Integrand size = 14, antiderivative size = 165 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^3} \, dx=-\frac {1}{4} \sqrt {3} b c^{2/3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )+\frac {1}{4} \sqrt {3} b c^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )+\frac {1}{2} b c^{2/3} \text {arctanh}\left (\sqrt [3]{c} x\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{2 x^2}-\frac {1}{8} b c^{2/3} \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{8} b c^{2/3} \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right ) \]
1/2*b*c^(2/3)*arctanh(c^(1/3)*x)+1/2*(-a-b*arctanh(c*x^3))/x^2-1/8*b*c^(2/ 3)*ln(1-c^(1/3)*x+c^(2/3)*x^2)+1/8*b*c^(2/3)*ln(1+c^(1/3)*x+c^(2/3)*x^2)+1 /4*b*c^(2/3)*arctan(-1/3*3^(1/2)+2/3*c^(1/3)*x*3^(1/2))*3^(1/2)+1/4*b*c^(2 /3)*arctan(1/3*3^(1/2)+2/3*c^(1/3)*x*3^(1/2))*3^(1/2)
Time = 0.06 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.13 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^3} \, dx=-\frac {a}{2 x^2}+\frac {1}{4} \sqrt {3} b c^{2/3} \arctan \left (\frac {-1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )+\frac {1}{4} \sqrt {3} b c^{2/3} \arctan \left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )-\frac {b \text {arctanh}\left (c x^3\right )}{2 x^2}-\frac {1}{4} b c^{2/3} \log \left (1-\sqrt [3]{c} x\right )+\frac {1}{4} b c^{2/3} \log \left (1+\sqrt [3]{c} x\right )-\frac {1}{8} b c^{2/3} \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{8} b c^{2/3} \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right ) \]
-1/2*a/x^2 + (Sqrt[3]*b*c^(2/3)*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/4 + (S qrt[3]*b*c^(2/3)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/4 - (b*ArcTanh[c*x^3]) /(2*x^2) - (b*c^(2/3)*Log[1 - c^(1/3)*x])/4 + (b*c^(2/3)*Log[1 + c^(1/3)*x ])/4 - (b*c^(2/3)*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/8 + (b*c^(2/3)*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/8
Time = 0.37 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6452, 754, 27, 219, 1142, 25, 27, 1082, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {3}{2} b c \int \frac {1}{1-c^2 x^6}dx-\frac {a+b \text {arctanh}\left (c x^3\right )}{2 x^2}\) |
| \(\Big \downarrow \) 754 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{3} \int \frac {1}{1-c^{2/3} x^2}dx+\frac {1}{3} \int \frac {2-\sqrt [3]{c} x}{2 \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}dx+\frac {1}{3} \int \frac {\sqrt [3]{c} x+2}{2 \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}dx\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{3} \int \frac {1}{1-c^{2/3} x^2}dx+\frac {1}{6} \int \frac {2-\sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx+\frac {1}{6} \int \frac {\sqrt [3]{c} x+2}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{2 x^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \int \frac {2-\sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx+\frac {1}{6} \int \frac {\sqrt [3]{c} x+2}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{2 x^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx-\frac {\int -\frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx+\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} x+1\right )}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{2 x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx+\frac {\int \frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx+\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} x+1\right )}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx+\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx\right )+\frac {1}{6} \left (\frac {3}{2} \int \frac {1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx+\frac {1}{2} \int \frac {2 \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{2 x^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx+\frac {3 \int \frac {1}{-\left (1-2 \sqrt [3]{c} x\right )^2-3}d\left (1-2 \sqrt [3]{c} x\right )}{\sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx-\frac {3 \int \frac {1}{-\left (2 \sqrt [3]{c} x+1\right )^2-3}d\left (2 \sqrt [3]{c} x+1\right )}{\sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{2 x^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{\sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {2 \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx+\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{2 x^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \left (-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{\sqrt [3]{c}}-\frac {\log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{2 \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}+\frac {\log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{2 \sqrt [3]{c}}\right )+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{2 x^2}\) |
-1/2*(a + b*ArcTanh[c*x^3])/x^2 + (3*b*c*(ArcTanh[c^(1/3)*x]/(3*c^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - 2*c^(1/3)*x)/Sqrt[3]])/c^(1/3)) - Log[1 - c^(1/3) *x + c^(2/3)*x^2]/(2*c^(1/3)))/6 + ((Sqrt[3]*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt [3]])/c^(1/3) + Log[1 + c^(1/3)*x + c^(2/3)*x^2]/(2*c^(1/3)))/6))/2
3.2.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a /b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 *k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] / ; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(-\frac {a}{2 x^{2}}-\frac {b \,\operatorname {arctanh}\left (c \,x^{3}\right )}{2 x^{2}}-\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}\) | \(159\) |
| parts | \(-\frac {a}{2 x^{2}}-\frac {b \,\operatorname {arctanh}\left (c \,x^{3}\right )}{2 x^{2}}-\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}\) | \(159\) |
| risch | \(-\frac {b \ln \left (c \,x^{3}+1\right )}{4 x^{2}}-\frac {a}{2 x^{2}}+\frac {b \ln \left (-c \,x^{3}+1\right )}{4 x^{2}}-\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}\) | \(176\) |
-1/2*a/x^2-1/2*b/x^2*arctanh(c*x^3)-1/4*b/(1/c)^(2/3)*ln(x-(1/c)^(1/3))+1/ 8*b/(1/c)^(2/3)*ln(x^2+(1/c)^(1/3)*x+(1/c)^(2/3))+1/4*b/(1/c)^(2/3)*3^(1/2 )*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x+1))+1/4*b/(1/c)^(2/3)*ln(x+(1/c)^(1/ 3))-1/8*b/(1/c)^(2/3)*ln(x^2-(1/c)^(1/3)*x+(1/c)^(2/3))+1/4*b/(1/c)^(2/3)* 3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x-1))
Time = 0.27 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.38 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^3} \, dx=-\frac {2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {1}{3}} b x^{2} \arctan \left (\frac {2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {2}{3}} x + \sqrt {3} c}{3 \, c}\right ) - 2 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{3}} x^{2} \arctan \left (\frac {2 \, \sqrt {3} {\left (c^{2}\right )}^{\frac {2}{3}} x - \sqrt {3} c}{3 \, c}\right ) + \left (-c^{2}\right )^{\frac {1}{3}} b x^{2} \log \left (c^{2} x^{2} - \left (-c^{2}\right )^{\frac {1}{3}} c x + \left (-c^{2}\right )^{\frac {2}{3}}\right ) + b {\left (c^{2}\right )}^{\frac {1}{3}} x^{2} \log \left (c^{2} x^{2} - {\left (c^{2}\right )}^{\frac {1}{3}} c x + {\left (c^{2}\right )}^{\frac {2}{3}}\right ) - 2 \, \left (-c^{2}\right )^{\frac {1}{3}} b x^{2} \log \left (c x + \left (-c^{2}\right )^{\frac {1}{3}}\right ) - 2 \, b {\left (c^{2}\right )}^{\frac {1}{3}} x^{2} \log \left (c x + {\left (c^{2}\right )}^{\frac {1}{3}}\right ) + 2 \, b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a}{8 \, x^{2}} \]
-1/8*(2*sqrt(3)*(-c^2)^(1/3)*b*x^2*arctan(1/3*(2*sqrt(3)*(-c^2)^(2/3)*x + sqrt(3)*c)/c) - 2*sqrt(3)*b*(c^2)^(1/3)*x^2*arctan(1/3*(2*sqrt(3)*(c^2)^(2 /3)*x - sqrt(3)*c)/c) + (-c^2)^(1/3)*b*x^2*log(c^2*x^2 - (-c^2)^(1/3)*c*x + (-c^2)^(2/3)) + b*(c^2)^(1/3)*x^2*log(c^2*x^2 - (c^2)^(1/3)*c*x + (c^2)^ (2/3)) - 2*(-c^2)^(1/3)*b*x^2*log(c*x + (-c^2)^(1/3)) - 2*b*(c^2)^(1/3)*x^ 2*log(c*x + (c^2)^(1/3)) + 2*b*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 4*a)/x^2
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^3} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^3} \, dx=\frac {1}{8} \, {\left ({\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {\log \left (c^{\frac {2}{3}} x^{2} + c^{\frac {1}{3}} x + 1\right )}{c^{\frac {1}{3}}} - \frac {\log \left (c^{\frac {2}{3}} x^{2} - c^{\frac {1}{3}} x + 1\right )}{c^{\frac {1}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} x + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} x - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}\right )} c - \frac {4 \, \operatorname {artanh}\left (c x^{3}\right )}{x^{2}}\right )} b - \frac {a}{2 \, x^{2}} \]
1/8*((2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*x + c^(1/3))/c^(1/3))/c^(1/3 ) + 2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*x - c^(1/3))/c^(1/3))/c^(1/3) + log(c^(2/3)*x^2 + c^(1/3)*x + 1)/c^(1/3) - log(c^(2/3)*x^2 - c^(1/3)*x + 1)/c^(1/3) + 2*log((c^(1/3)*x + 1)/c^(1/3))/c^(1/3) - 2*log((c^(1/3)*x - 1)/c^(1/3))/c^(1/3))*c - 4*arctanh(c*x^3)/x^2)*b - 1/2*a/x^2
Time = 0.31 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^3} \, dx=\frac {1}{8} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {\log \left (x^{2} + \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} - \frac {\log \left (x^{2} - \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \log \left ({\left | x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{{\left | c \right |}^{\frac {1}{3}}} - \frac {2 \, \log \left ({\left | x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{{\left | c \right |}^{\frac {1}{3}}}\right )} b c - \frac {b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{4 \, x^{2}} - \frac {a}{2 \, x^{2}} \]
1/8*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1/abs(c)^(1/3))*abs(c)^(1/3))/abs (c)^(1/3) + 2*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1/abs(c)^(1/3))*abs(c)^(1/ 3))/abs(c)^(1/3) + log(x^2 + x/abs(c)^(1/3) + 1/abs(c)^(2/3))/abs(c)^(1/3) - log(x^2 - x/abs(c)^(1/3) + 1/abs(c)^(2/3))/abs(c)^(1/3) + 2*log(abs(x + 1/abs(c)^(1/3)))/abs(c)^(1/3) - 2*log(abs(x - 1/abs(c)^(1/3)))/abs(c)^(1/ 3))*b*c - 1/4*b*log(-(c*x^3 + 1)/(c*x^3 - 1))/x^2 - 1/2*a/x^2
Time = 4.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^3} \, dx=\frac {b\,\ln \left (1-c\,x^3\right )}{4\,x^2}-\frac {b\,c^{2/3}\,\left (-\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )}{2}+\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )}{2}+\mathrm {atan}\left (c^{1/3}\,x\,1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{2}-\frac {b\,\ln \left (c\,x^3+1\right )}{4\,x^2}-\frac {a}{2\,x^2}+\frac {\sqrt {3}\,b\,c^{2/3}\,\left (\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )+\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )\right )}{4} \]