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

Optimal. Leaf size=40 \[ x \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{\sqrt{d+e x^2}}{\sqrt{e}} \]

[Out]

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

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Rubi [A]  time = 0.0086467, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {6217, 261} \[ x \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{\sqrt{d+e x^2}}{\sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6217

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

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.010424, size = 40, normalized size = 1. \[ x \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{\sqrt{d+e x^2}}{\sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.029, size = 76, normalized size = 1.9 \begin{align*} x{\it Artanh} \left ({x\sqrt{e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) +{\frac{1}{d}{e}^{{\frac{3}{2}}} \left ({\frac{{x}^{2}}{3\,e}\sqrt{e{x}^{2}+d}}-{\frac{2\,d}{3\,{e}^{2}}\sqrt{e{x}^{2}+d}} \right ) }-{\frac{1}{3\,d} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{e}}}} \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)

[Out]

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

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Maxima [B]  time = 0.976917, size = 88, normalized size = 2.2 \begin{align*} x \operatorname{artanh}\left (\frac{\sqrt{e} x}{\sqrt{e x^{2} + d}}\right ) - \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}{3 \, d \sqrt{e}} + \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x^{2} + d} d}{3 \, d \sqrt{e}} \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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.13618, size = 124, normalized size = 3.1 \begin{align*} \frac{e x \log \left (\frac{2 \, e x^{2} + 2 \, \sqrt{e x^{2} + d} \sqrt{e} x + d}{d}\right ) - 2 \, \sqrt{e x^{2} + d} \sqrt{e}}{2 \, e} \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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.49326, size = 36, normalized size = 0.9 \begin{align*} \begin{cases} x \operatorname{atanh}{\left (\frac{\sqrt{e} x}{\sqrt{d + e x^{2}}} \right )} - \frac{\sqrt{d + e x^{2}}}{\sqrt{e}} & \text{for}\: e \neq 0 \\0 & \text{otherwise} \end{cases} \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)

[Out]

Piecewise((x*atanh(sqrt(e)*x/sqrt(d + e*x**2)) - sqrt(d + e*x**2)/sqrt(e), Ne(e, 0)), (0, True))

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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, algorithm="giac")

[Out]

Exception raised: NotImplementedError