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

Optimal. Leaf size=142 \[ \frac{2 d^{3/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right ),\frac{1}{2}\right )}{9 e^{3/4} \sqrt{d+e x^2}}-\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{e}}+\frac{2}{3} x^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \]

[Out]

(-4*Sqrt[x]*Sqrt[d + e*x^2])/(9*Sqrt[e]) + (2*x^(3/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/3 + (2*d^(3/4)*(Sq
rt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/
2])/(9*e^(3/4)*Sqrt[d + e*x^2])

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Rubi [A]  time = 0.0705534, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {6221, 321, 329, 220} \[ \frac{2 d^{3/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )|\frac{1}{2}\right )}{9 e^{3/4} \sqrt{d+e x^2}}-\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{e}}+\frac{2}{3} x^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*Sqrt[x]*Sqrt[d + e*x^2])/(9*Sqrt[e]) + (2*x^(3/2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/3 + (2*d^(3/4)*(Sq
rt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/
2])/(9*e^(3/4)*Sqrt[d + e*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 321

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

Rule 329

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

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \, dx &=\frac{2}{3} x^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{1}{3} \left (2 \sqrt{e}\right ) \int \frac{x^{3/2}}{\sqrt{d+e x^2}} \, dx\\ &=-\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{e}}+\frac{2}{3} x^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )+\frac{(2 d) \int \frac{1}{\sqrt{x} \sqrt{d+e x^2}} \, dx}{9 \sqrt{e}}\\ &=-\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{e}}+\frac{2}{3} x^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )+\frac{(4 d) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d+e x^4}} \, dx,x,\sqrt{x}\right )}{9 \sqrt{e}}\\ &=-\frac{4 \sqrt{x} \sqrt{d+e x^2}}{9 \sqrt{e}}+\frac{2}{3} x^{3/2} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )+\frac{2 d^{3/4} \left (\sqrt{d}+\sqrt{e} x\right ) \sqrt{\frac{d+e x^2}{\left (\sqrt{d}+\sqrt{e} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{e} \sqrt{x}}{\sqrt [4]{d}}\right )|\frac{1}{2}\right )}{9 e^{3/4} \sqrt{d+e x^2}}\\ \end{align*}

Mathematica [C]  time = 0.245893, size = 135, normalized size = 0.95 \[ \frac{2}{9} \sqrt{x} \left (3 x \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{2 \sqrt{d+e x^2}}{\sqrt{e}}\right )+\frac{4 \sqrt{d} x \sqrt{\frac{i \sqrt{d}}{\sqrt{e}}} \sqrt{\frac{d}{e x^2}+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{d}}{\sqrt{e}}}}{\sqrt{x}}\right ),-1\right )}{9 \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*((-2*Sqrt[d + e*x^2])/Sqrt[e] + 3*x*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]]))/9 + (4*Sqrt[d]*Sqrt[(I*S
qrt[d])/Sqrt[e]]*Sqrt[1 + d/(e*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d])/Sqrt[e]]/Sqrt[x]], -1])/(9*Sqrt[d
+ e*x^2])

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Maple [F]  time = 1.036, size = 0, normalized size = 0. \begin{align*} \int \sqrt{x}{\it Artanh} \left ({x\sqrt{e}{\frac{1}{\sqrt{e{x}^{2}+d}}}} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -2 \, d \sqrt{e} \int -\frac{x e^{\left (\frac{1}{2} \, \log \left (e x^{2} + d\right ) + \frac{1}{2} \, \log \left (x\right )\right )}}{3 \,{\left (e^{2} x^{4} + d e x^{2} -{\left (e x^{2} + d\right )}^{2}\right )}}\,{d x} + \frac{1}{3} \, x^{\frac{3}{2}} \log \left (\sqrt{e} x + \sqrt{e x^{2} + d}\right ) - \frac{1}{3} \, x^{\frac{3}{2}} \log \left (-\sqrt{e} x + \sqrt{e x^{2} + d}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*d*sqrt(e)*integrate(-1/3*x*e^(1/2*log(e*x^2 + d) + 1/2*log(x))/(e^2*x^4 + d*e*x^2 - (e*x^2 + d)^2), x) + 1/
3*x^(3/2)*log(sqrt(e)*x + sqrt(e*x^2 + d)) - 1/3*x^(3/2)*log(-sqrt(e)*x + sqrt(e*x^2 + d))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{x} \operatorname{artanh}\left (\frac{\sqrt{e} x}{\sqrt{e x^{2} + d}}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x)*arctanh(sqrt(e)*x/sqrt(e*x^2 + d)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 4 \, d e^{\frac{1}{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[undef, undef, undef, undef, 4*d*e^(1/2)]