3.9.83 \(\int \frac {1}{x^{11} \sqrt {1-x^4}} \, dx\) [883]

Optimal. Leaf size=55 \[ -\frac {\sqrt {1-x^4}}{10 x^{10}}-\frac {2 \sqrt {1-x^4}}{15 x^6}-\frac {4 \sqrt {1-x^4}}{15 x^2} \]

[Out]

-1/10*(-x^4+1)^(1/2)/x^10-2/15*(-x^4+1)^(1/2)/x^6-4/15*(-x^4+1)^(1/2)/x^2

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Rubi [A]
time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, 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}}{10 x^{10}}-\frac {2 \sqrt {1-x^4}}{15 x^6}-\frac {4 \sqrt {1-x^4}}{15 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^11*Sqrt[1 - x^4]),x]

[Out]

-1/10*Sqrt[1 - x^4]/x^10 - (2*Sqrt[1 - x^4])/(15*x^6) - (4*Sqrt[1 - x^4])/(15*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^{11} \sqrt {1-x^4}} \, dx &=-\frac {\sqrt {1-x^4}}{10 x^{10}}+\frac {4}{5} \int \frac {1}{x^7 \sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{10 x^{10}}-\frac {2 \sqrt {1-x^4}}{15 x^6}+\frac {8}{15} \int \frac {1}{x^3 \sqrt {1-x^4}} \, dx\\ &=-\frac {\sqrt {1-x^4}}{10 x^{10}}-\frac {2 \sqrt {1-x^4}}{15 x^6}-\frac {4 \sqrt {1-x^4}}{15 x^2}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 30, normalized size = 0.55 \begin {gather*} \frac {\sqrt {1-x^4} \left (-3-4 x^4-8 x^8\right )}{30 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^11*Sqrt[1 - x^4]),x]

[Out]

(Sqrt[1 - x^4]*(-3 - 4*x^4 - 8*x^8))/(30*x^10)

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Maple [A]
time = 0.17, size = 37, normalized size = 0.67

method	result	size
trager	\(-\frac {\left (8 x^{8}+4 x^{4}+3\right ) \sqrt {-x^{4}+1}}{30 x^{10}}\)	\(27\)
meijerg	\(-\frac {\left (\frac {8}{3} x^{8}+\frac {4}{3} x^{4}+1\right ) \sqrt {-x^{4}+1}}{10 x^{10}}\)	\(27\)
risch	\(\frac {8 x^{12}-4 x^{8}-x^{4}-3}{30 x^{10} \sqrt {-x^{4}+1}}\)	\(32\)
default	\(\frac {\left (x^{2}+1\right ) \left (x^{2}-1\right ) \left (8 x^{8}+4 x^{4}+3\right )}{30 x^{10} \sqrt {-x^{4}+1}}\)	\(37\)
elliptic	\(\frac {\left (x^{2}+1\right ) \left (x^{2}-1\right ) \left (8 x^{8}+4 x^{4}+3\right )}{30 x^{10} \sqrt {-x^{4}+1}}\)	\(37\)
gosper	\(\frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (8 x^{8}+4 x^{4}+3\right )}{30 x^{10} \sqrt {-x^{4}+1}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^11/(-x^4+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/30*(x^2+1)*(x^2-1)*(8*x^8+4*x^4+3)/x^10/(-x^4+1)^(1/2)

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Maxima [A]
time = 0.29, size = 43, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {-x^{4} + 1}}{2 \, x^{2}} - \frac {{\left (-x^{4} + 1\right )}^{\frac {3}{2}}}{3 \, x^{6}} - \frac {{\left (-x^{4} + 1\right )}^{\frac {5}{2}}}{10 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(-x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(-x^4 + 1)/x^2 - 1/3*(-x^4 + 1)^(3/2)/x^6 - 1/10*(-x^4 + 1)^(5/2)/x^10

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Fricas [A]
time = 0.39, size = 26, normalized size = 0.47 \begin {gather*} -\frac {{\left (8 \, x^{8} + 4 \, x^{4} + 3\right )} \sqrt {-x^{4} + 1}}{30 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(-x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-1/30*(8*x^8 + 4*x^4 + 3)*sqrt(-x^4 + 1)/x^10

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Sympy [C] Result contains complex when optimal does not.
time = 0.66, size = 104, normalized size = 1.89 \begin {gather*} \begin {cases} - \frac {4 \sqrt {-1 + \frac {1}{x^{4}}}}{15} - \frac {2 \sqrt {-1 + \frac {1}{x^{4}}}}{15 x^{4}} - \frac {\sqrt {-1 + \frac {1}{x^{4}}}}{10 x^{8}} & \text {for}\: \frac {1}{\left |{x^{4}}\right |} > 1 \\- \frac {4 i \sqrt {1 - \frac {1}{x^{4}}}}{15} - \frac {2 i \sqrt {1 - \frac {1}{x^{4}}}}{15 x^{4}} - \frac {i \sqrt {1 - \frac {1}{x^{4}}}}{10 x^{8}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**11/(-x**4+1)**(1/2),x)

[Out]

Piecewise((-4*sqrt(-1 + x**(-4))/15 - 2*sqrt(-1 + x**(-4))/(15*x**4) - sqrt(-1 + x**(-4))/(10*x**8), 1/Abs(x**
4) > 1), (-4*I*sqrt(1 - 1/x**4)/15 - 2*I*sqrt(1 - 1/x**4)/(15*x**4) - I*sqrt(1 - 1/x**4)/(10*x**8), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (43) = 86\).
time = 0.76, size = 109, normalized size = 1.98 \begin {gather*} \frac {x^{10} {\left (\frac {25 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{2}}{x^{4}} + \frac {150 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{4}}{x^{8}} + 3\right )}}{960 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{5}} - \frac {5 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}}{32 \, x^{2}} - \frac {5 \, {\left (\sqrt {-x^{4} + 1} - 1\right )}^{3}}{192 \, x^{6}} - \frac {{\left (\sqrt {-x^{4} + 1} - 1\right )}^{5}}{320 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^11/(-x^4+1)^(1/2),x, algorithm="giac")

[Out]

1/960*x^10*(25*(sqrt(-x^4 + 1) - 1)^2/x^4 + 150*(sqrt(-x^4 + 1) - 1)^4/x^8 + 3)/(sqrt(-x^4 + 1) - 1)^5 - 5/32*
(sqrt(-x^4 + 1) - 1)/x^2 - 5/192*(sqrt(-x^4 + 1) - 1)^3/x^6 - 1/320*(sqrt(-x^4 + 1) - 1)^5/x^10

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Mupad [B]
time = 1.19, size = 26, normalized size = 0.47 \begin {gather*} -\frac {\sqrt {1-x^4}\,\left (8\,x^8+4\,x^4+3\right )}{30\,x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^11*(1 - x^4)^(1/2)),x)

[Out]

-((1 - x^4)^(1/2)*(4*x^4 + 8*x^8 + 3))/(30*x^10)

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