3.9.82 \(\int \frac {1}{x^7 \sqrt {1-x^4}} \, dx\) [882]

Optimal. Leaf size=37 \[ -\frac {\sqrt {1-x^4}}{6 x^6}-\frac {\sqrt {1-x^4}}{3 x^2} \]

[Out]

-1/6*(-x^4+1)^(1/2)/x^6-1/3*(-x^4+1)^(1/2)/x^2

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \begin {gather*} -\frac {\sqrt {1-x^4}}{6 x^6}-\frac {\sqrt {1-x^4}}{3 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^7*Sqrt[1 - x^4]),x]

[Out]

-1/6*Sqrt[1 - x^4]/x^6 - Sqrt[1 - x^4]/(3*x^2)

Rule 270

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 277

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

Rubi steps

\begin {align*} \int \frac {1}{x^7 \sqrt {1-x^4}} \, dx &=-\frac {\sqrt {1-x^4}}{6 x^6}+\frac {2}{3} \int \frac {1}{x^3 \sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{6 x^6}-\frac {\sqrt {1-x^4}}{3 x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 25, normalized size = 0.68 \begin {gather*} \frac {\left (-1-2 x^4\right ) \sqrt {1-x^4}}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*Sqrt[1 - x^4]),x]

[Out]

((-1 - 2*x^4)*Sqrt[1 - x^4])/(6*x^6)

________________________________________________________________________________________

Maple [A]
time = 0.15, size = 32, normalized size = 0.86

method	result	size
trager	\(-\frac {\left (2 x^{4}+1\right ) \sqrt {-x^{4}+1}}{6 x^{6}}\)	\(22\)
meijerg	\(-\frac {\left (2 x^{4}+1\right ) \sqrt {-x^{4}+1}}{6 x^{6}}\)	\(22\)
risch	\(\frac {2 x^{8}-x^{4}-1}{6 x^{6} \sqrt {-x^{4}+1}}\)	\(27\)
default	\(\frac {\left (x^{2}+1\right ) \left (x^{2}-1\right ) \left (2 x^{4}+1\right )}{6 x^{6} \sqrt {-x^{4}+1}}\)	\(32\)
elliptic	\(\frac {\left (x^{2}+1\right ) \left (x^{2}-1\right ) \left (2 x^{4}+1\right )}{6 x^{6} \sqrt {-x^{4}+1}}\)	\(32\)
gosper	\(\frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (2 x^{4}+1\right )}{6 x^{6} \sqrt {-x^{4}+1}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(-x^4+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 29, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {-x^{4} + 1}}{2 \, x^{2}} - \frac {{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(-x^4 + 1)/x^2 - 1/6*(-x^4 + 1)^(3/2)/x^6

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 21, normalized size = 0.57 \begin {gather*} -\frac {{\left (2 \, x^{4} + 1\right )} \sqrt {-x^{4} + 1}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-1/6*(2*x^4 + 1)*sqrt(-x^4 + 1)/x^6

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.43, size = 63, normalized size = 1.70 \begin {gather*} \begin {cases} - \frac {i \sqrt {x^{4} - 1}}{3 x^{2}} - \frac {i \sqrt {x^{4} - 1}}{6 x^{6}} & \text {for}\: \left |{x^{4}}\right | > 1 \\- \frac {\sqrt {1 - x^{4}}}{3 x^{2}} - \frac {\sqrt {1 - x^{4}}}{6 x^{6}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(-x**4+1)**(1/2),x)

[Out]

Piecewise((-I*sqrt(x**4 - 1)/(3*x**2) - I*sqrt(x**4 - 1)/(6*x**6), Abs(x**4) > 1), (-sqrt(1 - x**4)/(3*x**2) -
sqrt(1 - x**4)/(6*x**6), True))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (29) = 58\).
time = 1.23, size = 73, normalized size = 1.97 \begin {gather*} \frac {x^{6} {\left (\frac {9 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{2}}{x^{4}} + 1\right )}}{48 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{3}} - \frac {3 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}}{16 \, x^{2}} - \frac {{\left (\sqrt {-x^{4} + 1} - 1\right )}^{3}}{48 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

1/48*x^6*(9*(sqrt(-x^4 + 1) - 1)^2/x^4 + 1)/(sqrt(-x^4 + 1) - 1)^3 - 3/16*(sqrt(-x^4 + 1) - 1)/x^2 - 1/48*(sqr
t(-x^4 + 1) - 1)^3/x^6

________________________________________________________________________________________

Mupad [B]
time = 1.15, size = 21, normalized size = 0.57 \begin {gather*} -\frac {\sqrt {1-x^4}\,\left (2\,x^4+1\right )}{6\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(1 - x^4)^(1/2)),x)

[Out]

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

________________________________________________________________________________________