Integrand size = 14, antiderivative size = 174 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^5} \, dx=-\frac {3 b c}{4 x}+\frac {1}{8} \sqrt {3} b c^{4/3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )-\frac {1}{8} \sqrt {3} b c^{4/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )+\frac {1}{4} b c^{4/3} \text {arctanh}\left (\sqrt [3]{c} x\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{4 x^4}-\frac {1}{16} b c^{4/3} \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{16} b c^{4/3} \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right ) \]
-3/4*b*c/x+1/4*b*c^(4/3)*arctanh(c^(1/3)*x)+1/4*(-a-b*arctanh(c*x^3))/x^4- 1/16*b*c^(4/3)*ln(1-c^(1/3)*x+c^(2/3)*x^2)+1/16*b*c^(4/3)*ln(1+c^(1/3)*x+c ^(2/3)*x^2)-1/8*b*c^(4/3)*arctan(-1/3*3^(1/2)+2/3*c^(1/3)*x*3^(1/2))*3^(1/ 2)-1/8*b*c^(4/3)*arctan(1/3*3^(1/2)+2/3*c^(1/3)*x*3^(1/2))*3^(1/2)
Time = 0.05 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.13 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^5} \, dx=-\frac {a}{4 x^4}-\frac {3 b c}{4 x}-\frac {1}{8} \sqrt {3} b c^{4/3} \arctan \left (\frac {-1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )-\frac {1}{8} \sqrt {3} b c^{4/3} \arctan \left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )-\frac {b \text {arctanh}\left (c x^3\right )}{4 x^4}-\frac {1}{8} b c^{4/3} \log \left (1-\sqrt [3]{c} x\right )+\frac {1}{8} b c^{4/3} \log \left (1+\sqrt [3]{c} x\right )-\frac {1}{16} b c^{4/3} \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{16} b c^{4/3} \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right ) \]
-1/4*a/x^4 - (3*b*c)/(4*x) - (Sqrt[3]*b*c^(4/3)*ArcTan[(-1 + 2*c^(1/3)*x)/ Sqrt[3]])/8 - (Sqrt[3]*b*c^(4/3)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/8 - (b *ArcTanh[c*x^3])/(4*x^4) - (b*c^(4/3)*Log[1 - c^(1/3)*x])/8 + (b*c^(4/3)*L og[1 + c^(1/3)*x])/8 - (b*c^(4/3)*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/16 + ( b*c^(4/3)*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/16
Time = 0.40 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6452, 847, 825, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^5} \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {3}{4} b c \int \frac {1}{x^2 \left (1-c^2 x^6\right )}dx-\frac {a+b \text {arctanh}\left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \int \frac {x^4}{1-c^2 x^6}dx-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 825 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \left (\frac {\int \frac {1}{1-c^{2/3} x^2}dx}{3 c^{4/3}}+\frac {\int -\frac {\sqrt [3]{c} x+1}{2 \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}dx}{3 c^{4/3}}+\frac {\int -\frac {1-\sqrt [3]{c} x}{2 \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}dx}{3 c^{4/3}}\right )-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \left (\frac {\int \frac {1}{1-c^{2/3} x^2}dx}{3 c^{4/3}}-\frac {\int \frac {\sqrt [3]{c} x+1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\int \frac {1-\sqrt [3]{c} x}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{6 c^{4/3}}\right )-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \left (-\frac {\int \frac {\sqrt [3]{c} x+1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\int \frac {1-\sqrt [3]{c} x}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \left (-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx+\frac {\int -\frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx-\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} x+1\right )}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \left (-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx-\frac {\int \frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx-\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} x+1\right )}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \left (-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx-\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx-\frac {1}{2} \int \frac {2 \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \left (-\frac {\frac {3 \int \frac {1}{-\left (1-2 \sqrt [3]{c} x\right )^2-3}d\left (1-2 \sqrt [3]{c} x\right )}{\sqrt [3]{c}}-\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {-\frac {1}{2} \int \frac {2 \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx-\frac {3 \int \frac {1}{-\left (2 \sqrt [3]{c} x+1\right )^2-3}d\left (2 \sqrt [3]{c} x+1\right )}{\sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \left (-\frac {-\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{\sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}-\frac {1}{2} \int \frac {2 \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{4 x^4}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3}{4} b c \left (c^2 \left (-\frac {\frac {\log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{\sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}-\frac {\log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{2 \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{4 x^4}\) |
-1/4*(a + b*ArcTanh[c*x^3])/x^4 + (3*b*c*(-x^(-1) + c^2*(ArcTanh[c^(1/3)*x ]/(3*c^(5/3)) - (-((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*c^(4/3)) - ((Sqrt[3]*ArcT an[(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*c^(4/3)))))/4
3.2.15.3.1 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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k *m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m)) Int[1/ (r^2 - s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1 ] && NegQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
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 = 0.38 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(-\frac {a}{4 x^{4}}-\frac {b \,\operatorname {arctanh}\left (c \,x^{3}\right )}{4 x^{4}}-\frac {3 b c}{4 x}-\frac {b c \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(172\) |
| parts | \(-\frac {a}{4 x^{4}}-\frac {b \,\operatorname {arctanh}\left (c \,x^{3}\right )}{4 x^{4}}-\frac {3 b c}{4 x}-\frac {b c \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(172\) |
| risch | \(-\frac {b \ln \left (c \,x^{3}+1\right )}{8 x^{4}}-\frac {a}{4 x^{4}}+\frac {b \ln \left (-c \,x^{3}+1\right )}{8 x^{4}}-\frac {3 b c}{4 x}-\frac {b c \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{16 \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{8 \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(189\) |
-1/4*a/x^4-1/4*b/x^4*arctanh(c*x^3)-3/4*b*c/x-1/8*b*c/(1/c)^(1/3)*ln(x-(1/ c)^(1/3))+1/16*b*c/(1/c)^(1/3)*ln(x^2+(1/c)^(1/3)*x+(1/c)^(2/3))-1/8*b*c*3 ^(1/2)/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x+1))+1/8*b*c/(1/c)^( 1/3)*ln(x+(1/c)^(1/3))-1/16*b*c/(1/c)^(1/3)*ln(x^2-(1/c)^(1/3)*x+(1/c)^(2/ 3))-1/8*b*c*3^(1/2)/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x-1))
Time = 0.26 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.13 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^5} \, dx=-\frac {2 \, \sqrt {3} b \left (-c\right )^{\frac {1}{3}} c x^{4} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-c\right )^{\frac {1}{3}} x - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, \sqrt {3} b c^{\frac {4}{3}} x^{4} \arctan \left (\frac {2}{3} \, \sqrt {3} c^{\frac {1}{3}} x - \frac {1}{3} \, \sqrt {3}\right ) + b \left (-c\right )^{\frac {1}{3}} c x^{4} \log \left (c x^{2} + \left (-c\right )^{\frac {2}{3}} x - \left (-c\right )^{\frac {1}{3}}\right ) + b c^{\frac {4}{3}} x^{4} \log \left (c x^{2} - c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right ) - 2 \, b \left (-c\right )^{\frac {1}{3}} c x^{4} \log \left (c x - \left (-c\right )^{\frac {2}{3}}\right ) - 2 \, b c^{\frac {4}{3}} x^{4} \log \left (c x + c^{\frac {2}{3}}\right ) + 12 \, b c x^{3} + 2 \, b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a}{16 \, x^{4}} \]
-1/16*(2*sqrt(3)*b*(-c)^(1/3)*c*x^4*arctan(2/3*sqrt(3)*(-c)^(1/3)*x - 1/3* sqrt(3)) + 2*sqrt(3)*b*c^(4/3)*x^4*arctan(2/3*sqrt(3)*c^(1/3)*x - 1/3*sqrt (3)) + b*(-c)^(1/3)*c*x^4*log(c*x^2 + (-c)^(2/3)*x - (-c)^(1/3)) + b*c^(4/ 3)*x^4*log(c*x^2 - c^(2/3)*x + c^(1/3)) - 2*b*(-c)^(1/3)*c*x^4*log(c*x - ( -c)^(2/3)) - 2*b*c^(4/3)*x^4*log(c*x + c^(2/3)) + 12*b*c*x^3 + 2*b*log(-(c *x^3 + 1)/(c*x^3 - 1)) + 4*a)/x^4
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^5} \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^5} \, dx=-\frac {1}{16} \, {\left ({\left (2 \, \sqrt {3} c^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) + 2 \, \sqrt {3} c^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right ) - c^{\frac {1}{3}} \log \left (c^{\frac {2}{3}} x^{2} + c^{\frac {1}{3}} x + 1\right ) + c^{\frac {1}{3}} \log \left (c^{\frac {2}{3}} x^{2} - c^{\frac {1}{3}} x + 1\right ) - 2 \, c^{\frac {1}{3}} \log \left (\frac {c^{\frac {1}{3}} x + 1}{c^{\frac {1}{3}}}\right ) + 2 \, c^{\frac {1}{3}} \log \left (\frac {c^{\frac {1}{3}} x - 1}{c^{\frac {1}{3}}}\right ) + \frac {12}{x}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x^{3}\right )}{x^{4}}\right )} b - \frac {a}{4 \, x^{4}} \]
-1/16*((2*sqrt(3)*c^(1/3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*x + c^(1/3))/c^(1/ 3)) + 2*sqrt(3)*c^(1/3)*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) - 2*c^(1/3)*log((c^(1/3)*x + 1)/c^(1/3)) + 2*c^(1/3)*log((c^ (1/3)*x - 1)/c^(1/3)) + 12/x)*c + 4*arctanh(c*x^3)/x^4)*b - 1/4*a/x^4
Time = 0.51 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^5} \, dx=-\frac {\sqrt {3} b c^{3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {\sqrt {3} b c^{3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {b c^{3} \log \left (x^{2} + \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{16 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {b c^{3} \log \left (x^{2} - \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{16 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {b c^{3} \log \left ({\left | x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {b c^{3} \log \left ({\left | x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{8 \, x^{4}} - \frac {3 \, b c x^{3} + a}{4 \, x^{4}} \]
-1/8*sqrt(3)*b*c^3*arctan(1/3*sqrt(3)*(2*x + 1/abs(c)^(1/3))*abs(c)^(1/3)) /abs(c)^(5/3) - 1/8*sqrt(3)*b*c^3*arctan(1/3*sqrt(3)*(2*x - 1/abs(c)^(1/3) )*abs(c)^(1/3))/abs(c)^(5/3) + 1/16*b*c^3*log(x^2 + x/abs(c)^(1/3) + 1/abs (c)^(2/3))/abs(c)^(5/3) - 1/16*b*c^3*log(x^2 - x/abs(c)^(1/3) + 1/abs(c)^( 2/3))/abs(c)^(5/3) + 1/8*b*c^3*log(abs(x + 1/abs(c)^(1/3)))/abs(c)^(5/3) - 1/8*b*c^3*log(abs(x - 1/abs(c)^(1/3)))/abs(c)^(5/3) - 1/8*b*log(-(c*x^3 + 1)/(c*x^3 - 1))/x^4 - 1/4*(3*b*c*x^3 + a)/x^4
Time = 3.72 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^5} \, dx=\frac {b\,\ln \left (1-c\,x^3\right )}{8\,x^4}-\frac {b\,c^{4/3}\,\left (-\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )}{2}+\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )}{2}+\mathrm {atan}\left (c^{1/3}\,x\,1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{4}-\frac {3\,b\,c}{4\,x}-\frac {b\,\ln \left (c\,x^3+1\right )}{8\,x^4}-\frac {a}{4\,x^4}-\frac {\sqrt {3}\,b\,c^{4/3}\,\left (\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )+\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )\right )}{8} \]
(b*log(1 - c*x^3))/(8*x^4) - (b*c^(4/3)*(atan((c^(1/3)*x*(3^(1/2) + 1i))/2 )/2 - atan((c^(1/3)*x*(3^(1/2) - 1i))/2)/2 + atan(c^(1/3)*x*1i))*1i)/4 - ( 3*b*c)/(4*x) - (b*log(c*x^3 + 1))/(8*x^4) - a/(4*x^4) - (3^(1/2)*b*c^(4/3) *(atan((c^(1/3)*x*(3^(1/2) - 1i))/2) + atan((c^(1/3)*x*(3^(1/2) + 1i))/2)) )/8