\(\int \frac {1}{\sqrt {-1+x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx\) [302]

Optimal result

Integrand size = 26, antiderivative size = 46 \[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx=\frac {\sqrt {1-x^2} \operatorname {EllipticF}\left (\arcsin (x),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \sqrt {-1+x^2}} \]

[Out]

EllipticF(x,I*3^(1/2)+2*I)*(-x^2+1)^(1/2)/(x^2-1)^(1/2)/(2-3^(1/2))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {432, 430} \[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx=\frac {\sqrt {1-x^2} \operatorname {EllipticF}\left (\arcsin (x),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \sqrt {x^2-1}} \]

[In]

Int[1/(Sqrt[-1 + x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]),x]

[Out]

(Sqrt[1 - x^2]*EllipticF[ArcSin[x], -7 - 4*Sqrt[3]])/(Sqrt[7 - 4*Sqrt[3]]*Sqrt[-1 + x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-x^2} \int \frac {1}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx}{\sqrt {-1+x^2}} \\ & = \frac {\sqrt {1-x^2} F\left (\sin ^{-1}(x)|-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \sqrt {-1+x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx=\frac {\sqrt {1-x^2} \operatorname {EllipticF}\left (\arcsin (x),\frac {1}{-7+4 \sqrt {3}}\right )}{\sqrt {7-4 \sqrt {3}} \sqrt {-1+x^2}} \]

[In]

Integrate[1/(Sqrt[-1 + x^2]*Sqrt[7 - 4*Sqrt[3] + x^2]),x]

[Out]

(Sqrt[1 - x^2]*EllipticF[ArcSin[x], (-7 + 4*Sqrt[3])^(-1)])/(Sqrt[7 - 4*Sqrt[3]]*Sqrt[-1 + x^2])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (37 ) = 74\).

Time = 2.51 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.54

method	result	size
default	\(-\frac {i F\left (\frac {i x}{-2+\sqrt {3}}, 2 i-i \sqrt {3}\right ) \sqrt {-x^{2}+1}\, \sqrt {-\left (-x^{2}+4 \sqrt {3}-7\right ) \left (-4 \sqrt {3}+7\right )}\, \left (-2+\sqrt {3}\right ) \sqrt {x^{2}-1}\, \sqrt {7+x^{2}-4 \sqrt {3}}}{\left (4 \sqrt {3}-7\right ) \left (-x^{4}+4 x^{2} \sqrt {3}-6 x^{2}-4 \sqrt {3}+7\right )}\)	\(117\)
elliptic	\(-\frac {i \sqrt {-\left (x^{2}-1\right ) \left (-x^{2}+4 \sqrt {3}-7\right )}\, \sqrt {-4 \sqrt {3}+7}\, \sqrt {1-\frac {x^{2}}{4 \sqrt {3}-7}}\, \sqrt {-x^{2}+1}\, F\left (\frac {i x}{\sqrt {-4 \sqrt {3}+7}}, 2 i-i \sqrt {3}\right )}{\sqrt {x^{2}-1}\, \sqrt {7+x^{2}-4 \sqrt {3}}\, \sqrt {6 x^{2}-7+x^{4}-4 x^{2} \sqrt {3}+4 \sqrt {3}}}\)	\(128\)

[In]

int(1/(x^2-1)^(1/2)/(7+x^2-4*3^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-I*EllipticF(I*x/(-2+3^(1/2)),2*I-I*3^(1/2))*(-x^2+1)^(1/2)*(-(-x^2+4*3^(1/2)-7)*(-4*3^(1/2)+7))^(1/2)/(4*3^(1
/2)-7)*(-2+3^(1/2))*(x^2-1)^(1/2)*(7+x^2-4*3^(1/2))^(1/2)/(-x^4+4*x^2*3^(1/2)-6*x^2-4*3^(1/2)+7)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (34) = 68\).

Time = 0.10 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.57 \[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx=-\frac {1}{2} \, {\left ({\left (2 \, \sqrt {3} \sqrt {2} + 3 \, \sqrt {2}\right )} \sqrt {4 \, \sqrt {3} - 7} + 2 \, \sqrt {2} \sqrt {4 \, \sqrt {3} + 7} \sqrt {4 \, \sqrt {3} - 7}\right )} \sqrt {-4 \, \sqrt {3} + 4 \, \sqrt {4 \, \sqrt {3} + 7} - 6} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x \sqrt {-4 \, \sqrt {3} + 4 \, \sqrt {4 \, \sqrt {3} + 7} - 6}\right )\,|\,-4 \, \sqrt {4 \, \sqrt {3} + 7} {\left (2 \, \sqrt {3} - 3\right )} - 7) \]

[In]

integrate(1/(x^2-1)^(1/2)/(7+x^2-4*3^(1/2))^(1/2),x, algorithm="fricas")

[Out]

-1/2*((2*sqrt(3)*sqrt(2) + 3*sqrt(2))*sqrt(4*sqrt(3) - 7) + 2*sqrt(2)*sqrt(4*sqrt(3) + 7)*sqrt(4*sqrt(3) - 7))
*sqrt(-4*sqrt(3) + 4*sqrt(4*sqrt(3) + 7) - 6)*elliptic_f(arcsin(1/2*sqrt(2)*x*sqrt(-4*sqrt(3) + 4*sqrt(4*sqrt(
3) + 7) - 6)), -4*sqrt(4*sqrt(3) + 7)*(2*sqrt(3) - 3) - 7)

Sympy [F]

\[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx=\int \frac {1}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \sqrt {x^{2} - 4 \sqrt {3} + 7}}\, dx \]

[In]

integrate(1/(x**2-1)**(1/2)/(7+x**2-4*3**(1/2))**(1/2),x)

[Out]

Integral(1/(sqrt((x - 1)*(x + 1))*sqrt(x**2 - 4*sqrt(3) + 7)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} - 4 \, \sqrt {3} + 7} \sqrt {x^{2} - 1}} \,d x } \]

[In]

integrate(1/(x^2-1)^(1/2)/(7+x^2-4*3^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^2 - 4*sqrt(3) + 7)*sqrt(x^2 - 1)), x)

Giac [F]

\[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx=\int { \frac {1}{\sqrt {x^{2} - 4 \, \sqrt {3} + 7} \sqrt {x^{2} - 1}} \,d x } \]

[In]

integrate(1/(x^2-1)^(1/2)/(7+x^2-4*3^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^2 - 4*sqrt(3) + 7)*sqrt(x^2 - 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-1+x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx=\int \frac {1}{\sqrt {x^2-1}\,\sqrt {x^2-4\,\sqrt {3}+7}} \,d x \]

[In]

int(1/((x^2 - 1)^(1/2)*(x^2 - 4*3^(1/2) + 7)^(1/2)),x)

[Out]

int(1/((x^2 - 1)^(1/2)*(x^2 - 4*3^(1/2) + 7)^(1/2)), x)