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]

-(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

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Rubi [A]  time = 0.197406, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 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]

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 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 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 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 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 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])

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 = 0.0523718, 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]

-a/(2*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

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Maple [A]  time = 0.01, size = 159, normalized size = 1. \begin{align*} -{\frac{a}{2\,{x}^{2}}}-{\frac{b{\it Artanh} \left ( c{x}^{3} \right ) }{2\,{x}^{2}}}-{\frac{b}{4}\ln \left ( x-\sqrt [3]{{c}^{-1}} \right ) \left ({c}^{-1} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{8}\ln \left ({x}^{2}+\sqrt [3]{{c}^{-1}}x+ \left ({c}^{-1} \right ) ^{{\frac{2}{3}}} \right ) \left ({c}^{-1} \right ) ^{-{\frac{2}{3}}}}+{\frac{b\sqrt{3}}{4}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{x}{\sqrt [3]{{c}^{-1}}}}+1 \right ) } \right ) \left ({c}^{-1} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{4}\ln \left ( x+\sqrt [3]{{c}^{-1}} \right ) \left ({c}^{-1} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{8}\ln \left ({x}^{2}-\sqrt [3]{{c}^{-1}}x+ \left ({c}^{-1} \right ) ^{{\frac{2}{3}}} \right ) \left ({c}^{-1} \right ) ^{-{\frac{2}{3}}}}+{\frac{b\sqrt{3}}{4}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{x}{\sqrt [3]{{c}^{-1}}}}-1 \right ) } \right ) \left ({c}^{-1} \right ) ^{-{\frac{2}{3}}}} \end{align*}

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))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.9275, size = 598, normalized size = 3.62 \begin{align*} -\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}} \end{align*}

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: KeyError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: KeyError

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Giac [A]  time = 1.29789, size = 223, normalized size = 1.35 \begin{align*} \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}} \end{align*}

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