3.10.25 \(\int \frac {1}{x^7 \sqrt {1+x^4}} \, dx\) [925]

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

[Out]

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

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Rubi [A]
time = 0.00, 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 = {277, 270} \begin {gather*} \frac {\sqrt {x^4+1}}{3 x^2}-\frac {\sqrt {x^4+1}}{6 x^6} \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*}

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Mathematica [A]
time = 0.09, size = 23, normalized size = 0.70 \begin {gather*} \frac {\sqrt {1+x^4} \left (-1+2 x^4\right )}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*Sqrt[1 + x^4]),x]

[Out]

(Sqrt[1 + x^4]*(-1 + 2*x^4))/(6*x^6)

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Maple [A]
time = 0.16, size = 20, normalized size = 0.61

method	result	size
gosper	\(\frac {\sqrt {x^{4}+1}\, \left (2 x^{4}-1\right )}{6 x^{6}}\)	\(20\)
default	\(\frac {\sqrt {x^{4}+1}\, \left (2 x^{4}-1\right )}{6 x^{6}}\)	\(20\)
trager	\(\frac {\sqrt {x^{4}+1}\, \left (2 x^{4}-1\right )}{6 x^{6}}\)	\(20\)
meijerg	\(-\frac {\left (-2 x^{4}+1\right ) \sqrt {x^{4}+1}}{6 x^{6}}\)	\(20\)
elliptic	\(\frac {\sqrt {x^{4}+1}\, \left (2 x^{4}-1\right )}{6 x^{6}}\)	\(20\)
risch	\(\frac {2 x^{8}+x^{4}-1}{6 x^{6} \sqrt {x^{4}+1}}\)	\(23\)

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

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Maxima [A]
time = 0.28, size = 25, normalized size = 0.76 \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

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Fricas [A]
time = 0.36, size = 26, normalized size = 0.79 \begin {gather*} \frac {2 \, x^{6} + {\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^6 + (2*x^4 - 1)*sqrt(x^4 + 1))/x^6

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Sympy [A]
time = 0.43, size = 26, normalized size = 0.79 \begin {gather*} \frac {\sqrt {1 + \frac {1}{x^{4}}}}{3} - \frac {\sqrt {1 + \frac {1}{x^{4}}}}{6 x^{4}} \end {gather*}

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)

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Giac [A]
time = 1.30, size = 40, normalized size = 1.21 \begin {gather*} \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}} \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]

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

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Mupad [B]
time = 1.15, size = 19, normalized size = 0.58 \begin {gather*} \frac {\sqrt {x^4+1}\,\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*(x^4 + 1)^(1/2)),x)

[Out]

((x^4 + 1)^(1/2)*(2*x^4 - 1))/(6*x^6)

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