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 {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (\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 {Li}_2\left (\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]

1/2*(-b*x-a+1)*ln(-b*x-a+1)/b/c+1/2*(b*x+a+1)*ln(b*x+a+1)/b/c-1/6*d^(1/3)*ln(b*x+a+1)*ln(-b*(d^(1/3)+c^(1/3)*x
)/((1+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)+1/6*d^(1/3)*ln(-b*x-a+1)*ln(b*(d^(1/3)+c^(1/3)*x)/((1-a)*c^(1/3)+b*d^(1/3
)))/c^(4/3)+1/6*(-1)^(2/3)*d^(1/3)*ln(-b*x-a+1)*ln(-b*(d^(1/3)-(-1)^(1/3)*c^(1/3)*x)/((-1)^(1/3)*(1-a)*c^(1/3)
-b*d^(1/3)))/c^(4/3)-1/6*(-1)^(2/3)*d^(1/3)*ln(b*x+a+1)*ln(b*(d^(1/3)-(-1)^(1/3)*c^(1/3)*x)/((-1)^(1/3)*(1+a)*
c^(1/3)+b*d^(1/3)))/c^(4/3)+1/6*(-1)^(1/3)*d^(1/3)*ln(b*x+a+1)*ln(-b*(d^(1/3)+(-1)^(2/3)*c^(1/3)*x)/((-1)^(2/3
)*(1+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)-1/6*(-1)^(1/3)*d^(1/3)*ln(-b*x-a+1)*ln(b*(d^(1/3)+(-1)^(2/3)*c^(1/3)*x)/((
-1)^(2/3)*(1-a)*c^(1/3)+b*d^(1/3)))/c^(4/3)+1/6*(-1)^(2/3)*d^(1/3)*polylog(2,(-1)^(1/3)*c^(1/3)*(-b*x-a+1)/((-
1)^(1/3)*(1-a)*c^(1/3)-b*d^(1/3)))/c^(4/3)+1/6*d^(1/3)*polylog(2,c^(1/3)*(-b*x-a+1)/((1-a)*c^(1/3)+b*d^(1/3)))
/c^(4/3)-1/6*(-1)^(1/3)*d^(1/3)*polylog(2,(-1)^(2/3)*c^(1/3)*(-b*x-a+1)/((-1)^(2/3)*(1-a)*c^(1/3)+b*d^(1/3)))/
c^(4/3)-1/6*d^(1/3)*polylog(2,c^(1/3)*(b*x+a+1)/((1+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)+1/6*(-1)^(1/3)*d^(1/3)*poly
log(2,(-1)^(2/3)*c^(1/3)*(b*x+a+1)/((-1)^(2/3)*(1+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)-1/6*(-1)^(2/3)*d^(1/3)*polylo
g(2,(-1)^(1/3)*c^(1/3)*(b*x+a+1)/((-1)^(1/3)*(1+a)*c^(1/3)+b*d^(1/3)))/c^(4/3)

________________________________________________________________________________________

Rubi [A]  time = 1.43, antiderivative size = 832, 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 = {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 2295

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

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 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]

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 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 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 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]

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*}

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Mathematica [C]  time = 8.06, size = 917, normalized size = 1.10 \[ -\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 {Li}_2\left (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 {Li}_2\left (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]

-1/6*(-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*ArcTa
nh[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^ArcT
anh[(1 - #1)/(1 + #1)] + (4*ArcTanh[a + b*x]^2*#1*Sqrt[#1/(1 + #1)^2])/E^ArcTanh[(1 - #1)/(1 + #1)] + (2*ArcTa
nh[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) & ])/(b*c)

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \operatorname {artanh}\left (b x + a\right )}{c x^{3} + d}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [C]  time = 0.89, size = 650, normalized size = 0.78 \[ \frac {\arctanh \left (b x +a \right ) x}{c}+\frac {\arctanh \left (b x +a \right ) a}{b c}+\frac {2 b^{2} d \left (\munderset {\textit {\_R1} =\RootOf \left (\left (a^{3} c -d \,b^{3}-3 a^{2} c +3 a c -c \right ) \textit {\_Z}^{6}+\left (3 a^{3} c -3 d \,b^{3}-3 a^{2} c -3 a c +3 c \right ) \textit {\_Z}^{4}+\left (3 a^{3} c -3 d \,b^{3}+3 a^{2} c -3 a c -3 c \right ) \textit {\_Z}^{2}+a^{3} c -d \,b^{3}+3 a^{2} c +3 a c +c \right )}{\sum }\frac {\arctanh \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1}^{4} a^{3} c -\textit {\_R1}^{4} b^{3} d -3 \textit {\_R1}^{4} a^{2} c +3 \textit {\_R1}^{4} a c +2 \textit {\_R1}^{2} a^{3} c -2 \textit {\_R1}^{2} b^{3} d -\textit {\_R1}^{4} c -2 \textit {\_R1}^{2} a^{2} c -2 \textit {\_R1}^{2} a c +a^{3} c -d \,b^{3}+2 \textit {\_R1}^{2} c +a^{2} c -a c -c}\right )}{3 c}+\frac {2 b^{2} d \left (\munderset {\textit {\_R1} =\RootOf \left (\left (a^{3} c -d \,b^{3}-3 a^{2} c +3 a c -c \right ) \textit {\_Z}^{6}+\left (3 a^{3} c -3 d \,b^{3}-3 a^{2} c -3 a c +3 c \right ) \textit {\_Z}^{4}+\left (3 a^{3} c -3 d \,b^{3}+3 a^{2} c -3 a c -3 c \right ) \textit {\_Z}^{2}+a^{3} c -d \,b^{3}+3 a^{2} c +3 a c +c \right )}{\sum }\frac {\textit {\_R1}^{2} \left (\arctanh \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{4} a^{3} c -\textit {\_R1}^{4} b^{3} d -3 \textit {\_R1}^{4} a^{2} c +3 \textit {\_R1}^{4} a c +2 \textit {\_R1}^{2} a^{3} c -2 \textit {\_R1}^{2} b^{3} d -\textit {\_R1}^{4} c -2 \textit {\_R1}^{2} a^{2} c -2 \textit {\_R1}^{2} a c +a^{3} c -d \,b^{3}+2 \textit {\_R1}^{2} c +a^{2} c -a c -c}\right )}{3 c}+\frac {\ln \left (b x +a -1\right )}{2 b c}+\frac {\ln \left (b x +a +1\right )}{2 b c} \]

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]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {artanh}\left (b x + a\right )}{c + \frac {d}{x^{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {atanh}\left (a+b\,x\right )}{c+\frac {d}{x^3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a + b*x)/(c + d/x^3),x)

[Out]

int(atanh(a + b*x)/(c + d/x^3), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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