3.3.34 \(\int \frac {1+x^6}{x^{10} \sqrt {-1+x^6}} \, dx\)

Optimal. Leaf size=23 \[ \frac {\sqrt {x^6-1} \left (5 x^6+1\right )}{9 x^9} \]

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {453, 264} \begin {gather*} \frac {\sqrt {x^6-1}}{9 x^9}+\frac {5 \sqrt {x^6-1}}{9 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^6)/(x^10*Sqrt[-1 + x^6]),x]

[Out]

Sqrt[-1 + x^6]/(9*x^9) + (5*Sqrt[-1 + x^6])/(9*x^3)

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 453

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] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1+x^6}{x^{10} \sqrt {-1+x^6}} \, dx &=\frac {\sqrt {-1+x^6}}{9 x^9}+\frac {5}{3} \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{9 x^9}+\frac {5 \sqrt {-1+x^6}}{9 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^6-1} \left (5 x^6+1\right )}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^6)/(x^10*Sqrt[-1 + x^6]),x]

[Out]

(Sqrt[-1 + x^6]*(1 + 5*x^6))/(9*x^9)

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IntegrateAlgebraic [A]  time = 0.20, size = 23, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (1+5 x^6\right )}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^6)/(x^10*Sqrt[-1 + x^6]),x]

[Out]

(Sqrt[-1 + x^6]*(1 + 5*x^6))/(9*x^9)

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fricas [A]  time = 0.45, size = 26, normalized size = 1.13 \begin {gather*} \frac {5 \, x^{9} + {\left (5 \, x^{6} + 1\right )} \sqrt {x^{6} - 1}}{9 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)/x^10/(x^6-1)^(1/2),x, algorithm="fricas")

[Out]

1/9*(5*x^9 + (5*x^6 + 1)*sqrt(x^6 - 1))/x^9

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)/x^10/(x^6-1)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is
real):Check [abs(x)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Val
ue

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maple [A]  time = 0.08, size = 20, normalized size = 0.87

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}\, \left (5 x^{6}+1\right )}{9 x^{9}}\)	\(20\)
risch	\(\frac {5 x^{12}-4 x^{6}-1}{9 x^{9} \sqrt {x^{6}-1}}\)	\(25\)
gosper	\(\frac {\left (5 x^{6}+1\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{9 \sqrt {x^{6}-1}\, x^{9}}\)	\(40\)
meijerg	\(-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {-x^{6}+1}}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{3}}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (2 x^{6}+1\right ) \sqrt {-x^{6}+1}}{9 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{9}}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+1)/x^10/(x^6-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/9*(x^6-1)^(1/2)*(5*x^6+1)/x^9

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maxima [A]  time = 0.67, size = 25, normalized size = 1.09 \begin {gather*} \frac {2 \, \sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+1)/x^10/(x^6-1)^(1/2),x, algorithm="maxima")

[Out]

2/3*sqrt(x^6 - 1)/x^3 - 1/9*(x^6 - 1)^(3/2)/x^9

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mupad [B]  time = 0.17, size = 25, normalized size = 1.09 \begin {gather*} \frac {\sqrt {x^6-1}+5\,x^6\,\sqrt {x^6-1}}{9\,x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 + 1)/(x^10*(x^6 - 1)^(1/2)),x)

[Out]

((x^6 - 1)^(1/2) + 5*x^6*(x^6 - 1)^(1/2))/(9*x^9)

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sympy [A]  time = 2.89, size = 49, normalized size = 2.13 \begin {gather*} \frac {\begin {cases} \frac {\sqrt {x^{6} - 1}}{x^{3}} & \text {for}\: x > -1 \wedge x < 1 \end {cases}}{3} + \frac {\begin {cases} \frac {\sqrt {x^{6} - 1}}{x^{3}} - \frac {\left (x^{6} - 1\right )^{\frac {3}{2}}}{3 x^{9}} & \text {for}\: x > -1 \wedge x < 1 \end {cases}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6+1)/x**10/(x**6-1)**(1/2),x)

[Out]

Piecewise((sqrt(x**6 - 1)/x**3, (x > -1) & (x < 1)))/3 + Piecewise((sqrt(x**6 - 1)/x**3 - (x**6 - 1)**(3/2)/(3
*x**9), (x > -1) & (x < 1)))/3

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