3.2.85 \(\int \frac {1+x^4}{x^2 \sqrt [4]{-x^2+x^6}} \, dx\)

Optimal. Leaf size=20 \[ \frac {2 \left (x^6-x^2\right )^{3/4}}{3 x^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1590} \begin {gather*} \frac {2 \left (x^6-x^2\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^4)/(x^2*(-x^2 + x^6)^(1/4)),x]

[Out]

(2*(-x^2 + x^6)^(3/4))/(3*x^3)

Rule 1590

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x
, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps

\begin {align*} \int \frac {1+x^4}{x^2 \sqrt [4]{-x^2+x^6}} \, dx &=\frac {2 \left (-x^2+x^6\right )^{3/4}}{3 x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^2 \left (x^4-1\right )\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)/(x^2*(-x^2 + x^6)^(1/4)),x]

[Out]

(2*(x^2*(-1 + x^4))^(3/4))/(3*x^3)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.23, size = 20, normalized size = 1.00 \begin {gather*} \frac {2 \left (-x^2+x^6\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^4)/(x^2*(-x^2 + x^6)^(1/4)),x]

[Out]

(2*(-x^2 + x^6)^(3/4))/(3*x^3)

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 16, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (x^{6} - x^{2}\right )}^{\frac {3}{4}}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/x^2/(x^6-x^2)^(1/4),x, algorithm="fricas")

[Out]

2/3*(x^6 - x^2)^(3/4)/x^3

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/x^2/(x^6-x^2)^(1/4),x, algorithm="giac")

[Out]

integrate((x^4 + 1)/((x^6 - x^2)^(1/4)*x^2), x)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 17, normalized size = 0.85

method	result	size
trager	\(\frac {2 \left (x^{6}-x^{2}\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(17\)
risch	\(\frac {\frac {2 x^{4}}{3}-\frac {2}{3}}{x \left (x^{2} \left (x^{4}-1\right )\right )^{\frac {1}{4}}}\)	\(22\)
gosper	\(\frac {2 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}{3 x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}\)	\(28\)
meijerg	\(-\frac {2 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {3}{8}, \frac {1}{4}\right ], \left [\frac {5}{8}\right ], x^{4}\right )}{3 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {3}{2}}}+\frac {2 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {1}{4}, \frac {5}{8}\right ], \left [\frac {13}{8}\right ], x^{4}\right ) x^{\frac {5}{2}}}{5 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)/x^2/(x^6-x^2)^(1/4),x,method=_RETURNVERBOSE)

[Out]

2/3*(x^6-x^2)^(3/4)/x^3

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{4}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)/x^2/(x^6-x^2)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^4 + 1)/((x^6 - x^2)^(1/4)*x^2), x)

________________________________________________________________________________________

mupad [B]  time = 0.21, size = 16, normalized size = 0.80 \begin {gather*} \frac {2\,{\left (x^6-x^2\right )}^{3/4}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 + 1)/(x^2*(x^6 - x^2)^(1/4)),x)

[Out]

(2*(x^6 - x^2)^(3/4))/(3*x^3)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} + 1}{x^{2} \sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)/x**2/(x**6-x**2)**(1/4),x)

[Out]

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

________________________________________________________________________________________