\(\int \frac {1-c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx\) [25]

Optimal result

Integrand size = 25, antiderivative size = 23 \[ \int \frac {1-c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=-\frac {E(\arcsin (c x)|-1)}{c}+\frac {2 \operatorname {EllipticF}(\arcsin (c x),-1)}{c} \]

[Out]

-EllipticE(c*x,I)/c+2*EllipticF(c*x,I)/c

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1213, 434, 435, 254, 227} \[ \int \frac {1-c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=\frac {2 \operatorname {EllipticF}(\arcsin (c x),-1)}{c}-\frac {E(\arcsin (c x)|-1)}{c} \]

[In]

Int[(1 - c^2*x^2)/Sqrt[1 - c^4*x^4],x]

[Out]

-(EllipticE[ArcSin[c*x], -1]/c) + (2*EllipticF[ArcSin[c*x], -1])/c

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 254

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_.)*((a2_.) + (b2_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[(a1*a2 + b1*b2*x^(2*
n))^p, x] /; FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a
2, 0]))

Rule 434

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

Rule 435

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

Rule 1213

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

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {1-c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^4 x^4\right )-\frac {1}{3} c^2 x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},c^4 x^4\right ) \]

[In]

Integrate[(1 - c^2*x^2)/Sqrt[1 - c^4*x^4],x]

[Out]

x*Hypergeometric2F1[1/4, 1/2, 5/4, c^4*x^4] - (c^2*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, c^4*x^4])/3

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 1.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65

method	result	size
meijerg	\(-\frac {c^{2} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};c^{4} x^{4}\right )}{3}+x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^{4} x^{4}\right )\)	\(38\)
default	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, F\left (x \sqrt {c^{2}}, i\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, \left (F\left (x \sqrt {c^{2}}, i\right )-E\left (x \sqrt {c^{2}}, i\right )\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}\)	\(117\)
elliptic	\(\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, F\left (x \sqrt {c^{2}}, i\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}\, \left (F\left (x \sqrt {c^{2}}, i\right )-E\left (x \sqrt {c^{2}}, i\right )\right )}{\sqrt {c^{2}}\, \sqrt {-c^{4} x^{4}+1}}\)	\(117\)

[In]

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

[Out]

-1/3*c^2*x^3*hypergeom([1/2,3/4],[7/4],c^4*x^4)+x*hypergeom([1/4,1/2],[5/4],c^4*x^4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (21) = 42\).

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.70 \[ \int \frac {1-c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=\frac {\sqrt {-c^{4} x^{4} + 1} c^{3} + \sqrt {-c^{4}} {\left ({\left (c^{2} - 1\right )} x F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1) + x E(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)\right )}}{c^{5} x} \]

[In]

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

[Out]

(sqrt(-c^4*x^4 + 1)*c^3 + sqrt(-c^4)*((c^2 - 1)*x*elliptic_f(arcsin(1/(c*x)), -1) + x*elliptic_e(arcsin(1/(c*x
)), -1)))/(c^5*x)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (14) = 28\).

Time = 0.86 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.09 \[ \int \frac {1-c^2 x^2}{\sqrt {1-c^4 x^4}} \, dx=- \frac {c^{2} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {c^{4} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {c^{4} x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]

[In]

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

[Out]

-c**2*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c**4*x**4*exp_polar(2*I*pi))/(4*gamma(7/4)) + x*gamma(1/4)*hyp
er((1/4, 1/2), (5/4,), c**4*x**4*exp_polar(2*I*pi))/(4*gamma(5/4))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(-(c^2*x^2 - 1)/(1 - c^4*x^4)^(1/2),x)

[Out]

-int((c^2*x^2 - 1)/(1 - c^4*x^4)^(1/2), x)