3.1.12 \(\int \tanh ^{-1}(\frac {\sqrt {e} x}{\sqrt {d+e x^2}}) \, dx\) [12]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 = {6352, 267} \begin {gather*} x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )-\frac {\sqrt {d+e x^2}}{\sqrt {e}} \end {gather*}

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 267

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]

Rule 6352

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]

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*}

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Mathematica [A]
time = 0.01, size = 40, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {d+e x^2}}{\sqrt {e}}+x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(32)=64\).
time = 0.00, size = 76, normalized size = 1.90

method	result	size
default	\(x \arctanh \left (\frac {x \sqrt {e}}{\sqrt {e \,x^{2}+d}}\right )+\frac {e^{\frac {3}{2}} \left (\frac {x^{2} \sqrt {e \,x^{2}+d}}{3 e}-\frac {2 d \sqrt {e \,x^{2}+d}}{3 e^{2}}\right )}{d}-\frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{3 \sqrt {e}\, d}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (32) = 64\).
time = 0.28, size = 66, normalized size = 1.65 \begin {gather*} x \operatorname {artanh}\left (\frac {x e^{\frac {1}{2}}}{\sqrt {x^{2} e + d}}\right ) - \frac {{\left (x^{2} e + d\right )}^{\frac {3}{2}} e^{\left (-\frac {1}{2}\right )}}{3 \, d} + \frac {{\left ({\left (x^{2} e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x^{2} e + d} d\right )} e^{\left (-\frac {1}{2}\right )}}{3 \, d} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (32) = 64\).
time = 0.35, size = 129, normalized size = 3.22 \begin {gather*} \frac {{\left (x \cosh \left (\frac {1}{2}\right ) + x \sinh \left (\frac {1}{2}\right )\right )} \log \left (\frac {2 \, x^{2} \cosh \left (\frac {1}{2}\right )^{2} + 4 \, x^{2} \cosh \left (\frac {1}{2}\right ) \sinh \left (\frac {1}{2}\right ) + 2 \, x^{2} \sinh \left (\frac {1}{2}\right )^{2} + 2 \, {\left (x \cosh \left (\frac {1}{2}\right ) + x \sinh \left (\frac {1}{2}\right )\right )} \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}} + d}{d}\right ) - 2 \, \sqrt {\frac {{\left (x^{2} + d\right )} \cosh \left (\frac {1}{2}\right ) + {\left (x^{2} - d\right )} \sinh \left (\frac {1}{2}\right )}{\cosh \left (\frac {1}{2}\right ) - \sinh \left (\frac {1}{2}\right )}}}{2 \, {\left (\cosh \left (\frac {1}{2}\right ) + \sinh \left (\frac {1}{2}\right )\right )}} \end {gather*}

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

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Sympy [A]
time = 0.33, size = 36, normalized size = 0.90 \begin {gather*} \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 {gather*}

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 [A]
time = 0.42, size = 59, normalized size = 1.48 \begin {gather*} \frac {1}{2} \, x \log \left (-\frac {\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}} + 1}{\frac {\sqrt {e} x}{\sqrt {e x^{2} + d}} - 1}\right ) - \frac {\sqrt {e^{2} x^{2} + d e}}{e} \end {gather*}

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]

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

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Mupad [B]
time = 1.06, size = 32, normalized size = 0.80 \begin {gather*} x\,\mathrm {atanh}\left (\frac {\sqrt {e}\,x}{\sqrt {e\,x^2+d}}\right )-\frac {\sqrt {e\,x^2+d}}{\sqrt {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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