Integrand size = 14, antiderivative size = 117 \[ \int x^4 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {3 b x^2}{10 c}-\frac {\sqrt {3} b \arctan \left (\frac {1+2 c^{2/3} x^2}{\sqrt {3}}\right )}{10 c^{5/3}}+\frac {1}{5} x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )+\frac {b \log \left (1-c^{2/3} x^2\right )}{10 c^{5/3}}-\frac {b \log \left (1+c^{2/3} x^2+c^{4/3} x^4\right )}{20 c^{5/3}} \]
3/10*b*x^2/c+1/5*x^5*(a+b*arctanh(c*x^3))+1/10*b*ln(1-c^(2/3)*x^2)/c^(5/3) -1/20*b*ln(1+c^(2/3)*x^2+c^(4/3)*x^4)/c^(5/3)-1/10*b*arctan(1/3*(1+2*c^(2/ 3)*x^2)*3^(1/2))*3^(1/2)/c^(5/3)
Time = 0.03 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.69 \[ \int x^4 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {3 b x^2}{10 c}+\frac {a x^5}{5}-\frac {\sqrt {3} b \arctan \left (\frac {-1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{10 c^{5/3}}+\frac {\sqrt {3} b \arctan \left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{10 c^{5/3}}+\frac {1}{5} b x^5 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-\sqrt [3]{c} x\right )}{10 c^{5/3}}+\frac {b \log \left (1+\sqrt [3]{c} x\right )}{10 c^{5/3}}-\frac {b \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{20 c^{5/3}}-\frac {b \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{20 c^{5/3}} \]
(3*b*x^2)/(10*c) + (a*x^5)/5 - (Sqrt[3]*b*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3 ]])/(10*c^(5/3)) + (Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/(10*c^(5/ 3)) + (b*x^5*ArcTanh[c*x^3])/5 + (b*Log[1 - c^(1/3)*x])/(10*c^(5/3)) + (b* Log[1 + c^(1/3)*x])/(10*c^(5/3)) - (b*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(2 0*c^(5/3)) - (b*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(20*c^(5/3))
Time = 0.33 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {6452, 807, 843, 750, 16, 1142, 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.
| \(\displaystyle \int x^4 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{5} x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{5} b c \int \frac {x^7}{1-c^2 x^6}dx\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {1}{5} x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{10} b c \int \frac {x^6}{1-c^2 x^6}dx^2\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {1}{5} x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{10} b c \left (\frac {\int \frac {1}{1-c^2 x^6}dx^2}{c^2}-\frac {x^2}{c^2}\right )\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{5} x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{10} b c \left (\frac {\frac {1}{3} \int \frac {1}{1-c^{2/3} x^2}dx^2+\frac {1}{3} \int \frac {c^{2/3} x^2+2}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2}{c^2}-\frac {x^2}{c^2}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{5} x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{10} b c \left (\frac {\frac {1}{3} \int \frac {c^{2/3} x^2+2}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2-\frac {\log \left (1-c^{2/3} x^2\right )}{3 c^{2/3}}}{c^2}-\frac {x^2}{c^2}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{5} x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{10} b c \left (\frac {\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2+\frac {\int \frac {c^{2/3} \left (2 c^{2/3} x^2+1\right )}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2}{2 c^{2/3}}\right )-\frac {\log \left (1-c^{2/3} x^2\right )}{3 c^{2/3}}}{c^2}-\frac {x^2}{c^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{10} b c \left (\frac {\frac {1}{3} \left (\frac {3}{2} \int \frac {1}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2+\frac {1}{2} \int \frac {2 c^{2/3} x^2+1}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2\right )-\frac {\log \left (1-c^{2/3} x^2\right )}{3 c^{2/3}}}{c^2}-\frac {x^2}{c^2}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{5} x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{10} b c \left (\frac {\frac {1}{3} \left (\frac {1}{2} \int \frac {2 c^{2/3} x^2+1}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2-\frac {3 \int \frac {1}{-x^4-3}d\left (2 c^{2/3} x^2+1\right )}{c^{2/3}}\right )-\frac {\log \left (1-c^{2/3} x^2\right )}{3 c^{2/3}}}{c^2}-\frac {x^2}{c^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{5} x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{10} b c \left (\frac {\frac {1}{3} \left (\frac {1}{2} \int \frac {2 c^{2/3} x^2+1}{c^{4/3} x^4+c^{2/3} x^2+1}dx^2+\frac {\sqrt {3} \arctan \left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right )}{c^{2/3}}\right )-\frac {\log \left (1-c^{2/3} x^2\right )}{3 c^{2/3}}}{c^2}-\frac {x^2}{c^2}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{5} x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{10} b c \left (\frac {\frac {1}{3} \left (\frac {\sqrt {3} \arctan \left (\frac {2 c^{2/3} x^2+1}{\sqrt {3}}\right )}{c^{2/3}}+\frac {\log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )}{2 c^{2/3}}\right )-\frac {\log \left (1-c^{2/3} x^2\right )}{3 c^{2/3}}}{c^2}-\frac {x^2}{c^2}\right )\) |
(x^5*(a + b*ArcTanh[c*x^3]))/5 - (3*b*c*(-(x^2/c^2) + (-1/3*Log[1 - c^(2/3 )*x^2]/c^(2/3) + ((Sqrt[3]*ArcTan[(1 + 2*c^(2/3)*x^2)/Sqrt[3]])/c^(2/3) + Log[1 + c^(2/3)*x^2 + c^(4/3)*x^4]/(2*c^(2/3)))/3)/c^2))/10
3.2.12.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && 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 = 0.46 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(\frac {a \,x^{5}}{5}+\frac {b \,x^{5} \operatorname {arctanh}\left (c \,x^{3}\right )}{5}+\frac {3 b \,x^{2}}{10 c}+\frac {b \ln \left (x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{10 c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x^{4}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{20 c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{10 c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}\) | \(114\) |
| parts | \(\frac {a \,x^{5}}{5}+\frac {b \,x^{5} \operatorname {arctanh}\left (c \,x^{3}\right )}{5}+\frac {3 b \,x^{2}}{10 c}+\frac {b \ln \left (x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{10 c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x^{4}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}} x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}\right )}{20 c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{10 c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {2}{3}}}\) | \(114\) |
| risch | \(\frac {b \,x^{5} \ln \left (c \,x^{3}+1\right )}{10}+\frac {a \,x^{5}}{5}-\frac {b \,x^{5} \ln \left (-c \,x^{3}+1\right )}{10}+\frac {3 b \,x^{2}}{10 c}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{10 c^{2} \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{20 c^{2} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{10 c^{2} \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{10 c^{2} \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{20 c^{2} \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{10 c^{2} \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(203\) |
1/5*a*x^5+1/5*b*x^5*arctanh(c*x^3)+3/10*b*x^2/c+1/10*b/c^3/(1/c^2)^(2/3)*l n(x^2-(1/c^2)^(1/3))-1/20*b/c^3/(1/c^2)^(2/3)*ln(x^4+(1/c^2)^(1/3)*x^2+(1/ c^2)^(2/3))-1/10*b/c^3/(1/c^2)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c^2) ^(1/3)*x^2+1))
Time = 0.25 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.42 \[ \int x^4 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {2 \, b c^{3} x^{5} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a c^{3} x^{5} + 6 \, b c^{2} x^{2} - 2 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{6}} c \arctan \left (-\frac {\sqrt {3} {\left (4 \, c^{2} x^{4} - 2 \, {\left (c^{2}\right )}^{\frac {2}{3}} x^{2} + {\left (c^{2}\right )}^{\frac {1}{3}}\right )} {\left (c^{2}\right )}^{\frac {1}{6}}}{8 \, c^{3} x^{6} + c}\right ) - b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left (c^{2} x^{4} + {\left (c^{2}\right )}^{\frac {2}{3}} x^{2} + {\left (c^{2}\right )}^{\frac {1}{3}}\right ) + 2 \, b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left (c^{2} x^{2} - {\left (c^{2}\right )}^{\frac {2}{3}}\right )}{20 \, c^{3}} \]
1/20*(2*b*c^3*x^5*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 4*a*c^3*x^5 + 6*b*c^2*x^ 2 - 2*sqrt(3)*b*(c^2)^(1/6)*c*arctan(-sqrt(3)*(4*c^2*x^4 - 2*(c^2)^(2/3)*x ^2 + (c^2)^(1/3))*(c^2)^(1/6)/(8*c^3*x^6 + c)) - b*(c^2)^(2/3)*log(c^2*x^4 + (c^2)^(2/3)*x^2 + (c^2)^(1/3)) + 2*b*(c^2)^(2/3)*log(c^2*x^2 - (c^2)^(2 /3)))/c^3
Timed out. \[ \int x^4 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.88 \[ \int x^4 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {1}{5} \, a x^{5} + \frac {1}{20} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x^{3}\right ) + c {\left (\frac {6 \, x^{2}}{c^{2}} - \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {4}{3}} x^{2} + c^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {8}{3}}} - \frac {\log \left (c^{\frac {4}{3}} x^{4} + c^{\frac {2}{3}} x^{2} + 1\right )}{c^{\frac {8}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {2}{3}} x^{2} - 1}{c^{\frac {2}{3}}}\right )}{c^{\frac {8}{3}}}\right )}\right )} b \]
1/5*a*x^5 + 1/20*(4*x^5*arctanh(c*x^3) + c*(6*x^2/c^2 - 2*sqrt(3)*arctan(1 /3*sqrt(3)*(2*c^(4/3)*x^2 + c^(2/3))/c^(2/3))/c^(8/3) - log(c^(4/3)*x^4 + c^(2/3)*x^2 + 1)/c^(8/3) + 2*log((c^(2/3)*x^2 - 1)/c^(2/3))/c^(8/3)))*b
Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.08 \[ \int x^4 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=-\frac {1}{20} \, b c^{9} {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )} {\left | c \right |}^{\frac {2}{3}}\right )}{c^{10} {\left | c \right |}^{\frac {2}{3}}} + \frac {\log \left (x^{4} + \frac {x^{2}}{{\left | c \right |}^{\frac {2}{3}}} + \frac {1}{{\left | c \right |}^{\frac {4}{3}}}\right )}{c^{10} {\left | c \right |}^{\frac {2}{3}}} - \frac {2 \, \log \left ({\left | x^{2} - \frac {1}{{\left | c \right |}^{\frac {2}{3}}} \right |}\right )}{c^{10} {\left | c \right |}^{\frac {2}{3}}}\right )} + \frac {1}{10} \, b x^{5} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + \frac {1}{5} \, a x^{5} + \frac {3 \, b x^{2}}{10 \, c} \]
-1/20*b*c^9*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1/abs(c)^(2/3))*abs(c)^ (2/3))/(c^10*abs(c)^(2/3)) + log(x^4 + x^2/abs(c)^(2/3) + 1/abs(c)^(4/3))/ (c^10*abs(c)^(2/3)) - 2*log(abs(x^2 - 1/abs(c)^(2/3)))/(c^10*abs(c)^(2/3)) ) + 1/10*b*x^5*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 1/5*a*x^5 + 3/10*b*x^2/c
Time = 4.95 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.06 \[ \int x^4 \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {a\,x^5}{5}+\frac {b\,\ln \left (1-c^{2/3}\,x^2\right )}{10\,c^{5/3}}+\frac {3\,b\,x^2}{10\,c}-\frac {\ln \left (2\,c^{2/3}\,x^2+1-\sqrt {3}\,1{}\mathrm {i}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{20\,c^{5/3}}-\frac {\ln \left (2\,c^{2/3}\,x^2+1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{20\,c^{5/3}}+\frac {b\,x^5\,\ln \left (c\,x^3+1\right )}{10}-\frac {b\,x^5\,\ln \left (1-c\,x^3\right )}{10} \]