Integrand size = 12, antiderivative size = 131 \[ \int x \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=-\frac {\sqrt {3} b \arctan \left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}+\frac {\sqrt {3} b \arctan \left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}-\frac {b \text {arctanh}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {b \text {arctanh}\left (\frac {\sqrt [3]{c} x}{1+c^{2/3} x^2}\right )}{4 c^{2/3}} \] Output:
-1/4*3^(1/2)*b*arctan(1/3*(1-2*c^(1/3)*x)*3^(1/2))/c^(2/3)+1/4*3^(1/2)*b*a rctan(1/3*(1+2*c^(1/3)*x)*3^(1/2))/c^(2/3)-1/2*b*arctanh(c^(1/3)*x)/c^(2/3 )+1/2*x^2*(a+b*arctanh(c*x^3))-1/4*b*arctanh(c^(1/3)*x/(1+c^(2/3)*x^2))/c^ (2/3)
Time = 0.03 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.43 \[ \int x \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {a x^2}{2}+\frac {\sqrt {3} b \arctan \left (\frac {-1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}+\frac {\sqrt {3} b \arctan \left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 c^{2/3}}+\frac {1}{2} b x^2 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-\sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b \log \left (1+\sqrt [3]{c} x\right )}{4 c^{2/3}}+\frac {b \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}} \] Input:
Integrate[x*(a + b*ArcTanh[c*x^3]),x]
Output:
(a*x^2)/2 + (Sqrt[3]*b*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3)) + ( Sqrt[3]*b*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/(4*c^(2/3)) + (b*x^2*ArcTanh[ c*x^3])/2 + (b*Log[1 - c^(1/3)*x])/(4*c^(2/3)) - (b*Log[1 + c^(1/3)*x])/(4 *c^(2/3)) + (b*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3)) - (b*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(8*c^(2/3))
Time = 0.55 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.34, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6452, 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.
| \(\displaystyle \int x \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{2} b c \int \frac {x^4}{1-c^2 x^6}dx\) |
| \(\Big \downarrow \) 825 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{2} b c \left (\frac {\int \frac {1}{1-c^{2/3} x^2}dx}{3 c^{4/3}}+\frac {\int -\frac {\sqrt [3]{c} x+1}{2 \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}dx}{3 c^{4/3}}+\frac {\int -\frac {1-\sqrt [3]{c} x}{2 \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}dx}{3 c^{4/3}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{2} b c \left (\frac {\int \frac {1}{1-c^{2/3} x^2}dx}{3 c^{4/3}}-\frac {\int \frac {\sqrt [3]{c} x+1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\int \frac {1-\sqrt [3]{c} x}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{6 c^{4/3}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {\int \frac {\sqrt [3]{c} x+1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\int \frac {1-\sqrt [3]{c} x}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx+\frac {\int -\frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx-\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} x+1\right )}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx-\frac {\int \frac {\sqrt [3]{c} \left (1-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx-\frac {\int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} x+1\right )}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx-\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\frac {3}{2} \int \frac {1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx-\frac {1}{2} \int \frac {2 \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {\frac {3 \int \frac {1}{-\left (1-2 \sqrt [3]{c} x\right )^2-3}d\left (1-2 \sqrt [3]{c} x\right )}{\sqrt [3]{c}}-\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {-\frac {1}{2} \int \frac {2 \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx-\frac {3 \int \frac {1}{-\left (2 \sqrt [3]{c} x+1\right )^2-3}d\left (2 \sqrt [3]{c} x+1\right )}{\sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {-\frac {1}{2} \int \frac {1-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt [3]{c} x+1}dx-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{\sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}-\frac {1}{2} \int \frac {2 \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {\frac {\log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{2 \sqrt [3]{c}}-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{\sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right )}{\sqrt [3]{c}}-\frac {\log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{2 \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\text {arctanh}\left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
Input:
Int[x*(a + b*ArcTanh[c*x^3]),x]
Output:
(x^2*(a + b*ArcTanh[c*x^3]))/2 - (3*b*c*(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]*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))))/2
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[((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 = 0.28 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.35
| method | result | size |
| default | \(\frac {a \,x^{2}}{2}+\frac {b \,x^{2} \operatorname {arctanh}\left (c \,x^{3}\right )}{2}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(177\) |
| parts | \(\frac {a \,x^{2}}{2}+\frac {b \,x^{2} \operatorname {arctanh}\left (c \,x^{3}\right )}{2}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(177\) |
| risch | \(\frac {x^{2} b \ln \left (c \,x^{3}+1\right )}{4}+\frac {a \,x^{2}}{2}-\frac {b \,x^{2} \ln \left (-c \,x^{3}+1\right )}{4}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(194\) |
Input:
int(x*(a+b*arctanh(c*x^3)),x,method=_RETURNVERBOSE)
Output:
1/2*a*x^2+1/2*b*x^2*arctanh(c*x^3)+1/4*b/c/(1/c)^(1/3)*ln(x-(1/c)^(1/3))-1 /8*b/c/(1/c)^(1/3)*ln(x^2+(1/c)^(1/3)*x+(1/c)^(2/3))+1/4*b*3^(1/2)/c/(1/c) ^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x+1))-1/4*b/c/(1/c)^(1/3)*ln(x+(1 /c)^(1/3))+1/8*b/c/(1/c)^(1/3)*ln(x^2-(1/c)^(1/3)*x+(1/c)^(2/3))+1/4*b*3^( 1/2)/c/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x-1))
Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (97) = 194\).
Time = 0.08 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.82 \[ \int x \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {2 \, b c^{2} x^{2} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a c^{2} x^{2} + 2 \, \sqrt {3} b c \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}} \arctan \left (\frac {\sqrt {3} {\left (2 \, c x + \left (-c^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\left (-c^{2}\right )^{\frac {1}{3}}}}{3 \, c}\right ) + 2 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{6}} c \arctan \left (\frac {\sqrt {3} {\left (c^{2}\right )}^{\frac {1}{6}} {\left (2 \, c x + {\left (c^{2}\right )}^{\frac {1}{3}}\right )}}{3 \, c}\right ) + \left (-c^{2}\right )^{\frac {2}{3}} b \log \left (c^{2} x^{2} + \left (-c^{2}\right )^{\frac {1}{3}} c x + \left (-c^{2}\right )^{\frac {2}{3}}\right ) - b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left (c^{2} x^{2} + {\left (c^{2}\right )}^{\frac {1}{3}} c x + {\left (c^{2}\right )}^{\frac {2}{3}}\right ) - 2 \, \left (-c^{2}\right )^{\frac {2}{3}} b \log \left (c x - \left (-c^{2}\right )^{\frac {1}{3}}\right ) + 2 \, b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left (c x - {\left (c^{2}\right )}^{\frac {1}{3}}\right )}{8 \, c^{2}} \] Input:
integrate(x*(a+b*arctanh(c*x^3)),x, algorithm="fricas")
Output:
1/8*(2*b*c^2*x^2*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 4*a*c^2*x^2 + 2*sqrt(3)*b *c*sqrt(-(-c^2)^(1/3))*arctan(1/3*sqrt(3)*(2*c*x + (-c^2)^(1/3))*sqrt(-(-c ^2)^(1/3))/c) + 2*sqrt(3)*b*(c^2)^(1/6)*c*arctan(1/3*sqrt(3)*(c^2)^(1/6)*( 2*c*x + (c^2)^(1/3))/c) + (-c^2)^(2/3)*b*log(c^2*x^2 + (-c^2)^(1/3)*c*x + (-c^2)^(2/3)) - b*(c^2)^(2/3)*log(c^2*x^2 + (c^2)^(1/3)*c*x + (c^2)^(2/3)) - 2*(-c^2)^(2/3)*b*log(c*x - (-c^2)^(1/3)) + 2*b*(c^2)^(2/3)*log(c*x - (c ^2)^(1/3)))/c^2
Timed out. \[ \int x \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\text {Timed out} \] Input:
integrate(x*(a+b*atanh(c*x**3)),x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.18 \[ \int x \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {1}{2} \, a x^{2} + \frac {1}{8} \, {\left (4 \, x^{2} \operatorname {artanh}\left (c x^{3}\right ) + c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {\log \left (c^{\frac {2}{3}} x^{2} + c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} + \frac {\log \left (c^{\frac {2}{3}} x^{2} - c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} x + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} x - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )}\right )} b \] Input:
integrate(x*(a+b*arctanh(c*x^3)),x, algorithm="maxima")
Output:
1/2*a*x^2 + 1/8*(4*x^2*arctanh(c*x^3) + c*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2 *c^(2/3)*x + c^(1/3))/c^(1/3))/c^(5/3) + 2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c ^(2/3)*x - c^(1/3))/c^(1/3))/c^(5/3) - log(c^(2/3)*x^2 + c^(1/3)*x + 1)/c^ (5/3) + log(c^(2/3)*x^2 - c^(1/3)*x + 1)/c^(5/3) - 2*log((c^(1/3)*x + 1)/c ^(1/3))/c^(5/3) + 2*log((c^(1/3)*x - 1)/c^(1/3))/c^(5/3)))*b
Time = 0.20 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.32 \[ \int x \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {1}{4} \, b x^{2} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + \frac {1}{2} \, a x^{2} + \frac {\sqrt {3} b c \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {\sqrt {3} b c \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {b c \log \left (x^{2} + \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {b c \log \left (x^{2} - \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{8 \, {\left | c \right |}^{\frac {5}{3}}} - \frac {b c \log \left ({\left | x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} + \frac {b c \log \left ({\left | x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{4 \, {\left | c \right |}^{\frac {5}{3}}} \] Input:
integrate(x*(a+b*arctanh(c*x^3)),x, algorithm="giac")
Output:
1/4*b*x^2*log(-(c*x^3 + 1)/(c*x^3 - 1)) + 1/2*a*x^2 + 1/4*sqrt(3)*b*c*arct an(1/3*sqrt(3)*(2*x + 1/abs(c)^(1/3))*abs(c)^(1/3))/abs(c)^(5/3) + 1/4*sqr t(3)*b*c*arctan(1/3*sqrt(3)*(2*x - 1/abs(c)^(1/3))*abs(c)^(1/3))/abs(c)^(5 /3) - 1/8*b*c*log(x^2 + x/abs(c)^(1/3) + 1/abs(c)^(2/3))/abs(c)^(5/3) + 1/ 8*b*c*log(x^2 - x/abs(c)^(1/3) + 1/abs(c)^(2/3))/abs(c)^(5/3) - 1/4*b*c*lo g(abs(x + 1/abs(c)^(1/3)))/abs(c)^(5/3) + 1/4*b*c*log(abs(x - 1/abs(c)^(1/ 3)))/abs(c)^(5/3)
Time = 4.01 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.90 \[ \int x \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {a\,x^2}{2}+\frac {b\,\left (-\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )}{2}+\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )}{2}+\mathrm {atan}\left (c^{1/3}\,x\,1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{2\,c^{2/3}}+\frac {b\,x^2\,\ln \left (c\,x^3+1\right )}{4}-\frac {b\,x^2\,\ln \left (1-c\,x^3\right )}{4}+\frac {\sqrt {3}\,b\,\left (\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )+\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )\right )}{4\,c^{2/3}} \] Input:
int(x*(a + b*atanh(c*x^3)),x)
Output:
(a*x^2)/2 + (b*(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)/(2*c^(2/3)) + (b*x^2*log(c*x^3 + 1))/4 - (b*x^2*log(1 - c*x^3))/4 + (3^(1/2)*b*(atan((c^(1/3)*x*(3^(1/2) - 1i))/2) + atan((c^(1/3)*x*(3^(1/2) + 1i))/2)))/(4*c^(2/3))
Time = 0.16 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.80 \[ \int x \left (a+b \text {arctanh}\left (c x^3\right )\right ) \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {2 c^{\frac {1}{3}} x -1}{\sqrt {3}}\right ) b +2 \sqrt {3}\, \mathit {atan} \left (\frac {2 c^{\frac {1}{3}} x +1}{\sqrt {3}}\right ) b +4 c^{\frac {2}{3}} \mathit {atanh} \left (c \,x^{3}\right ) b \,x^{2}+2 \mathit {atanh} \left (c \,x^{3}\right ) b +4 c^{\frac {2}{3}} a \,x^{2}-3 \,\mathrm {log}\left (c^{\frac {2}{3}} x +c^{\frac {1}{3}}\right ) b +3 \,\mathrm {log}\left (c^{\frac {2}{3}} x -c^{\frac {1}{3}}\right ) b}{8 c^{\frac {2}{3}}} \] Input:
int(x*(a+b*atanh(c*x^3)),x)
Output:
(2*sqrt(3)*atan((2*c**(1/3)*x - 1)/sqrt(3))*b + 2*sqrt(3)*atan((2*c**(1/3) *x + 1)/sqrt(3))*b + 4*c**(2/3)*atanh(c*x**3)*b*x**2 + 2*atanh(c*x**3)*b + 4*c**(2/3)*a*x**2 - 3*log(c**(2/3)*x + c**(1/3))*b + 3*log(c**(2/3)*x - c **(1/3))*b)/(8*c**(2/3))