\(\int \frac {a+b \arctan (c x^3)}{x^3} \, dx\) [106]

Optimal result

Integrand size = 14, antiderivative size = 127 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^3} \, dx=\frac {1}{2} b c^{2/3} \arctan \left (\sqrt [3]{c} x\right )-\frac {a+b \arctan \left (c x^3\right )}{2 x^2}-\frac {1}{4} b c^{2/3} \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )+\frac {1}{4} b c^{2/3} \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )+\frac {1}{4} \sqrt {3} b c^{2/3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [3]{c} x}{1+c^{2/3} x^2}\right ) \] Output:

1/2*b*c^(2/3)*arctan(c^(1/3)*x)-1/2*(a+b*arctan(c*x^3))/x^2+1/4*b*c^(2/3)* 
arctan(-3^(1/2)+2*c^(1/3)*x)+1/4*b*c^(2/3)*arctan(3^(1/2)+2*c^(1/3)*x)+1/4 
*3^(1/2)*b*c^(2/3)*arctanh(3^(1/2)*c^(1/3)*x/(1+c^(2/3)*x^2))
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.34 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^3} \, dx=-\frac {a}{2 x^2}+\frac {1}{2} b c^{2/3} \arctan \left (\sqrt [3]{c} x\right )-\frac {b \arctan \left (c x^3\right )}{2 x^2}-\frac {1}{4} b c^{2/3} \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )+\frac {1}{4} b c^{2/3} \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )-\frac {1}{8} \sqrt {3} b c^{2/3} \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{8} \sqrt {3} b c^{2/3} \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right ) \] Input:

Integrate[(a + b*ArcTan[c*x^3])/x^3,x]
 

Output:

-1/2*a/x^2 + (b*c^(2/3)*ArcTan[c^(1/3)*x])/2 - (b*ArcTan[c*x^3])/(2*x^2) - 
 (b*c^(2/3)*ArcTan[Sqrt[3] - 2*c^(1/3)*x])/4 + (b*c^(2/3)*ArcTan[Sqrt[3] + 
 2*c^(1/3)*x])/4 - (Sqrt[3]*b*c^(2/3)*Log[1 - Sqrt[3]*c^(1/3)*x + c^(2/3)* 
x^2])/8 + (Sqrt[3]*b*c^(2/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/8
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.46, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5361, 753, 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.

\(\displaystyle \int \frac {a+b \arctan \left (c x^3\right )}{x^3} \, dx\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {3}{2} b c \int \frac {1}{c^2 x^6+1}dx-\frac {a+b \arctan \left (c x^3\right )}{2 x^2}\)

\(\Big \downarrow \) 753

\(\displaystyle \frac {3}{2} b c \left (\frac {1}{3} \int \frac {1}{c^{2/3} x^2+1}dx+\frac {1}{3} \int \frac {2-\sqrt {3} \sqrt [3]{c} x}{2 \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}dx+\frac {1}{3} \int \frac {\sqrt {3} \sqrt [3]{c} x+2}{2 \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}dx\right )-\frac {a+b \arctan \left (c x^3\right )}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{2} b c \left (\frac {1}{3} \int \frac {1}{c^{2/3} x^2+1}dx+\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{c} x+2}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx\right )-\frac {a+b \arctan \left (c x^3\right )}{2 x^2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \int \frac {2-\sqrt {3} \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {1}{6} \int \frac {\sqrt {3} \sqrt [3]{c} x+2}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \arctan \left (c x^3\right )}{2 x^2}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {\sqrt {3} \int -\frac {\sqrt [3]{c} \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {\sqrt {3} \int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}\right )+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \arctan \left (c x^3\right )}{2 x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {\sqrt {3} \int \frac {\sqrt [3]{c} \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {\sqrt {3} \int \frac {\sqrt [3]{c} \left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx}{2 \sqrt [3]{c}}\right )+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \arctan \left (c x^3\right )}{2 x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{c} x+\sqrt {3}}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx\right )+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \arctan \left (c x^3\right )}{2 x^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {\int \frac {1}{-\left (1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{c} x+\sqrt {3}}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {\int \frac {1}{-\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+1\right )}{\sqrt {3} \sqrt [3]{c}}\right )+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \arctan \left (c x^3\right )}{2 x^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-2 \sqrt [3]{c} x}{c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1}dx-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )\right )}{\sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [3]{c} x+\sqrt {3}}{c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1}dx+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+1\right )\right )}{\sqrt [3]{c}}\right )+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \arctan \left (c x^3\right )}{2 x^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {3}{2} b c \left (\frac {1}{6} \left (-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )\right )}{\sqrt [3]{c}}-\frac {\sqrt {3} \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{2 \sqrt [3]{c}}\right )+\frac {1}{6} \left (\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+1\right )\right )}{\sqrt [3]{c}}+\frac {\sqrt {3} \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{2 \sqrt [3]{c}}\right )+\frac {\arctan \left (\sqrt [3]{c} x\right )}{3 \sqrt [3]{c}}\right )-\frac {a+b \arctan \left (c x^3\right )}{2 x^2}\)

Input:

Int[(a + b*ArcTan[c*x^3])/x^3,x]
 

Output:

-1/2*(a + b*ArcTan[c*x^3])/x^2 + (3*b*c*(ArcTan[c^(1/3)*x]/(3*c^(1/3)) + ( 
-(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 + (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))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 216

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])
 

rule 217

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])
 

rule 753

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ 
b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[(2*k 
 - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[ 
(r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(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] && PosQ[a 
/b]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1142

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]
 

rule 5361

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]
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.13

method	result	size
default	\(-\frac {a}{2 x^{2}}+b \left (-\frac {\arctan \left (c \,x^{3}\right )}{2 x^{2}}+\frac {3 c \left (-\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6}+\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{3}\right )}{2}\right )\)	\(144\)
parts	\(-\frac {a}{2 x^{2}}+b \left (-\frac {\arctan \left (c \,x^{3}\right )}{2 x^{2}}+\frac {3 c \left (-\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6}+\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{3}\right )}{2}\right )\)	\(144\)

Input:

int((a+b*arctan(c*x^3))/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)^(1/6)*ln 
(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))+1/6*(1/c^2)^(1/6)*arctan(2*x/( 
1/c^2)^(1/6)-3^(1/2))+1/12*3^(1/2)*(1/c^2)^(1/6)*ln(x^2+3^(1/2)*(1/c^2)^(1 
/6)*x+(1/c^2)^(1/3))+1/6*(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)+3^(1/2))+1 
/3*(1/c^2)^(1/6)*arctan(x/(1/c^2)^(1/6))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (93) = 186\).

Time = 0.12 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.13 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^3} \, dx=\frac {2 \, \left (-b^{6} c^{4}\right )^{\frac {1}{6}} x^{2} \log \left (b c x + \left (-b^{6} c^{4}\right )^{\frac {1}{6}}\right ) - 2 \, \left (-b^{6} c^{4}\right )^{\frac {1}{6}} x^{2} \log \left (b c x - \left (-b^{6} c^{4}\right )^{\frac {1}{6}}\right ) + \left (-b^{6} c^{4}\right )^{\frac {1}{6}} {\left (\sqrt {-3} x^{2} + x^{2}\right )} \log \left (2 \, b c x + \left (-b^{6} c^{4}\right )^{\frac {1}{6}} {\left (\sqrt {-3} + 1\right )}\right ) - \left (-b^{6} c^{4}\right )^{\frac {1}{6}} {\left (\sqrt {-3} x^{2} + x^{2}\right )} \log \left (2 \, b c x - \left (-b^{6} c^{4}\right )^{\frac {1}{6}} {\left (\sqrt {-3} + 1\right )}\right ) + \left (-b^{6} c^{4}\right )^{\frac {1}{6}} {\left (\sqrt {-3} x^{2} - x^{2}\right )} \log \left (2 \, b c x + \left (-b^{6} c^{4}\right )^{\frac {1}{6}} {\left (\sqrt {-3} - 1\right )}\right ) - \left (-b^{6} c^{4}\right )^{\frac {1}{6}} {\left (\sqrt {-3} x^{2} - x^{2}\right )} \log \left (2 \, b c x - \left (-b^{6} c^{4}\right )^{\frac {1}{6}} {\left (\sqrt {-3} - 1\right )}\right ) - 4 \, b \arctan \left (c x^{3}\right ) - 4 \, a}{8 \, x^{2}} \] Input:

integrate((a+b*arctan(c*x^3))/x^3,x, algorithm="fricas")
 

Output:

1/8*(2*(-b^6*c^4)^(1/6)*x^2*log(b*c*x + (-b^6*c^4)^(1/6)) - 2*(-b^6*c^4)^( 
1/6)*x^2*log(b*c*x - (-b^6*c^4)^(1/6)) + (-b^6*c^4)^(1/6)*(sqrt(-3)*x^2 + 
x^2)*log(2*b*c*x + (-b^6*c^4)^(1/6)*(sqrt(-3) + 1)) - (-b^6*c^4)^(1/6)*(sq 
rt(-3)*x^2 + x^2)*log(2*b*c*x - (-b^6*c^4)^(1/6)*(sqrt(-3) + 1)) + (-b^6*c 
^4)^(1/6)*(sqrt(-3)*x^2 - x^2)*log(2*b*c*x + (-b^6*c^4)^(1/6)*(sqrt(-3) - 
1)) - (-b^6*c^4)^(1/6)*(sqrt(-3)*x^2 - x^2)*log(2*b*c*x - (-b^6*c^4)^(1/6) 
*(sqrt(-3) - 1)) - 4*b*arctan(c*x^3) - 4*a)/x^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (117) = 234\).

Time = 27.39 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.93 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^3} \, dx=\begin {cases} - \frac {a}{2 x^{2}} + \frac {b \operatorname {atan}{\left (c x^{3} \right )}}{2 \sqrt [3]{- \frac {1}{c^{2}}}} - \frac {b \operatorname {atan}{\left (c x^{3} \right )}}{2 x^{2}} + \frac {3 b \log {\left (4 x^{2} - 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{8 c \left (- \frac {1}{c^{2}}\right )^{\frac {5}{6}}} - \frac {3 b \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{8 c \left (- \frac {1}{c^{2}}\right )^{\frac {5}{6}}} - \frac {\sqrt {3} b \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{4 c \left (- \frac {1}{c^{2}}\right )^{\frac {5}{6}}} - \frac {\sqrt {3} b \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{4 c \left (- \frac {1}{c^{2}}\right )^{\frac {5}{6}}} & \text {for}\: c \neq 0 \\- \frac {a}{2 x^{2}} & \text {otherwise} \end {cases} \] Input:

integrate((a+b*atan(c*x**3))/x**3,x)
 

Output:

Piecewise((-a/(2*x**2) + b*atan(c*x**3)/(2*(-1/c**2)**(1/3)) - b*atan(c*x* 
*3)/(2*x**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)**(5/6)) - 3*b*log(4*x**2 + 4*x*(-1/c**2)**(1/6) + 4*(-1/c* 
*2)**(1/3))/(8*c*(-1/c**2)**(5/6)) - sqrt(3)*b*atan(2*sqrt(3)*x/(3*(-1/c** 
2)**(1/6)) - sqrt(3)/3)/(4*c*(-1/c**2)**(5/6)) - sqrt(3)*b*atan(2*sqrt(3)* 
x/(3*(-1/c**2)**(1/6)) + sqrt(3)/3)/(4*c*(-1/c**2)**(5/6)), Ne(c, 0)), (-a 
/(2*x**2), True))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.08 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^3} \, dx=\frac {1}{8} \, {\left ({\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {1}{3}}} + \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {1}{3}}} + \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}\right )} c - \frac {4 \, \arctan \left (c x^{3}\right )}{x^{2}}\right )} b - \frac {a}{2 \, x^{2}} \] Input:

integrate((a+b*arctan(c*x^3))/x^3,x, algorithm="maxima")
 

Output:

1/8*((sqrt(3)*log(c^(2/3)*x^2 + sqrt(3)*c^(1/3)*x + 1)/c^(1/3) - sqrt(3)*l 
og(c^(2/3)*x^2 - sqrt(3)*c^(1/3)*x + 1)/c^(1/3) + 4*arctan(c^(1/3)*x)/c^(1 
/3) + 2*arctan((2*c^(2/3)*x + sqrt(3)*c^(1/3))/c^(1/3))/c^(1/3) + 2*arctan 
((2*c^(2/3)*x - sqrt(3)*c^(1/3))/c^(1/3))/c^(1/3))*c - 4*arctan(c*x^3)/x^2 
)*b - 1/2*a/x^2
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.08 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^3} \, dx=\frac {1}{8} \, {\left (\frac {\sqrt {3} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {4 \, \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}}\right )} b c - \frac {b \arctan \left (c x^{3}\right ) + a}{2 \, x^{2}} \] Input:

integrate((a+b*arctan(c*x^3))/x^3,x, algorithm="giac")
 

Output:

1/8*(sqrt(3)*log(x^2 + sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/abs(c)^(1/ 
3) - sqrt(3)*log(x^2 - sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/abs(c)^(1/ 
3) + 2*arctan((2*x + sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/abs(c)^(1/3) + 2* 
arctan((2*x - sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/abs(c)^(1/3) + 4*arctan( 
x*abs(c)^(1/3))/abs(c)^(1/3))*b*c - 1/2*(b*arctan(c*x^3) + a)/x^2
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 1.21 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^3} \, dx=-\frac {a}{2\,x^2}-\frac {b\,c^{2/3}\,\left (\frac {\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )}{2}-\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )}{2}+\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )}{2}-\frac {b\,\mathrm {atan}\left (c\,x^3\right )}{2\,x^2}-\frac {\sqrt {3}\,b\,c^{2/3}\,\left (\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )+\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )\right )\,1{}\mathrm {i}}{4} \] Input:

int((a + b*atan(c*x^3))/x^3,x)
 

Output:

- a/(2*x^2) - (b*c^(2/3)*(atan((-1)^(2/3)*c^(1/3)*x)/2 - atan((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)))/ 
2 - (b*atan(c*x^3))/(2*x^2) - (3^(1/2)*b*c^(2/3)*(atan((c^(1/3)*x*(3^(1/2) 
*1i + 1))/2) + atan((-1)^(2/3)*c^(1/3)*x))*1i)/4
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \arctan \left (c x^3\right )}{x^3} \, dx=\frac {6 \mathit {atan} \left (c^{\frac {1}{3}} x \right ) b c \,x^{2}-4 c^{\frac {1}{3}} \mathit {atan} \left (c \,x^{3}\right ) b +2 \mathit {atan} \left (c \,x^{3}\right ) b c \,x^{2}-4 c^{\frac {1}{3}} a -\sqrt {3}\, \mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b c \,x^{2}+\sqrt {3}\, \mathrm {log}\left (c^{\frac {2}{3}} x^{2}+c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b c \,x^{2}}{8 c^{\frac {1}{3}} x^{2}} \] Input:

int((a+b*atan(c*x^3))/x^3,x)
 

Output:

(6*atan(c**(1/3)*x)*b*c*x**2 - 4*c**(1/3)*atan(c*x**3)*b + 2*atan(c*x**3)* 
b*c*x**2 - 4*c**(1/3)*a - sqrt(3)*log(c**(2/3)*x**2 - c**(1/3)*sqrt(3)*x + 
 1)*b*c*x**2 + sqrt(3)*log(c**(2/3)*x**2 + c**(1/3)*sqrt(3)*x + 1)*b*c*x** 
2)/(8*c**(1/3)*x**2)