 c^{4/3}}-\frac{\sqrt [3]{-1} \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} (1-a)+b \sqrt [3]{d}}\right ) \log (-a-b x+1)}{6 c^{4/3}}+\frac{(a+b x+1) \log (a+b x+1)}{2 b c}-\frac{\sqrt [3]{d} \log (a+b x+1) \log \left (-\frac{b \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{(a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{d} \log (a+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+1)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{d} \log (a+b x+1) \log \left (-\frac{b \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{(-1)^{2/3} (a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{(-1)^{2/3} \sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{c} (-a-b x+1)}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [3]{c} (-a-b x+1)}{\sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{d} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{c} (-a-b x+1)}{(-1)^{2/3} \sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [3]{c} (a+b x+1)}{(a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{d} \text{PolyLog}\left (2,\frac{(-1)^{2/3} \sqrt [3]{c} (a+b x+1)}{(-1)^{2/3} (a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{d} \text{PolyLog}\left (2,\frac{\sqrt [3]{-1} \sqrt [3]{c} (a+b x+1)}{\sqrt [3]{-1} \sqrt [3]{c} (a+1)+b \sqrt [3]{d}}\right )}{6 c^{4/3}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a + b*x]/(c + d/x^3),x]

[Out]

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

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 2389

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 2295

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

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+\frac{d}{x^3}} \, dx &=-\left (\frac{1}{2} \int \frac{\log (1-a-b x)}{c+\frac{d}{x^3}} \, dx\right )+\frac{1}{2} \int \frac{\log (1+a+b x)}{c+\frac{d}{x^3}} \, dx\\ &=-\left (\frac{1}{2} \int \left (\frac{\log (1-a-b x)}{c}-\frac{d \log (1-a-b x)}{c \left (d+c x^3\right )}\right ) \, dx\right )+\frac{1}{2} \int \left (\frac{\log (1+a+b x)}{c}-\frac{d \log (1+a+b x)}{c \left (d+c x^3\right )}\right ) \, dx\\ &=-\frac{\int \log (1-a-b x) \, dx}{2 c}+\frac{\int \log (1+a+b x) \, dx}{2 c}+\frac{d \int \frac{\log (1-a-b x)}{d+c x^3} \, dx}{2 c}-\frac{d \int \frac{\log (1+a+b x)}{d+c x^3} \, dx}{2 c}\\ &=\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1-a-b x)}{2 b c}+\frac{\operatorname{Subst}(\int \log (x) \, dx,x,1+a+b x)}{2 b c}+\frac{d \int \left (-\frac{\log (1-a-b x)}{3 d^{2/3} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}-\frac{\log (1-a-b x)}{3 d^{2/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}-\frac{\log (1-a-b x)}{3 d^{2/3} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}\right ) \, dx}{2 c}-\frac{d \int \left (-\frac{\log (1+a+b x)}{3 d^{2/3} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}-\frac{\log (1+a+b x)}{3 d^{2/3} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}-\frac{\log (1+a+b x)}{3 d^{2/3} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}\right ) \, dx}{2 c}\\ &=\frac{(1-a-b x) \log (1-a-b x)}{2 b c}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}-\frac{\sqrt [3]{d} \int \frac{\log (1-a-b x)}{-\sqrt [3]{d}-\sqrt [3]{c} x} \, dx}{6 c}-\frac{\sqrt [3]{d} \int \frac{\log (1-a-b x)}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x} \, dx}{6 c}-\frac{\sqrt [3]{d} \int \frac{\log (1-a-b x)}{-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x} \, dx}{6 c}+\frac{\sqrt [3]{d} \int \frac{\log (1+a+b x)}{-\sqrt [3]{d}-\sqrt [3]{c} x} \, dx}{6 c}+\frac{\sqrt [3]{d} \int \frac{\log (1+a+b x)}{-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x} \, dx}{6 c}+\frac{\sqrt [3]{d} \int \frac{\log (1+a+b x)}{-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x} \, dx}{6 c}\\ &=\frac{(1-a-b x) \log (1-a-b x)}{2 b c}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}-\frac{\sqrt [3]{d} \log (1+a+b x) \log \left (-\frac{b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{d} \log (1-a-b x) \log \left (\frac{b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{(-1)^{2/3} \sqrt [3]{d} \log (1-a-b x) \log \left (-\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{d} \log (1+a+b x) \log \left (\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{d} \log (1+a+b x) \log \left (-\frac{b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{d} \log (1-a-b x) \log \left (\frac{b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1-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{b \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{1+a+b x} \, dx}{6 c^{4/3}}+\frac{\left (b \sqrt [3]{d}\right ) \int \frac{\log \left (-\frac{b \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}{(1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{1-a-b x} \, dx}{6 c^{4/3}}-\frac{\left (\sqrt [3]{-1} b \sqrt [3]{d}\right ) \int \frac{\log \left (\frac{b \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{1+a+b x} \, dx}{6 c^{4/3}}-\frac{\left (\sqrt [3]{-1} b \sqrt [3]{d}\right ) \int \frac{\log \left (-\frac{b \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{1-a-b x} \, dx}{6 c^{4/3}}+\frac{\left ((-1)^{2/3} b \sqrt [3]{d}\right ) \int \frac{\log \left (\frac{b \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}{-\sqrt [3]{-1} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{1+a+b x} \, dx}{6 c^{4/3}}+\frac{\left ((-1)^{2/3} b \sqrt [3]{d}\right ) \int \frac{\log \left (-\frac{b \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}{-\sqrt [3]{-1} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{1-a-b x} \, dx}{6 c^{4/3}}\\ &=\frac{(1-a-b x) \log (1-a-b x)}{2 b c}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}-\frac{\sqrt [3]{d} \log (1+a+b x) \log \left (-\frac{b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{d} \log (1-a-b x) \log \left (\frac{b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{(-1)^{2/3} \sqrt [3]{d} \log (1-a-b x) \log \left (-\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{d} \log (1+a+b x) \log \left (\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{d} \log (1+a+b x) \log \left (-\frac{b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{d} \log (1-a-b x) \log \left (\frac{b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{d} \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [3]{c} x}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{x} \, dx,x,1+a+b x\right )}{6 c^{4/3}}-\frac{\sqrt [3]{d} \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [3]{c} x}{(1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{x} \, dx,x,1-a-b x\right )}{6 c^{4/3}}-\frac{\left (\sqrt [3]{-1} \sqrt [3]{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{(-1)^{2/3} \sqrt [3]{c} x}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{x} \, dx,x,1+a+b x\right )}{6 c^{4/3}}+\frac{\left (\sqrt [3]{-1} \sqrt [3]{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{(-1)^{2/3} \sqrt [3]{c} x}{(-1)^{2/3} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{x} \, dx,x,1-a-b x\right )}{6 c^{4/3}}+\frac{\left ((-1)^{2/3} \sqrt [3]{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{-1} \sqrt [3]{c} x}{-\sqrt [3]{-1} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{x} \, dx,x,1+a+b x\right )}{6 c^{4/3}}-\frac{\left ((-1)^{2/3} \sqrt [3]{d}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [3]{-1} \sqrt [3]{c} x}{-\sqrt [3]{-1} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{x} \, dx,x,1-a-b x\right )}{6 c^{4/3}}\\ &=\frac{(1-a-b x) \log (1-a-b x)}{2 b c}+\frac{(1+a+b x) \log (1+a+b x)}{2 b c}-\frac{\sqrt [3]{d} \log (1+a+b x) \log \left (-\frac{b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{d} \log (1-a-b x) \log \left (\frac{b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{(1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{(-1)^{2/3} \sqrt [3]{d} \log (1-a-b x) \log \left (-\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{d} \log (1+a+b x) \log \left (\frac{b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{d} \log (1+a+b x) \log \left (-\frac{b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{d} \log (1-a-b x) \log \left (\frac{b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{(-1)^{2/3} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{(-1)^{2/3} \sqrt [3]{d} \text{Li}_2\left (\frac{\sqrt [3]{-1} \sqrt [3]{c} (1-a-b x)}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{d} \text{Li}_2\left (\frac{\sqrt [3]{c} (1-a-b x)}{(1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{d} \text{Li}_2\left (\frac{(-1)^{2/3} \sqrt [3]{c} (1-a-b x)}{(-1)^{2/3} (1-a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{\sqrt [3]{d} \text{Li}_2\left (\frac{\sqrt [3]{c} (1+a+b x)}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{d} \text{Li}_2\left (\frac{(-1)^{2/3} \sqrt [3]{c} (1+a+b x)}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{6 c^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{d} \text{Li}_2\left (\frac{\sqrt [3]{-1} \sqrt [3]{c} (1+a+b x)}{\sqrt [3]{-1} (1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 c^{4/3}}\\ \end{align*}

Mathematica [C]  time = 7.93517, size = 917, normalized size = 1.1 \[ -\frac{d \text{RootSum}\left [c \text{$\#$1}^3 a^3+3 c \text{$\#$1}^2 a^3+c a^3+3 c \text{$\#$1} a^3-3 c \text{$\#$1}^3 a^2-3 c \text{$\#$1}^2 a^2+3 c a^2+3 c \text{$\#$1} a^2+3 c \text{$\#$1}^3 a-3 c \text{$\#$1}^2 a+3 c a-3 c \text{$\#$1} a-c \text{$\#$1}^3-b^3 d \text{$\#$1}^3+3 c \text{$\#$1}^2-3 b^3 d \text{$\#$1}^2+c-b^3 d-3 c \text{$\#$1}-3 b^3 d \text{$\#$1}\& ,\frac{-2 \text{$\#$1} \tanh ^{-1}(a+b x)^2+2 e^{-\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )} \sqrt{\frac{\text{$\#$1}}{(\text{$\#$1}+1)^2}} \tanh ^{-1}(a+b x)^2+2 e^{-\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )} \text{$\#$1}^2 \sqrt{\frac{\text{$\#$1}}{(\text{$\#$1}+1)^2}} \tanh ^{-1}(a+b x)^2+4 e^{-\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )} \text{$\#$1} \sqrt{\frac{\text{$\#$1}}{(\text{$\#$1}+1)^2}} \tanh ^{-1}(a+b x)^2-2 \tanh ^{-1}(a+b x)^2+2 \tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right ) \text{$\#$1}^2 \tanh ^{-1}(a+b x)+2 \log \left (1-e^{-2 \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )\right )}\right ) \text{$\#$1}^2 \tanh ^{-1}(a+b x)+i \pi \text{$\#$1}^2 \tanh ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right ) \tanh ^{-1}(a+b x)-2 \log \left (1-e^{-2 \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )\right )}\right ) \tanh ^{-1}(a+b x)-i \pi \tanh ^{-1}(a+b x)-i \pi \log \left (1+e^{2 \tanh ^{-1}(a+b x)}\right ) \text{$\#$1}^2+2 \tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )\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}}{\text{$\#$1}+1}\right ) \log \left (i \sinh \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )\right )\right ) \text{$\#$1}^2-\text{PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )\right )}\right ) \text{$\#$1}^2+i \pi \log \left (1+e^{2 \tanh ^{-1}(a+b x)}\right )-2 \tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )\right )}\right )-i \pi \log \left (\frac{1}{\sqrt{1-(a+b x)^2}}\right )+2 \tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right ) \log \left (i \sinh \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )\right )\right )+\text{PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}(a+b x)+\tanh ^{-1}\left (\frac{1-\text{$\#$1}}{\text{$\#$1}+1}\right )\right )}\right )}{c \text{$\#$1}^2 a^3+c a^3+2 c \text{$\#$1} a^3-2 c \text{$\#$1}^2 a^2+2 c a^2+c \text{$\#$1}^2 a+c a-2 c \text{$\#$1} a-b^3 d \text{$\#$1}^2-b^3 d-2 b^3 d \text{$\#$1}}\& \right ] b^3-6 (a+b x) \tanh ^{-1}(a+b x)+6 \log \left (\frac{1}{\sqrt{1-(a+b x)^2}}\right )}{6 b c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(-6*(a + b*x)*ArcTanh[a + b*x] + 6*Log[1/Sqrt[1 - (a + b*x)^2]] + b^3*d*RootSum[c + 3*a*c + 3*a^2*c + a^3*c -
b^3*d - 3*c*#1 - 3*a*c*#1 + 3*a^2*c*#1 + 3*a^3*c*#1 - 3*b^3*d*#1 + 3*c*#1^2 - 3*a*c*#1^2 - 3*a^2*c*#1^2 + 3*a
^3*c*#1^2 - 3*b^3*d*#1^2 - c*#1^3 + 3*a*c*#1^3 - 3*a^2*c*#1^3 + a^3*c*#1^3 - b^3*d*#1^3 & , ((-I)*Pi*ArcTanh[a
+ b*x] - 2*ArcTanh[a + b*x]^2 - 2*ArcTanh[a + b*x]*ArcTanh[(1 - #1)/(1 + #1)] + I*Pi*Log[1 + E^(2*ArcTanh[a +
b*x])] - 2*ArcTanh[a + b*x]*Log[1 - E^(-2*(ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]))] - 2*ArcTanh[(1 -
#1)/(1 + #1)]*Log[1 - E^(-2*(ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]))] - I*Pi*Log[1/Sqrt[1 - (a + b*x)^
2]] + 2*ArcTanh[(1 - #1)/(1 + #1)]*Log[I*Sinh[ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]]] + PolyLog[2, E^(
-2*(ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]))] - 2*ArcTanh[a + b*x]^2*#1 + I*Pi*ArcTanh[a + b*x]*#1^2 +
2*ArcTanh[a + b*x]*ArcTanh[(1 - #1)/(1 + #1)]*#1^2 - I*Pi*Log[1 + E^(2*ArcTanh[a + b*x])]*#1^2 + 2*ArcTanh[a +
b*x]*Log[1 - E^(-2*(ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]))]*#1^2 + 2*ArcTanh[(1 - #1)/(1 + #1)]*Log[
1 - E^(-2*(ArcTanh[a + b*x] + 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[I*Sinh[ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]]]*#1^2 - PolyLog[2, E^(-2*
(ArcTanh[a + b*x] + ArcTanh[(1 - #1)/(1 + #1)]))]*#1^2 + (2*ArcTanh[a + b*x]^2*Sqrt[#1/(1 + #1)^2])/E^ArcTanh[
(1 - #1)/(1 + #1)] + (4*ArcTanh[a + b*x]^2*#1*Sqrt[#1/(1 + #1)^2])/E^ArcTanh[(1 - #1)/(1 + #1)] + (2*ArcTanh[a
+ b*x]^2*#1^2*Sqrt[#1/(1 + #1)^2])/E^ArcTanh[(1 - #1)/(1 + #1)])/(a*c + 2*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*a^2*c*#1^2 + a^3*c*#1^2 - b^3*d*#1^2) & ])/(6*b*c)

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Maple [C]  time = 0.579, size = 650, normalized size = 0.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctanh(b*x+a)/c*x+1/b*arctanh(b*x+a)/c*a+2/3*b^2/c*d*sum(1/(_R1^4*a^3*c-_R1^4*b^3*d-3*_R1^4*a^2*c+3*_R1^4*a*c
+2*_R1^2*a^3*c-2*_R1^2*b^3*d-_R1^4*c-2*_R1^2*a^2*c-2*_R1^2*a*c+a^3*c-b^3*d+2*_R1^2*c+a^2*c-a*c-c)*(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*c-b^3*d-3*a^2*c+3*a*c-c)*_Z^6+(3*a^3*c-3*b^3*d-3*a^2*c-3*a*c+3*c)*_Z^4+(3*a^3*c-3*b^3*d+3*a^2*c-3*a*c-3*c)*_
Z^2+a^3*c-d*b^3+3*a^2*c+3*a*c+c))+2/3*b^2/c*d*sum(_R1^2/(_R1^4*a^3*c-_R1^4*b^3*d-3*_R1^4*a^2*c+3*_R1^4*a*c+2*_
R1^2*a^3*c-2*_R1^2*b^3*d-_R1^4*c-2*_R1^2*a^2*c-2*_R1^2*a*c+a^3*c-b^3*d+2*_R1^2*c+a^2*c-a*c-c)*(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*c-
b^3*d-3*a^2*c+3*a*c-c)*_Z^6+(3*a^3*c-3*b^3*d-3*a^2*c-3*a*c+3*c)*_Z^4+(3*a^3*c-3*b^3*d+3*a^2*c-3*a*c-3*c)*_Z^2+
a^3*c-d*b^3+3*a^2*c+3*a*c+c))+1/2/b/c*ln(b*x+a-1)+1/2/b/c*ln(b*x+a+1)

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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)/(c+d/x^3),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{x^{3} \operatorname{artanh}\left (b x + a\right )}{c x^{3} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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