\(\int \frac {1}{\sqrt {2+e x} \sqrt {12-3 e^2 x^2}} \, dx\) [915]

Optimal result

Integrand size = 24, antiderivative size = 25 \[ \int \frac {1}{\sqrt {2+e x} \sqrt {12-3 e^2 x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{\sqrt {3} e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {641, 65, 212} \[ \int \frac {1}{\sqrt {2+e x} \sqrt {12-3 e^2 x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {1}{2} \sqrt {2-e x}\right )}{\sqrt {3} e} \]

[In]

Int[1/(Sqrt[2 + e*x]*Sqrt[12 - 3*e^2*x^2]),x]

[Out]

-(ArcTanh[Sqrt[2 - e*x]/2]/(Sqrt[3]*e))

Rule 65

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {6-3 e x} (2+e x)} \, dx \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{4-\frac {x^2}{3}} \, dx,x,\sqrt {6-3 e x}\right )}{3 e} \\ & = -\frac {\tanh ^{-1}\left (\frac {1}{2} \sqrt {2-e x}\right )}{\sqrt {3} e} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(25)=50\).

Time = 0.35 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.84 \[ \int \frac {1}{\sqrt {2+e x} \sqrt {12-3 e^2 x^2}} \, dx=\frac {\log \left (-2 \sqrt {2+e x}+\sqrt {4-e^2 x^2}\right )-\log \left (e \left (2 \sqrt {2+e x}+\sqrt {4-e^2 x^2}\right )\right )}{2 \sqrt {3} e} \]

[In]

Integrate[1/(Sqrt[2 + e*x]*Sqrt[12 - 3*e^2*x^2]),x]

[Out]

(Log[-2*Sqrt[2 + e*x] + Sqrt[4 - e^2*x^2]] - Log[e*(2*Sqrt[2 + e*x] + Sqrt[4 - e^2*x^2])])/(2*Sqrt[3]*e)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(19)=38\).

Time = 2.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.00

method	result	size
default	\(-\frac {\sqrt {-x^{2} e^{2}+4}\, \sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {-3 e x +6}\, \sqrt {3}}{6}\right )}{3 \sqrt {e x +2}\, \sqrt {-e x +2}\, e}\)	\(50\)

[In]

int(1/(e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(-e^2*x^2+4)^(1/2)*3^(1/2)*arctanh(1/6*(-3*e*x+6)^(1/2)*3^(1/2))/(e*x+2)^(1/2)/(-e*x+2)^(1/2)/e

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (19) = 38\).

Time = 0.42 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int \frac {1}{\sqrt {2+e x} \sqrt {12-3 e^2 x^2}} \, dx=\frac {\sqrt {3} \log \left (-\frac {3 \, e^{2} x^{2} - 12 \, e x + 4 \, \sqrt {3} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2} - 36}{e^{2} x^{2} + 4 \, e x + 4}\right )}{6 \, e} \]

[In]

integrate(1/(e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/2),x, algorithm="fricas")

[Out]

1/6*sqrt(3)*log(-(3*e^2*x^2 - 12*e*x + 4*sqrt(3)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2) - 36)/(e^2*x^2 + 4*e*x +
4))/e

Sympy [F]

\[ \int \frac {1}{\sqrt {2+e x} \sqrt {12-3 e^2 x^2}} \, dx=\frac {\sqrt {3} \int \frac {1}{\sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}}\, dx}{3} \]

[In]

integrate(1/(e*x+2)**(1/2)/(-3*e**2*x**2+12)**(1/2),x)

[Out]

sqrt(3)*Integral(1/(sqrt(e*x + 2)*sqrt(-e**2*x**2 + 4)), x)/3

Maxima [F]

\[ \int \frac {1}{\sqrt {2+e x} \sqrt {12-3 e^2 x^2}} \, dx=\int { \frac {1}{\sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}} \,d x } \]

[In]

integrate(1/(e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\sqrt {2+e x} \sqrt {12-3 e^2 x^2}} \, dx=-\frac {\sqrt {3} {\left (\log \left (\sqrt {-e x + 2} + 2\right ) - \log \left (-\sqrt {-e x + 2} + 2\right )\right )}}{6 \, e} \]

[In]

integrate(1/(e*x+2)^(1/2)/(-3*e^2*x^2+12)^(1/2),x, algorithm="giac")

[Out]

-1/6*sqrt(3)*(log(sqrt(-e*x + 2) + 2) - log(-sqrt(-e*x + 2) + 2))/e

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {2+e x} \sqrt {12-3 e^2 x^2}} \, dx=\int \frac {1}{\sqrt {12-3\,e^2\,x^2}\,\sqrt {e\,x+2}} \,d x \]

[In]

int(1/((12 - 3*e^2*x^2)^(1/2)*(e*x + 2)^(1/2)),x)

[Out]

int(1/((12 - 3*e^2*x^2)^(1/2)*(e*x + 2)^(1/2)), x)