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

Optimal. Leaf size=55 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{x}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{\sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.0310416, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {6221, 266, 63, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{x}-\frac{\sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{\sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 266

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.0422285, size = 61, normalized size = 1.11 \[ \frac{\sqrt{e} \left (\log (x)-\log \left (\sqrt{d} \sqrt{d+e x^2}+d\right )\right )}{\sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.031, size = 53, normalized size = 1. \begin{align*} -{\frac{1}{x}{\it Artanh} \left ({x\sqrt{e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) }-{\sqrt{e}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{e{x}^{2}+d} \right ) } \right ){\frac{1}{\sqrt{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^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} d \sqrt{e} \int -\frac{\sqrt{e x^{2} + d}}{e^{2} x^{5} + d e x^{3} -{\left (e x^{3} + d x\right )}{\left (e x^{2} + d\right )}}\,{d x} - \frac{\log \left (\sqrt{e} x + \sqrt{e x^{2} + d}\right ) - \log \left (-\sqrt{e} x + \sqrt{e x^{2} + d}\right )}{2 \, 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^2,x, algorithm="maxima")

[Out]

d*sqrt(e)*integrate(-sqrt(e*x^2 + d)/(e^2*x^5 + d*e*x^3 - (e*x^3 + d*x)*(e*x^2 + d)), x) - 1/2*(log(sqrt(e)*x
+ sqrt(e*x^2 + d)) - log(-sqrt(e)*x + sqrt(e*x^2 + d)))/x

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Fricas [B]  time = 2.41289, size = 637, normalized size = 11.58 \begin{align*} \left [\frac{x \sqrt{\frac{e}{d}} \log \left (-\frac{e^{2} x^{2} - 2 \, \sqrt{e x^{2} + d} d \sqrt{e} \sqrt{\frac{e}{d}} + 2 \, d e}{x^{2}}\right ) +{\left (x - 1\right )} \log \left (\frac{2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x + d}{d}\right ) - x \log \left (\frac{e x + \sqrt{e x^{2} + d} \sqrt{e}}{x}\right ) + x \log \left (\frac{e x - \sqrt{e x^{2} + d} \sqrt{e}}{x}\right )}{2 \, x}, \frac{2 \, x \sqrt{-\frac{e}{d}} \arctan \left (\frac{\sqrt{e x^{2} + d} d \sqrt{e} \sqrt{-\frac{e}{d}}}{e^{2} x^{2} + d e}\right ) +{\left (x - 1\right )} \log \left (\frac{2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x + d}{d}\right ) - x \log \left (\frac{e x + \sqrt{e x^{2} + d} \sqrt{e}}{x}\right ) + x \log \left (\frac{e x - \sqrt{e x^{2} + d} \sqrt{e}}{x}\right )}{2 \, x}\right ] \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^2,x, algorithm="fricas")

[Out]

[1/2*(x*sqrt(e/d)*log(-(e^2*x^2 - 2*sqrt(e*x^2 + d)*d*sqrt(e)*sqrt(e/d) + 2*d*e)/x^2) + (x - 1)*log((2*e*x^2 +
2*sqrt(e*x^2 + d)*sqrt(e)*x + d)/d) - x*log((e*x + sqrt(e*x^2 + d)*sqrt(e))/x) + x*log((e*x - sqrt(e*x^2 + d)
*sqrt(e))/x))/x, 1/2*(2*x*sqrt(-e/d)*arctan(sqrt(e*x^2 + d)*d*sqrt(e)*sqrt(-e/d)/(e^2*x^2 + d*e)) + (x - 1)*lo
g((2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x + d)/d) - x*log((e*x + sqrt(e*x^2 + d)*sqrt(e))/x) + x*log((e*x - sqr
t(e*x^2 + d)*sqrt(e))/x))/x]

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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^{2}}\, 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**2,x)

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \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^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError