3.412 \(\int \frac {1}{\sqrt {\frac {a-b x^4}{x^2}}} \, dx\)

Optimal. Leaf size=33 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^2}-b x^2}}\right )}{2 \sqrt {b}} \]

[Out]

1/2*arctan(x*b^(1/2)/(a/x^2-b*x^2)^(1/2))/b^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1979, 2008, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^2}-b x^2}}\right )}{2 \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(a - b*x^4)/x^2],x]

[Out]

ArcTan[(Sqrt[b]*x)/Sqrt[a/x^2 - b*x^2]]/(2*Sqrt[b])

Rule 203

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

Rule 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2008

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[2/(2 - n), Subst[Int[1/(1 - a*x^2), x], x, x/Sq
rt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {\frac {a-b x^4}{x^2}}} \, dx &=\int \frac {1}{\sqrt {\frac {a}{x^2}-b x^2}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {x}{\sqrt {\frac {a}{x^2}-b x^2}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^2}-b x^2}}\right )}{2 \sqrt {b}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 62, normalized size = 1.88 \[ \frac {\sqrt {a-b x^4} \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}\right )}{2 \sqrt {b} x \sqrt {\frac {a-b x^4}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(a - b*x^4)/x^2],x]

[Out]

(Sqrt[a - b*x^4]*ArcTan[(Sqrt[b]*x^2)/Sqrt[a - b*x^4]])/(2*Sqrt[b]*x*Sqrt[(a - b*x^4)/x^2])

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 88, normalized size = 2.67 \[ \left [-\frac {\sqrt {-b} \log \left (2 \, b x^{4} - 2 \, \sqrt {-b} x^{3} \sqrt {-\frac {b x^{4} - a}{x^{2}}} - a\right )}{4 \, b}, -\frac {\arctan \left (\frac {\sqrt {b} x^{3} \sqrt {-\frac {b x^{4} - a}{x^{2}}}}{b x^{4} - a}\right )}{2 \, \sqrt {b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*sqrt(-b)*log(2*b*x^4 - 2*sqrt(-b)*x^3*sqrt(-(b*x^4 - a)/x^2) - a)/b, -1/2*arctan(sqrt(b)*x^3*sqrt(-(b*x^
4 - a)/x^2)/(b*x^4 - a))/sqrt(b)]

________________________________________________________________________________________

giac [A]  time = 0.19, size = 47, normalized size = 1.42 \[ \frac {\log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{4 \, \sqrt {-b}} - \frac {\log \left ({\left | -\sqrt {-b} x^{2} + \sqrt {-b x^{4} + a} \right |}\right )}{2 \, \sqrt {-b} \mathrm {sgn}\relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(abs(a))*sgn(x)/sqrt(-b) - 1/2*log(abs(-sqrt(-b)*x^2 + sqrt(-b*x^4 + a)))/(sqrt(-b)*sgn(x))

________________________________________________________________________________________

maple [B]  time = 0.19, size = 53, normalized size = 1.61 \[ \frac {\sqrt {-b \,x^{4}+a}\, \arctan \left (\frac {\sqrt {b}\, x^{2}}{\sqrt {-b \,x^{4}+a}}\right )}{2 \sqrt {-\frac {b \,x^{4}-a}{x^{2}}}\, \sqrt {b}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((-b*x^4+a)/x^2)^(1/2),x)

[Out]

1/2/(-(b*x^4-a)/x^2)^(1/2)/x*(-b*x^4+a)^(1/2)/b^(1/2)*arctan(1/(-b*x^4+a)^(1/2)*b^(1/2)*x^2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {x^{5}}{{\left (b x^{4} - a\right )} \sqrt {-b x^{4} + a}}\,{d x} + \frac {x^{2}}{2 \, \sqrt {-b x^{4} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*integrate(x^5/((b*x^4 - a)*sqrt(-b*x^4 + a)), x) + 1/2*x^2/sqrt(-b*x^4 + a)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\sqrt {\frac {a-b\,x^4}{x^2}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a - b*x^4)/x^2)^(1/2),x)

[Out]

int(1/((a - b*x^4)/x^2)^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\frac {a - b x^{4}}{x^{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-b*x**4+a)/x**2)**(1/2),x)

[Out]

Integral(1/sqrt((a - b*x**4)/x**2), x)

________________________________________________________________________________________