\(\int \frac {1}{x^{11} \sqrt {1+x^4}} \, dx\) [926]

Optimal result

Integrand size = 13, antiderivative size = 49 \[ \int \frac {1}{x^{11} \sqrt {1+x^4}} \, dx=-\frac {\sqrt {1+x^4}}{10 x^{10}}+\frac {2 \sqrt {1+x^4}}{15 x^6}-\frac {4 \sqrt {1+x^4}}{15 x^2} \]

[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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {277, 270} \[ \int \frac {1}{x^{11} \sqrt {1+x^4}} \, dx=-\frac {\sqrt {x^4+1}}{10 x^{10}}+\frac {2 \sqrt {x^4+1}}{15 x^6}-\frac {4 \sqrt {x^4+1}}{15 x^2} \]

[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*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^{11} \sqrt {1+x^4}} \, dx=\frac {\sqrt {1+x^4} \left (-3+4 x^4-8 x^8\right )}{30 x^{10}} \]

[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)

Maple [A] (verified)

Time = 4.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.51

method	result	size
gosper	\(-\frac {\sqrt {x^{4}+1}\, \left (8 x^{8}-4 x^{4}+3\right )}{30 x^{10}}\)	\(25\)
default	\(-\frac {\sqrt {x^{4}+1}\, \left (8 x^{8}-4 x^{4}+3\right )}{30 x^{10}}\)	\(25\)
trager	\(-\frac {\sqrt {x^{4}+1}\, \left (8 x^{8}-4 x^{4}+3\right )}{30 x^{10}}\)	\(25\)
meijerg	\(-\frac {\left (\frac {8}{3} x^{8}-\frac {4}{3} x^{4}+1\right ) \sqrt {x^{4}+1}}{10 x^{10}}\)	\(25\)
elliptic	\(-\frac {\sqrt {x^{4}+1}\, \left (8 x^{8}-4 x^{4}+3\right )}{30 x^{10}}\)	\(25\)
pseudoelliptic	\(-\frac {\sqrt {x^{4}+1}\, \left (8 x^{8}-4 x^{4}+3\right )}{30 x^{10}}\)	\(25\)
risch	\(-\frac {8 x^{12}+4 x^{8}-x^{4}+3}{30 x^{10} \sqrt {x^{4}+1}}\)	\(30\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.63 \[ \int \frac {1}{x^{11} \sqrt {1+x^4}} \, dx=-\frac {8 \, x^{10} + {\left (8 \, x^{8} - 4 \, x^{4} + 3\right )} \sqrt {x^{4} + 1}}{30 \, x^{10}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^{11} \sqrt {1+x^4}} \, dx=- \frac {4 \sqrt {1 + \frac {1}{x^{4}}}}{15} + \frac {2 \sqrt {1 + \frac {1}{x^{4}}}}{15 x^{4}} - \frac {\sqrt {1 + \frac {1}{x^{4}}}}{10 x^{8}} \]

[In]

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

[Out]

-4*sqrt(1 + x**(-4))/15 + 2*sqrt(1 + x**(-4))/(15*x**4) - sqrt(1 + x**(-4))/(10*x**8)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x^{11} \sqrt {1+x^4}} \, dx=-\frac {\sqrt {x^{4} + 1}}{2 \, x^{2}} + \frac {{\left (x^{4} + 1\right )}^{\frac {3}{2}}}{3 \, x^{6}} - \frac {{\left (x^{4} + 1\right )}^{\frac {5}{2}}}{10 \, x^{10}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^{11} \sqrt {1+x^4}} \, dx=\frac {8 \, {\left (10 \, {\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{4} - 5 \, {\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} + 1\right )}}{15 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1\right )}^{5}} \]

[In]

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

[Out]

8/15*(10*(x^2 - sqrt(x^4 + 1))^4 - 5*(x^2 - sqrt(x^4 + 1))^2 + 1)/((x^2 - sqrt(x^4 + 1))^2 - 1)^5

Mupad [B] (verification not implemented)

Time = 5.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.49 \[ \int \frac {1}{x^{11} \sqrt {1+x^4}} \, dx=-\frac {\sqrt {x^4+1}\,\left (8\,x^8-4\,x^4+3\right )}{30\,x^{10}} \]

[In]

int(1/(x^11*(x^4 + 1)^(1/2)),x)

[Out]

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