3.5 \(\int \frac{\tanh ^{-1}(\frac{\sqrt{e} x}{\sqrt{d+e x^2}})}{x^3} \, dx\)

Optimal. Leaf size=53 \[ -\frac{\sqrt{e} \sqrt{d+e x^2}}{2 d x}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 x^2} \]

[Out]

-(Sqrt[e]*Sqrt[d + e*x^2])/(2*d*x) - ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/(2*x^2)

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Rubi [A]  time = 0.0186087, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {6221, 264} \[ -\frac{\sqrt{e} \sqrt{d+e x^2}}{2 d x}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x^3,x]

[Out]

-(Sqrt[e]*Sqrt[d + e*x^2])/(2*d*x) - ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/(2*x^2)

Rule 6221

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

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{x^3} \, dx &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 x^2}+\frac{1}{2} \sqrt{e} \int \frac{1}{x^2 \sqrt{d+e x^2}} \, dx\\ &=-\frac{\sqrt{e} \sqrt{d+e x^2}}{2 d x}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.036218, size = 50, normalized size = 0.94 \[ -\frac{\sqrt{e} x \sqrt{d+e x^2}+d \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 d x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]/x^3,x]

[Out]

-(Sqrt[e]*x*Sqrt[d + e*x^2] + d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*d*x^2)

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Maple [A]  time = 0.031, size = 60, normalized size = 1.1 \begin{align*} -{\frac{1}{2\,{x}^{2}}{\it Artanh} \left ({x\sqrt{e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }-{\frac{1}{2\,{d}^{2}x}\sqrt{e} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{x}{2\,{d}^{2}}{e}^{{\frac{3}{2}}}\sqrt{e{x}^{2}+d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^3,x)

[Out]

-1/2*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^2-1/2*e^(1/2)/d^2/x*(e*x^2+d)^(3/2)+1/2*e^(3/2)/d^2*x*(e*x^2+d)^(1/2
)

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Maxima [A]  time = 1.10477, size = 69, normalized size = 1.3 \begin{align*} -\frac{\operatorname{artanh}\left (\frac{\sqrt{e} x}{\sqrt{e x^{2} + d}}\right )}{2 \, x^{2}} - \frac{e^{\frac{3}{2}} x^{2} + d \sqrt{e}}{2 \, \sqrt{e x^{2} + d} d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^3,x, algorithm="maxima")

[Out]

-1/2*arctanh(sqrt(e)*x/sqrt(e*x^2 + d))/x^2 - 1/2*(e^(3/2)*x^2 + d*sqrt(e))/(sqrt(e*x^2 + d)*d*x)

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Fricas [A]  time = 2.22698, size = 134, normalized size = 2.53 \begin{align*} -\frac{2 \, \sqrt{e x^{2} + d} \sqrt{e} x + d \log \left (\frac{2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x + d}{d}\right )}{4 \, d x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^3,x, algorithm="fricas")

[Out]

-1/4*(2*sqrt(e*x^2 + d)*sqrt(e)*x + d*log((2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x + d)/d))/(d*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(x*e**(1/2)/(e*x**2+d)**(1/2))/x**3,x)

[Out]

Integral(atanh(sqrt(e)*x/sqrt(d + e*x**2))/x**3, x)

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Giac [A]  time = 1.24982, size = 96, normalized size = 1.81 \begin{align*} \frac{e}{{\left (x e^{\frac{1}{2}} - \sqrt{x^{2} e + d}\right )}^{2} - d} - \frac{\log \left (-\frac{\frac{x e^{\frac{1}{2}}}{\sqrt{x^{2} e + d}} + 1}{\frac{x e^{\frac{1}{2}}}{\sqrt{x^{2} e + d}} - 1}\right )}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/x^3,x, algorithm="giac")

[Out]

e/((x*e^(1/2) - sqrt(x^2*e + d))^2 - d) - 1/4*log(-(x*e^(1/2)/sqrt(x^2*e + d) + 1)/(x*e^(1/2)/sqrt(x^2*e + d)
- 1))/x^2