3.925 \(\int \frac {1}{x^7 \sqrt {1+x^4}} \, dx\)

Optimal. Leaf size=33 \[ \frac {\sqrt {x^4+1}}{3 x^2}-\frac {\sqrt {x^4+1}}{6 x^6} \]

[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 = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {271, 264} \[ \frac {\sqrt {x^4+1}}{3 x^2}-\frac {\sqrt {x^4+1}}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*Sqrt[1 + x^4]),x]

[Out]

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

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 271

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 = 0.00, size = 23, normalized size = 0.70 \[ -\frac {\left (1-2 x^4\right ) \sqrt {x^4+1}}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*Sqrt[1 + x^4]),x]

[Out]

-1/6*((1 - 2*x^4)*Sqrt[1 + x^4])/x^6

________________________________________________________________________________________

fricas [A]  time = 0.77, size = 26, normalized size = 0.79 \[ \frac {2 \, x^{6} + {\left (2 \, x^{4} - 1\right )} \sqrt {x^{4} + 1}}{6 \, x^{6}} \]

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^6 + (2*x^4 - 1)*sqrt(x^4 + 1))/x^6

________________________________________________________________________________________

giac [A]  time = 0.16, size = 40, normalized size = 1.21 \[ \frac {2 \, {\left (3 \, {\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1\right )}}{3 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 20, normalized size = 0.61 \[ \frac {\sqrt {x^{4}+1}\, \left (2 x^{4}-1\right )}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(x^4+1)^(1/2),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 25, normalized size = 0.76 \[ \frac {\sqrt {x^{4} + 1}}{2 \, x^{2}} - \frac {{\left (x^{4} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \]

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

________________________________________________________________________________________

mupad [B]  time = 1.15, size = 19, normalized size = 0.58 \[ \frac {\sqrt {x^4+1}\,\left (2\,x^4-1\right )}{6\,x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(x^4 + 1)^(1/2)),x)

[Out]

((x^4 + 1)^(1/2)*(2*x^4 - 1))/(6*x^6)

________________________________________________________________________________________

sympy [A]  time = 1.07, size = 26, normalized size = 0.79 \[ \frac {\sqrt {1 + \frac {1}{x^{4}}}}{3} - \frac {\sqrt {1 + \frac {1}{x^{4}}}}{6 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________