3.1.43 \(\int \frac {-2+x^6}{x^3 \sqrt {1+x^6}} \, dx\)

Optimal. Leaf size=13 \[ \frac {\sqrt {x^6+1}}{x^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {449} \begin {gather*} \frac {\sqrt {x^6+1}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 + x^6)/(x^3*Sqrt[1 + x^6]),x]

[Out]

Sqrt[1 + x^6]/x^2

Rule 449

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[
a*d*(m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^6+1}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + x^6]/x^2

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IntegrateAlgebraic [A]  time = 1.71, size = 13, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^6}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + x^6]/x^2

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fricas [A]  time = 0.42, size = 11, normalized size = 0.85 \begin {gather*} \frac {\sqrt {x^{6} + 1}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-2)/x^3/(x^6+1)^(1/2),x, algorithm="fricas")

[Out]

sqrt(x^6 + 1)/x^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} - 2}{\sqrt {x^{6} + 1} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-2)/x^3/(x^6+1)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.08, size = 12, normalized size = 0.92

method	result	size
trager	\(\frac {\sqrt {x^{6}+1}}{x^{2}}\)	\(12\)
risch	\(\frac {\sqrt {x^{6}+1}}{x^{2}}\)	\(12\)
gosper	\(\frac {\left (x^{4}-x^{2}+1\right ) \left (x^{2}+1\right )}{x^{2} \sqrt {x^{6}+1}}\)	\(27\)
meijerg	\(\frac {\hypergeom \left (\left [\frac {1}{2}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{6}\right ) x^{4}}{4}+\frac {\hypergeom \left (\left [-\frac {1}{3}, \frac {1}{2}\right ], \left [\frac {2}{3}\right ], -x^{6}\right )}{x^{2}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-2)/x^3/(x^6+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(x^6+1)^(1/2)/x^2

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maxima [B]  time = 0.49, size = 23, normalized size = 1.77 \begin {gather*} \frac {\sqrt {x^{4} - x^{2} + 1} \sqrt {x^{2} + 1}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-2)/x^3/(x^6+1)^(1/2),x, algorithm="maxima")

[Out]

sqrt(x^4 - x^2 + 1)*sqrt(x^2 + 1)/x^2

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mupad [B]  time = 0.10, size = 11, normalized size = 0.85 \begin {gather*} \frac {\sqrt {x^6+1}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^6 + 1)^(1/2)/x^2

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sympy [C]  time = 2.07, size = 63, normalized size = 4.85 \begin {gather*} \frac {x^{4} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{6 \Gamma \left (\frac {5}{3}\right )} - \frac {\Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{2} \\ \frac {2}{3} \end {matrix}\middle | {x^{6} e^{i \pi }} \right )}}{3 x^{2} \Gamma \left (\frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-2)/x**3/(x**6+1)**(1/2),x)

[Out]

x**4*gamma(2/3)*hyper((1/2, 2/3), (5/3,), x**6*exp_polar(I*pi))/(6*gamma(5/3)) - gamma(-1/3)*hyper((-1/3, 1/2)
, (2/3,), x**6*exp_polar(I*pi))/(3*x**2*gamma(2/3))

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