3.114 \(\int \frac{a+b \tanh ^{-1}(c x^3)}{x^2} \, dx\)

Optimal. Leaf size=104 \[ -\frac{a+b \tanh ^{-1}\left (c x^3\right )}{x}-\frac{1}{2} b \sqrt [3]{c} \log \left (1-c^{2/3} x^2\right )+\frac{1}{4} b \sqrt [3]{c} \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )+\frac{1}{2} \sqrt{3} b \sqrt [3]{c} \tan ^{-1}\left (\frac{2 c^{2/3} x^2+1}{\sqrt{3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0827732, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {6097, 275, 200, 31, 634, 617, 204, 628} \[ -\frac{a+b \tanh ^{-1}\left (c x^3\right )}{x}-\frac{1}{2} b \sqrt [3]{c} \log \left (1-c^{2/3} x^2\right )+\frac{1}{4} b \sqrt [3]{c} \log \left (c^{4/3} x^4+c^{2/3} x^2+1\right )+\frac{1}{2} \sqrt{3} b \sqrt [3]{c} \tan ^{-1}\left (\frac{2 c^{2/3} x^2+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^3])/x^2,x]

[Out]

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

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 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[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])] /;
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]

Rubi steps

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

Mathematica [A]  time = 0.0290422, size = 183, normalized size = 1.76 \[ -\frac{a}{x}+\frac{1}{4} b \sqrt [3]{c} \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )+\frac{1}{4} b \sqrt [3]{c} \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )-\frac{b \tanh ^{-1}\left (c x^3\right )}{x}-\frac{1}{2} b \sqrt [3]{c} \log \left (1-\sqrt [3]{c} x\right )-\frac{1}{2} b \sqrt [3]{c} \log \left (\sqrt [3]{c} x+1\right )+\frac{1}{2} \sqrt{3} b \sqrt [3]{c} \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x-1}{\sqrt{3}}\right )-\frac{1}{2} \sqrt{3} b \sqrt [3]{c} \tan ^{-1}\left (\frac{2 \sqrt [3]{c} x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^3])/x^2,x]

[Out]

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

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Maple [A]  time = 0.005, size = 105, normalized size = 1. \begin{align*} -{\frac{a}{x}}-{\frac{b{\it Artanh} \left ( c{x}^{3} \right ) }{x}}-{\frac{b}{2\,c}\ln \left ({x}^{2}-\sqrt [3]{{c}^{-2}} \right ) \left ({c}^{-2} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{4\,c}\ln \left ({x}^{4}+\sqrt [3]{{c}^{-2}}{x}^{2}+ \left ({c}^{-2} \right ) ^{{\frac{2}{3}}} \right ) \left ({c}^{-2} \right ) ^{-{\frac{2}{3}}}}+{\frac{b\sqrt{3}}{2\,c}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{{x}^{2}}{\sqrt [3]{{c}^{-2}}}}+1 \right ) } \right ) \left ({c}^{-2} \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^2,x)

[Out]

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

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Maxima [A]  time = 1.52792, size = 149, normalized size = 1.43 \begin{align*} \frac{1}{4} \,{\left (c{\left (\frac{2 \, \sqrt{3}{\left (c^{2}\right )}^{\frac{2}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (c^{2}\right )}^{\frac{1}{3}}{\left (2 \, x^{2} + \frac{1}{c^{2}}^{\frac{1}{3}}\right )}\right )}{c^{2}} + \frac{{\left (c^{2}\right )}^{\frac{2}{3}} \log \left (x^{4} + \frac{1}{c^{2}}^{\frac{1}{3}} x^{2} + \frac{1}{c^{2}}^{\frac{2}{3}}\right )}{c^{2}} - \frac{2 \,{\left (c^{2}\right )}^{\frac{2}{3}} \log \left (x^{2} - \frac{1}{c^{2}}^{\frac{1}{3}}\right )}{c^{2}}\right )} - \frac{4 \, \operatorname{artanh}\left (c x^{3}\right )}{x}\right )} b - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.82913, size = 312, normalized size = 3. \begin{align*} -\frac{2 \, \sqrt{3} b \left (-c\right )^{\frac{1}{3}} x \arctan \left (\frac{2}{3} \, \sqrt{3} \left (-c\right )^{\frac{2}{3}} x^{2} + \frac{1}{3} \, \sqrt{3}\right ) + b \left (-c\right )^{\frac{1}{3}} x \log \left (c^{2} x^{4} - \left (-c\right )^{\frac{1}{3}} c x^{2} + \left (-c\right )^{\frac{2}{3}}\right ) - 2 \, b \left (-c\right )^{\frac{1}{3}} x \log \left (c x^{2} + \left (-c\right )^{\frac{1}{3}}\right ) + 2 \, b \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right ) + 4 \, a}{4 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))/x^2,x, algorithm="fricas")

[Out]

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

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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**2,x)

[Out]

Exception raised: KeyError

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Giac [A]  time = 1.18253, size = 143, normalized size = 1.38 \begin{align*} \frac{1}{4} \, b c{\left (\frac{2 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + \frac{1}{{\left | c \right |}^{\frac{2}{3}}}\right )}{\left | c \right |}^{\frac{2}{3}}\right )}{{\left | c \right |}^{\frac{2}{3}}} + \frac{\log \left (x^{4} + \frac{x^{2}}{{\left | c \right |}^{\frac{2}{3}}} + \frac{1}{{\left | c \right |}^{\frac{4}{3}}}\right )}{{\left | c \right |}^{\frac{2}{3}}} - \frac{2 \, \log \left ({\left | x^{2} - \frac{1}{{\left | c \right |}^{\frac{2}{3}}} \right |}\right )}{{\left | c \right |}^{\frac{2}{3}}}\right )} - \frac{b \log \left (-\frac{c x^{3} + 1}{c x^{3} - 1}\right )}{2 \, x} - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))/x^2,x, algorithm="giac")

[Out]

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