Integrand size = 12, antiderivative size = 127 \[ \int x \left (a+b \arctan \left (c x^3\right )\right ) \, dx=-\frac {b \arctan \left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {1}{2} x^2 \left (a+b \arctan \left (c x^3\right )\right )+\frac {b \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}+\frac {\sqrt {3} b \text {arctanh}\left (\frac {\sqrt {3} \sqrt [3]{c} x}{1+c^{2/3} x^2}\right )}{4 c^{2/3}} \] Output:
-1/2*b*arctan(c^(1/3)*x)/c^(2/3)+1/2*x^2*(a+b*arctan(c*x^3))-1/4*b*arctan( -3^(1/2)+2*c^(1/3)*x)/c^(2/3)-1/4*b*arctan(3^(1/2)+2*c^(1/3)*x)/c^(2/3)+1/ 4*3^(1/2)*b*arctanh(3^(1/2)*c^(1/3)*x/(1+c^(2/3)*x^2))/c^(2/3)
Time = 0.03 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.34 \[ \int x \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {a x^2}{2}-\frac {b \arctan \left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {1}{2} b x^2 \arctan \left (c x^3\right )+\frac {b \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}} \] Input:
Integrate[x*(a + b*ArcTan[c*x^3]),x]
Output:
(a*x^2)/2 - (b*ArcTan[c^(1/3)*x])/(2*c^(2/3)) + (b*x^2*ArcTan[c*x^3])/2 + (b*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/(4*c^(2/3)) - (b*ArcTan[Sqrt[3] + 2*c^(1 /3)*x])/(4*c^(2/3)) - (Sqrt[3]*b*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2]) /(8*c^(2/3)) + (Sqrt[3]*b*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/(8*c^( 2/3))
Time = 0.41 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.54, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5361, 824, 27, 216, 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.
| \(\displaystyle \int x \left (a+b \arctan \left (c x^3\right )\right ) \, dx\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (c x^3\right )\right )-\frac {3}{2} b c \int \frac {x^4}{c^2 x^6+1}dx\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (c x^3\right )\right )-\frac {3}{2} b c \left (\frac {\int \frac {1}{c^{2/3} x^2+1}dx}{3 c^{4/3}}+\frac {\int -\frac {1-\sqrt {3} \sqrt [3]{c} x}{2 \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}dx}{3 c^{4/3}}+\frac {\int -\frac {\sqrt {3} \sqrt [3]{c} x+1}{2 \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}dx}{3 c^{4/3}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (c x^3\right )\right )-\frac {3}{2} b c \left (\frac {\int \frac {1}{c^{2/3} x^2+1}dx}{3 c^{4/3}}-\frac {\int \frac {1-\sqrt {3} \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\int \frac {\sqrt {3} \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {\int \frac {1-\sqrt {3} \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\int \frac {\sqrt {3} \sqrt [3]{c} x+1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {-\frac {1}{2} \int \frac {1}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {\sqrt {3} \int -\frac {\sqrt [3]{c} \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}-\frac {1}{2} \int \frac {1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {\frac {\sqrt {3} \int \frac {\sqrt [3]{c} \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}-\frac {1}{2} \int \frac {1}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}-\frac {1}{2} \int \frac {1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {1}{2} \int \frac {1}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{c} x+\sqrt {3}}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {1}{2} \int \frac {1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {\int \frac {1}{-\left (1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{c} x+\sqrt {3}}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {\int \frac {1}{-\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+1\right )}{\sqrt {3} \sqrt [3]{c}}}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )\right )}{\sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{c} x+\sqrt {3}}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+1\right )\right )}{\sqrt [3]{c}}}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \arctan \left (c x^3\right )\right )-\frac {3}{2} b c \left (-\frac {\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )\right )}{\sqrt [3]{c}}-\frac {\sqrt {3} \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{2 \sqrt [3]{c}}}{6 c^{4/3}}-\frac {\frac {\sqrt {3} \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{2 \sqrt [3]{c}}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+1\right )\right )}{\sqrt [3]{c}}}{6 c^{4/3}}+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 c^{5/3}}\right )\) |
Input:
Int[x*(a + b*ArcTan[c*x^3]),x]
Output:
(x^2*(a + b*ArcTan[c*x^3]))/2 - (3*b*c*(ArcTan[c^(1/3)*x]/(3*c^(5/3)) - (A rcTan[Sqrt[3]*(1 - (2*c^(1/3)*x)/Sqrt[3])]/c^(1/3) - (Sqrt[3]*Log[1 - Sqrt [3]*c^(1/3)*x + c^(2/3)*x^2])/(2*c^(1/3)))/(6*c^(4/3)) - (-(ArcTan[Sqrt[3] *(1 + (2*c^(1/3)*x)/Sqrt[3])]/c^(1/3)) + (Sqrt[3]*Log[1 + Sqrt[3]*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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[(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 - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/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] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[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_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[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 = 0.24 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(\frac {a \,x^{2}}{2}+b \left (\frac {x^{2} \arctan \left (c \,x^{3}\right )}{2}-\frac {3 c \left (\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{3 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{2}\right )\) | \(153\) |
| parts | \(\frac {a \,x^{2}}{2}+b \left (\frac {x^{2} \arctan \left (c \,x^{3}\right )}{2}-\frac {3 c \left (\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{3 c^{2} \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{2}\right )\) | \(153\) |
Input:
int(x*(a+b*arctan(c*x^3)),x,method=_RETURNVERBOSE)
Output:
1/2*a*x^2+b*(1/2*x^2*arctan(c*x^3)-3/2*c*(1/12*3^(1/2)*(1/c^2)^(5/6)*ln(x^ 2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))+1/6/c^2/(1/c^2)^(1/6)*arctan(2*x/ (1/c^2)^(1/6)-3^(1/2))-1/12*3^(1/2)*(1/c^2)^(5/6)*ln(x^2+3^(1/2)*(1/c^2)^( 1/6)*x+(1/c^2)^(1/3))+1/6/c^2/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)+3^(1/ 2))+1/3/c^2/(1/c^2)^(1/6)*arctan(x/(1/c^2)^(1/6))))
Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (93) = 186\).
Time = 0.13 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.19 \[ \int x \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{2} \, b x^{2} \arctan \left (c x^{3}\right ) + \frac {1}{2} \, a x^{2} + \frac {1}{8} \, \left (-\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} - 1\right )} \log \left (b^{5} x + \frac {1}{2} \, {\left (\sqrt {-3} c^{3} + c^{3}\right )} \left (-\frac {b^{6}}{c^{4}}\right )^{\frac {5}{6}}\right ) - \frac {1}{8} \, \left (-\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} - 1\right )} \log \left (b^{5} x - \frac {1}{2} \, {\left (\sqrt {-3} c^{3} + c^{3}\right )} \left (-\frac {b^{6}}{c^{4}}\right )^{\frac {5}{6}}\right ) + \frac {1}{8} \, \left (-\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} + 1\right )} \log \left (b^{5} x + \frac {1}{2} \, {\left (\sqrt {-3} c^{3} - c^{3}\right )} \left (-\frac {b^{6}}{c^{4}}\right )^{\frac {5}{6}}\right ) - \frac {1}{8} \, \left (-\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} {\left (\sqrt {-3} + 1\right )} \log \left (b^{5} x - \frac {1}{2} \, {\left (\sqrt {-3} c^{3} - c^{3}\right )} \left (-\frac {b^{6}}{c^{4}}\right )^{\frac {5}{6}}\right ) - \frac {1}{4} \, \left (-\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} \log \left (b^{5} x + \left (-\frac {b^{6}}{c^{4}}\right )^{\frac {5}{6}} c^{3}\right ) + \frac {1}{4} \, \left (-\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} \log \left (b^{5} x - \left (-\frac {b^{6}}{c^{4}}\right )^{\frac {5}{6}} c^{3}\right ) \] Input:
integrate(x*(a+b*arctan(c*x^3)),x, algorithm="fricas")
Output:
1/2*b*x^2*arctan(c*x^3) + 1/2*a*x^2 + 1/8*(-b^6/c^4)^(1/6)*(sqrt(-3) - 1)* log(b^5*x + 1/2*(sqrt(-3)*c^3 + c^3)*(-b^6/c^4)^(5/6)) - 1/8*(-b^6/c^4)^(1 /6)*(sqrt(-3) - 1)*log(b^5*x - 1/2*(sqrt(-3)*c^3 + c^3)*(-b^6/c^4)^(5/6)) + 1/8*(-b^6/c^4)^(1/6)*(sqrt(-3) + 1)*log(b^5*x + 1/2*(sqrt(-3)*c^3 - c^3) *(-b^6/c^4)^(5/6)) - 1/8*(-b^6/c^4)^(1/6)*(sqrt(-3) + 1)*log(b^5*x - 1/2*( sqrt(-3)*c^3 - c^3)*(-b^6/c^4)^(5/6)) - 1/4*(-b^6/c^4)^(1/6)*log(b^5*x + ( -b^6/c^4)^(5/6)*c^3) + 1/4*(-b^6/c^4)^(1/6)*log(b^5*x - (-b^6/c^4)^(5/6)*c ^3)
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (116) = 232\).
Time = 15.63 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.94 \[ \int x \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\begin {cases} \frac {a x^{2}}{2} + \frac {b x^{2} \operatorname {atan}{\left (c x^{3} \right )}}{2} - \frac {3 b \log {\left (4 x^{2} - 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{8 c \sqrt [6]{- \frac {1}{c^{2}}}} + \frac {3 b \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{8 c \sqrt [6]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3} b \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{4 c \sqrt [6]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3} b \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{4 c \sqrt [6]{- \frac {1}{c^{2}}}} + \frac {b \operatorname {atan}{\left (c x^{3} \right )}}{2 c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {2}{3}}} & \text {for}\: c \neq 0 \\\frac {a x^{2}}{2} & \text {otherwise} \end {cases} \] Input:
integrate(x*(a+b*atan(c*x**3)),x)
Output:
Piecewise((a*x**2/2 + b*x**2*atan(c*x**3)/2 - 3*b*log(4*x**2 - 4*x*(-1/c** 2)**(1/6) + 4*(-1/c**2)**(1/3))/(8*c*(-1/c**2)**(1/6)) + 3*b*log(4*x**2 + 4*x*(-1/c**2)**(1/6) + 4*(-1/c**2)**(1/3))/(8*c*(-1/c**2)**(1/6)) - sqrt(3 )*b*atan(2*sqrt(3)*x/(3*(-1/c**2)**(1/6)) - sqrt(3)/3)/(4*c*(-1/c**2)**(1/ 6)) - sqrt(3)*b*atan(2*sqrt(3)*x/(3*(-1/c**2)**(1/6)) + sqrt(3)/3)/(4*c*(- 1/c**2)**(1/6)) + b*atan(c*x**3)/(2*c**2*(-1/c**2)**(2/3)), Ne(c, 0)), (a* x**2/2, True))
Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.08 \[ \int x \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{2} \, a x^{2} + \frac {1}{8} \, {\left (4 \, x^{2} \arctan \left (c x^{3}\right ) + c {\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )}\right )} b \] Input:
integrate(x*(a+b*arctan(c*x^3)),x, algorithm="maxima")
Output:
1/2*a*x^2 + 1/8*(4*x^2*arctan(c*x^3) + c*(sqrt(3)*log(c^(2/3)*x^2 + sqrt(3 )*c^(1/3)*x + 1)/c^(5/3) - sqrt(3)*log(c^(2/3)*x^2 - sqrt(3)*c^(1/3)*x + 1 )/c^(5/3) - 4*arctan(c^(1/3)*x)/c^(5/3) - 2*arctan((2*c^(2/3)*x + sqrt(3)* c^(1/3))/c^(1/3))/c^(5/3) - 2*arctan((2*c^(2/3)*x - sqrt(3)*c^(1/3))/c^(1/ 3))/c^(5/3)))*b
Time = 0.18 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.24 \[ \int x \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{8} \, b c^{5} {\left (\frac {\sqrt {3} {\left | c \right |}^{\frac {1}{3}} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{6}} - \frac {\sqrt {3} {\left | c \right |}^{\frac {1}{3}} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{6}} - \frac {2 \, \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{4} {\left | c \right |}^{\frac {5}{3}}} - \frac {2 \, \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{4} {\left | c \right |}^{\frac {5}{3}}} - \frac {4 \, \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{c^{4} {\left | c \right |}^{\frac {5}{3}}}\right )} + \frac {1}{2} \, b x^{2} \arctan \left (c x^{3}\right ) + \frac {1}{2} \, a x^{2} \] Input:
integrate(x*(a+b*arctan(c*x^3)),x, algorithm="giac")
Output:
1/8*b*c^5*(sqrt(3)*abs(c)^(1/3)*log(x^2 + sqrt(3)*x/abs(c)^(1/3) + 1/abs(c )^(2/3))/c^6 - sqrt(3)*abs(c)^(1/3)*log(x^2 - sqrt(3)*x/abs(c)^(1/3) + 1/a bs(c)^(2/3))/c^6 - 2*arctan((2*x + sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/(c^ 4*abs(c)^(5/3)) - 2*arctan((2*x - sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/(c^4 *abs(c)^(5/3)) - 4*arctan(x*abs(c)^(1/3))/(c^4*abs(c)^(5/3))) + 1/2*b*x^2* arctan(c*x^3) + 1/2*a*x^2
Time = 0.99 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.89 \[ \int x \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {a\,x^2}{2}+\frac {b\,\left (\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )+\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )+2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )}{4\,c^{2/3}}+\frac {b\,x^2\,\mathrm {atan}\left (c\,x^3\right )}{2}-\frac {\sqrt {3}\,b\,\left (\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )-\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )\,1{}\mathrm {i}}{4\,c^{2/3}} \] Input:
int(x*(a + b*atan(c*x^3)),x)
Output:
(a*x^2)/2 + (b*(atan((-1)^(2/3)*c^(1/3)*x) + atan(((-1)^(2/3)*c^(1/3)*x*(3 ^(1/2)*1i - 1))/2) + 2*atan(((-1)^(2/3)*c^(1/3)*x*(3^(1/2)*1i + 1))/2)))/( 4*c^(2/3)) + (b*x^2*atan(c*x^3))/2 - (3^(1/2)*b*(atan((-1)^(2/3)*c^(1/3)*x ) - atan(((-1)^(2/3)*c^(1/3)*x*(3^(1/2)*1i - 1))/2))*1i)/(4*c^(2/3))
Time = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.79 \[ \int x \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {-6 c^{\frac {1}{3}} \mathit {atan} \left (c^{\frac {1}{3}} x \right ) b -2 c^{\frac {1}{3}} \mathit {atan} \left (c \,x^{3}\right ) b +4 \mathit {atan} \left (c \,x^{3}\right ) b c \,x^{2}-c^{\frac {1}{3}} \sqrt {3}\, \mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b +c^{\frac {1}{3}} \sqrt {3}\, \mathrm {log}\left (c^{\frac {2}{3}} x^{2}+c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b +4 a c \,x^{2}}{8 c} \] Input:
int(x*(a+b*atan(c*x^3)),x)
Output:
( - 6*c**(1/3)*atan(c**(1/3)*x)*b - 2*c**(1/3)*atan(c*x**3)*b + 4*atan(c*x **3)*b*c*x**2 - c**(1/3)*sqrt(3)*log(c**(2/3)*x**2 - c**(1/3)*sqrt(3)*x + 1)*b + c**(1/3)*sqrt(3)*log(c**(2/3)*x**2 + c**(1/3)*sqrt(3)*x + 1)*b + 4* a*c*x**2)/(8*c)