\(\int \frac {1}{(a+b x^2) \sqrt {4-d x^4}} \, dx\) [172]

Optimal result

Integrand size = 22, antiderivative size = 40 \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-d x^4}} \, dx=\frac {\operatorname {EllipticPi}\left (-\frac {2 b}{a \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} x}{\sqrt {2}}\right ),-1\right )}{\sqrt {2} a \sqrt [4]{d}} \]

[Out]

1/2*EllipticPi(1/2*d^(1/4)*x*2^(1/2),-2*b/a/d^(1/2),I)/a/d^(1/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1232} \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-d x^4}} \, dx=\frac {\operatorname {EllipticPi}\left (-\frac {2 b}{a \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} x}{\sqrt {2}}\right ),-1\right )}{\sqrt {2} a \sqrt [4]{d}} \]

[In]

Int[1/((a + b*x^2)*Sqrt[4 - d*x^4]),x]

[Out]

EllipticPi[(-2*b)/(a*Sqrt[d]), ArcSin[(d^(1/4)*x)/Sqrt[2]], -1]/(Sqrt[2]*a*d^(1/4))

Rule 1232

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[
a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\Pi \left (-\frac {2 b}{a \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} x}{\sqrt {2}}\right )\right |-1\right )}{\sqrt {2} a \sqrt [4]{d}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-d x^4}} \, dx=-\frac {i \operatorname {EllipticPi}\left (-\frac {2 b}{a \sqrt {d}},i \text {arcsinh}\left (\frac {\sqrt {-\sqrt {d}} x}{\sqrt {2}}\right ),-1\right )}{\sqrt {2} a \sqrt {-\sqrt {d}}} \]

[In]

Integrate[1/((a + b*x^2)*Sqrt[4 - d*x^4]),x]

[Out]

((-I)*EllipticPi[(-2*b)/(a*Sqrt[d]), I*ArcSinh[(Sqrt[-Sqrt[d]]*x)/Sqrt[2]], -1])/(Sqrt[2]*a*Sqrt[-Sqrt[d]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(77\) vs. \(2(32)=64\).

Time = 1.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.95

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {1-\frac {x^{2} \sqrt {d}}{2}}\, \sqrt {1+\frac {x^{2} \sqrt {d}}{2}}\, \Pi \left (\frac {d^{\frac {1}{4}} x \sqrt {2}}{2}, -\frac {2 b}{a \sqrt {d}}, \frac {\sqrt {-\frac {\sqrt {d}}{2}}\, \sqrt {2}}{d^{\frac {1}{4}}}\right )}{a \,d^{\frac {1}{4}} \sqrt {-d \,x^{4}+4}}\)	\(78\)
elliptic	\(\frac {\sqrt {2}\, \sqrt {1-\frac {x^{2} \sqrt {d}}{2}}\, \sqrt {1+\frac {x^{2} \sqrt {d}}{2}}\, \Pi \left (\frac {d^{\frac {1}{4}} x \sqrt {2}}{2}, -\frac {2 b}{a \sqrt {d}}, \frac {\sqrt {-\frac {\sqrt {d}}{2}}\, \sqrt {2}}{d^{\frac {1}{4}}}\right )}{a \,d^{\frac {1}{4}} \sqrt {-d \,x^{4}+4}}\)	\(78\)

[In]

int(1/(b*x^2+a)/(-d*x^4+4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(-sqrt(-d*x^4 + 4)/(b*d*x^6 + a*d*x^4 - 4*b*x^2 - 4*a), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-d x^4}} \, dx=\int \frac {1}{\left (b\,x^2+a\right )\,\sqrt {4-d\,x^4}} \,d x \]

[In]

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

[Out]

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