3.3.69 \(\int \frac {-1+x^4}{x^8 \sqrt [4]{-1+2 x^4}} \, dx\)

Optimal. Leaf size=25 \[ \frac {\left (-x^4-3\right ) \left (2 x^4-1\right )^{3/4}}{21 x^7} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {453, 264} \begin {gather*} -\frac {\left (2 x^4-1\right )^{3/4}}{7 x^7}-\frac {\left (2 x^4-1\right )^{3/4}}{21 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^4)/(x^8*(-1 + 2*x^4)^(1/4)),x]

[Out]

-1/7*(-1 + 2*x^4)^(3/4)/x^7 - (-1 + 2*x^4)^(3/4)/(21*x^3)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {-1+x^4}{x^8 \sqrt [4]{-1+2 x^4}} \, dx &=-\frac {\left (-1+2 x^4\right )^{3/4}}{7 x^7}-\frac {1}{7} \int \frac {1}{x^4 \sqrt [4]{-1+2 x^4}} \, dx\\ &=-\frac {\left (-1+2 x^4\right )^{3/4}}{7 x^7}-\frac {\left (-1+2 x^4\right )^{3/4}}{21 x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 23, normalized size = 0.92 \begin {gather*} -\frac {\left (x^4+3\right ) \left (2 x^4-1\right )^{3/4}}{21 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^4)/(x^8*(-1 + 2*x^4)^(1/4)),x]

[Out]

-1/21*((3 + x^4)*(-1 + 2*x^4)^(3/4))/x^7

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.17, size = 25, normalized size = 1.00 \begin {gather*} \frac {\left (-3-x^4\right ) \left (-1+2 x^4\right )^{3/4}}{21 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^4)/(x^8*(-1 + 2*x^4)^(1/4)),x]

[Out]

((-3 - x^4)*(-1 + 2*x^4)^(3/4))/(21*x^7)

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 19, normalized size = 0.76 \begin {gather*} -\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} + 3\right )}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)/x^8/(2*x^4-1)^(1/4),x, algorithm="fricas")

[Out]

-1/21*(2*x^4 - 1)^(3/4)*(x^4 + 3)/x^7

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} - 1}{{\left (2 \, x^{4} - 1\right )}^{\frac {1}{4}} x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)/x^8/(2*x^4-1)^(1/4),x, algorithm="giac")

[Out]

integrate((x^4 - 1)/((2*x^4 - 1)^(1/4)*x^8), x)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 20, normalized size = 0.80

method	result	size
gosper	\(-\frac {\left (x^{4}+3\right ) \left (2 x^{4}-1\right )^{\frac {3}{4}}}{21 x^{7}}\)	\(20\)
trager	\(-\frac {\left (x^{4}+3\right ) \left (2 x^{4}-1\right )^{\frac {3}{4}}}{21 x^{7}}\)	\(20\)
risch	\(-\frac {2 x^{8}+5 x^{4}-3}{21 x^{7} \left (2 x^{4}-1\right )^{\frac {1}{4}}}\)	\(27\)
meijerg	\(\frac {\left (-\mathrm {signum}\left (2 x^{4}-1\right )\right )^{\frac {1}{4}} \left (1+\frac {8 x^{4}}{3}\right ) \left (-2 x^{4}+1\right )^{\frac {3}{4}}}{7 \mathrm {signum}\left (2 x^{4}-1\right )^{\frac {1}{4}} x^{7}}-\frac {\left (-\mathrm {signum}\left (2 x^{4}-1\right )\right )^{\frac {1}{4}} \left (-2 x^{4}+1\right )^{\frac {3}{4}}}{3 \mathrm {signum}\left (2 x^{4}-1\right )^{\frac {1}{4}} x^{3}}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-1)/x^8/(2*x^4-1)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-1/21*(x^4+3)*(2*x^4-1)^(3/4)/x^7

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 29, normalized size = 1.16 \begin {gather*} -\frac {{\left (2 \, x^{4} - 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} + \frac {{\left (2 \, x^{4} - 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)/x^8/(2*x^4-1)^(1/4),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 30, normalized size = 1.20 \begin {gather*} -\frac {x^4\,{\left (2\,x^4-1\right )}^{3/4}+3\,{\left (2\,x^4-1\right )}^{3/4}}{21\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 - 1)/(x^8*(2*x^4 - 1)^(1/4)),x)

[Out]

-(x^4*(2*x^4 - 1)^(3/4) + 3*(2*x^4 - 1)^(3/4))/(21*x^7)

________________________________________________________________________________________

sympy [C]  time = 2.11, size = 233, normalized size = 9.32 \begin {gather*} \begin {cases} - \frac {2^{\frac {3}{4}} \left (-1 + \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {3}{4}\right )}{4 \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \frac {1}{2 \left |{x^{4}}\right |} > 1 \\- \frac {2^{\frac {3}{4}} \left (1 - \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{4 \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} - \begin {cases} - \frac {2^{\frac {3}{4}} \left (-1 + \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{4}\right )}{2 \Gamma \left (\frac {1}{4}\right )} - \frac {3 \cdot 2^{\frac {3}{4}} \left (-1 + \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{4}\right )}{16 x^{4} \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \frac {1}{2 \left |{x^{4}}\right |} > 1 \\\frac {2^{\frac {3}{4}} \left (1 - \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{2 \Gamma \left (\frac {1}{4}\right )} + \frac {3 \cdot 2^{\frac {3}{4}} \left (1 - \frac {1}{2 x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {7}{4}\right )}{16 x^{4} \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-1)/x**8/(2*x**4-1)**(1/4),x)

[Out]

Piecewise((-2**(3/4)*(-1 + 1/(2*x**4))**(3/4)*exp(3*I*pi/4)*gamma(-3/4)/(4*gamma(1/4)), 1/(2*Abs(x**4)) > 1),
(-2**(3/4)*(1 - 1/(2*x**4))**(3/4)*gamma(-3/4)/(4*gamma(1/4)), True)) - Piecewise((-2**(3/4)*(-1 + 1/(2*x**4))
**(3/4)*exp(-I*pi/4)*gamma(-7/4)/(2*gamma(1/4)) - 3*2**(3/4)*(-1 + 1/(2*x**4))**(3/4)*exp(-I*pi/4)*gamma(-7/4)
/(16*x**4*gamma(1/4)), 1/(2*Abs(x**4)) > 1), (2**(3/4)*(1 - 1/(2*x**4))**(3/4)*gamma(-7/4)/(2*gamma(1/4)) + 3*
2**(3/4)*(1 - 1/(2*x**4))**(3/4)*gamma(-7/4)/(16*x**4*gamma(1/4)), True))

________________________________________________________________________________________