3.109 \(\int \frac {a+b \tanh ^{-1}(c x^3)}{x^3} \, dx\)

Optimal. Leaf size=165 \[ -\frac {a+b \tanh ^{-1}\left (c x^3\right )}{2 x^2}-\frac {1}{8} b c^{2/3} \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )+\frac {1}{8} b c^{2/3} \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )-\frac {1}{4} \sqrt {3} b c^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )+\frac {1}{4} \sqrt {3} b c^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )+\frac {1}{2} b c^{2/3} \tanh ^{-1}\left (\sqrt [3]{c} x\right ) \]

[Out]

1/2*b*c^(2/3)*arctanh(c^(1/3)*x)+1/2*(-a-b*arctanh(c*x^3))/x^2-1/8*b*c^(2/3)*ln(1-c^(1/3)*x+c^(2/3)*x^2)+1/8*b
*c^(2/3)*ln(1+c^(1/3)*x+c^(2/3)*x^2)+1/4*b*c^(2/3)*arctan(-1/3*3^(1/2)+2/3*c^(1/3)*x*3^(1/2))*3^(1/2)+1/4*b*c^
(2/3)*arctan(1/3*3^(1/2)+2/3*c^(1/3)*x*3^(1/2))*3^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.20, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6097, 210, 634, 618, 204, 628, 206} \[ -\frac {a+b \tanh ^{-1}\left (c x^3\right )}{2 x^2}-\frac {1}{8} b c^{2/3} \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )+\frac {1}{8} b c^{2/3} \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )-\frac {1}{4} \sqrt {3} b c^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )+\frac {1}{4} \sqrt {3} b c^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )+\frac {1}{2} b c^{2/3} \tanh ^{-1}\left (\sqrt [3]{c} x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^3])/x^3,x]

[Out]

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 210

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-(a/b), n]], s = Denominator[Rt[-(a/b
), n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r +
s*Cos[(2*k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 - s^2*x^2), x])/(a*n) +
Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 6097

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
nh[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 - c^2*x^(2*n)), x], x
] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^3\right )}{x^3} \, dx &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{2 x^2}+\frac {1}{2} (3 b c) \int \frac {1}{1-c^2 x^6} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{2 x^2}+\frac {1}{2} (b c) \int \frac {1}{1-c^{2/3} x^2} \, dx+\frac {1}{2} (b c) \int \frac {1-\frac {\sqrt [3]{c} x}{2}}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac {1}{2} (b c) \int \frac {1+\frac {\sqrt [3]{c} x}{2}}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx\\ &=\frac {1}{2} b c^{2/3} \tanh ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{2 x^2}-\frac {1}{8} \left (b c^{2/3}\right ) \int \frac {-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac {1}{8} \left (b c^{2/3}\right ) \int \frac {\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac {1}{8} (3 b c) \int \frac {1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac {1}{8} (3 b c) \int \frac {1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx\\ &=\frac {1}{2} b c^{2/3} \tanh ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{2 x^2}-\frac {1}{8} b c^{2/3} \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{8} b c^{2/3} \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{4} \left (3 b c^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} x\right )-\frac {1}{4} \left (3 b c^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} x\right )\\ &=-\frac {1}{4} \sqrt {3} b c^{2/3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )+\frac {1}{4} \sqrt {3} b c^{2/3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )+\frac {1}{2} b c^{2/3} \tanh ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{2 x^2}-\frac {1}{8} b c^{2/3} \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{8} b c^{2/3} \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 187, normalized size = 1.13 \[ -\frac {a}{2 x^2}-\frac {1}{8} b c^{2/3} \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )+\frac {1}{8} b c^{2/3} \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )-\frac {1}{4} b c^{2/3} \log \left (1-\sqrt [3]{c} x\right )+\frac {1}{4} b c^{2/3} \log \left (\sqrt [3]{c} x+1\right )+\frac {1}{4} \sqrt {3} b c^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x-1}{\sqrt {3}}\right )+\frac {1}{4} \sqrt {3} b c^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right )-\frac {b \tanh ^{-1}\left (c x^3\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 228, normalized size = 1.38 \[ -\frac {2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {1}{3}} b x^{2} \arctan \left (\frac {2 \, \sqrt {3} \left (-c^{2}\right )^{\frac {2}{3}} x + \sqrt {3} c}{3 \, c}\right ) - 2 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{3}} x^{2} \arctan \left (\frac {2 \, \sqrt {3} {\left (c^{2}\right )}^{\frac {2}{3}} x - \sqrt {3} c}{3 \, c}\right ) + \left (-c^{2}\right )^{\frac {1}{3}} b x^{2} \log \left (c^{2} x^{2} - \left (-c^{2}\right )^{\frac {1}{3}} c x + \left (-c^{2}\right )^{\frac {2}{3}}\right ) + b {\left (c^{2}\right )}^{\frac {1}{3}} x^{2} \log \left (c^{2} x^{2} - {\left (c^{2}\right )}^{\frac {1}{3}} c x + {\left (c^{2}\right )}^{\frac {2}{3}}\right ) - 2 \, \left (-c^{2}\right )^{\frac {1}{3}} b x^{2} \log \left (c x + \left (-c^{2}\right )^{\frac {1}{3}}\right ) - 2 \, b {\left (c^{2}\right )}^{\frac {1}{3}} x^{2} \log \left (c x + {\left (c^{2}\right )}^{\frac {1}{3}}\right ) + 2 \, b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a}{8 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.21, size = 165, normalized size = 1.00 \[ \frac {1}{8} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {\log \left (x^{2} + \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} - \frac {\log \left (x^{2} - \frac {x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \log \left ({\left | x + \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{{\left | c \right |}^{\frac {1}{3}}} - \frac {2 \, \log \left ({\left | x - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{{\left | c \right |}^{\frac {1}{3}}}\right )} b c - \frac {b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{4 \, x^{2}} - \frac {a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.03, size = 159, normalized size = 0.96 \[ -\frac {a}{2 x^{2}}-\frac {b \arctanh \left (c \,x^{3}\right )}{2 x^{2}}-\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{8 \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{4 \left (\frac {1}{c}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a/x^2-1/2*b/x^2*arctanh(c*x^3)-1/4*b/(1/c)^(2/3)*ln(x-(1/c)^(1/3))+1/8*b/(1/c)^(2/3)*ln(x^2+(1/c)^(1/3)*x
+(1/c)^(2/3))+1/4*b/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x+1))+1/4*b/(1/c)^(2/3)*ln(x+(1/c)^(
1/3))-1/8*b/(1/c)^(2/3)*ln(x^2-(1/c)^(1/3)*x+(1/c)^(2/3))+1/4*b/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c
)^(1/3)*x-1))

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 155, normalized size = 0.94 \[ \frac {1}{8} \, {\left ({\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {\log \left (c^{\frac {2}{3}} x^{2} + c^{\frac {1}{3}} x + 1\right )}{c^{\frac {1}{3}}} - \frac {\log \left (c^{\frac {2}{3}} x^{2} - c^{\frac {1}{3}} x + 1\right )}{c^{\frac {1}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} x + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} x - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}\right )} c - \frac {4 \, \operatorname {artanh}\left (c x^{3}\right )}{x^{2}}\right )} b - \frac {a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.47, size = 118, normalized size = 0.72 \[ \frac {b\,\ln \left (1-c\,x^3\right )}{4\,x^2}-\frac {b\,c^{2/3}\,\left (-\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )}{2}+\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )}{2}+\mathrm {atan}\left (c^{1/3}\,x\,1{}\mathrm {i}\right )\right )\,1{}\mathrm {i}}{2}-\frac {b\,\ln \left (c\,x^3+1\right )}{4\,x^2}-\frac {a}{2\,x^2}+\frac {\sqrt {3}\,b\,c^{2/3}\,\left (\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}-\mathrm {i}\right )}{2}\right )+\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (\sqrt {3}+1{}\mathrm {i}\right )}{2}\right )\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^3))/x^3,x)

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________