3.4.42 \(\int \frac {1+x^{12}}{x^{16} \sqrt {-1+x^6}} \, dx\)

Optimal. Leaf size=28 \[ \frac {\sqrt {x^6-1} \left (23 x^{12}+4 x^6+3\right )}{45 x^{15}} \]

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Rubi [A]  time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1487, 453, 271, 264} \begin {gather*} \frac {\sqrt {x^6-1}}{15 x^{15}}+\frac {4 \sqrt {x^6-1}}{45 x^9}+\frac {23 \sqrt {x^6-1}}{45 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^12)/(x^16*Sqrt[-1 + x^6]),x]

[Out]

Sqrt[-1 + x^6]/(15*x^15) + (4*Sqrt[-1 + x^6])/(45*x^9) + (23*Sqrt[-1 + x^6])/(45*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 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]

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]

Rule 1487

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[(c^p
*(f*x)^(m + 2*n*p - n + 1)*(d + e*x^n)^(q + 1))/(e*f^(2*n*p - n + 1)*(m + 2*n*p + n*q + 1)), x] + Dist[1/(e*(m
+ 2*n*p + n*q + 1)), Int[(f*x)^m*(d + e*x^n)^q*ExpandToSum[e*(m + 2*n*p + n*q + 1)*((a + c*x^(2*n))^p - c^p*x
^(2*n*p)) - d*c^p*(m + 2*n*p - n + 1)*x^(2*n*p - n), x], x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && EqQ[n2,
2*n] && IGtQ[n, 0] && IGtQ[p, 0] && GtQ[2*n*p, n - 1] &&  !IntegerQ[q] && NeQ[m + 2*n*p + n*q + 1, 0]

Rubi steps

\begin {align*} \int \frac {1+x^{12}}{x^{16} \sqrt {-1+x^6}} \, dx &=-\frac {\sqrt {-1+x^6}}{6 x^9}-\frac {1}{6} \int \frac {-6-9 x^6}{x^{16} \sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{15 x^{15}}-\frac {\sqrt {-1+x^6}}{6 x^9}+\frac {23}{10} \int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{15 x^{15}}+\frac {4 \sqrt {-1+x^6}}{45 x^9}+\frac {23}{15} \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{15 x^{15}}+\frac {4 \sqrt {-1+x^6}}{45 x^9}+\frac {23 \sqrt {-1+x^6}}{45 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 33, normalized size = 1.18 \begin {gather*} \frac {23 x^{18}-19 x^{12}-x^6-3}{45 x^{15} \sqrt {x^6-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^12)/(x^16*Sqrt[-1 + x^6]),x]

[Out]

(-3 - x^6 - 19*x^12 + 23*x^18)/(45*x^15*Sqrt[-1 + x^6])

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IntegrateAlgebraic [A]  time = 0.28, size = 28, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (3+4 x^6+23 x^{12}\right )}{45 x^{15}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^12)/(x^16*Sqrt[-1 + x^6]),x]

[Out]

(Sqrt[-1 + x^6]*(3 + 4*x^6 + 23*x^12))/(45*x^15)

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fricas [A]  time = 0.48, size = 31, normalized size = 1.11 \begin {gather*} \frac {23 \, x^{15} + {\left (23 \, x^{12} + 4 \, x^{6} + 3\right )} \sqrt {x^{6} - 1}}{45 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^12+1)/x^16/(x^6-1)^(1/2),x, algorithm="fricas")

[Out]

1/45*(23*x^15 + (23*x^12 + 4*x^6 + 3)*sqrt(x^6 - 1))/x^15

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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^12+1)/x^16/(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 = 0.10, size = 25, normalized size = 0.89

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}\, \left (23 x^{12}+4 x^{6}+3\right )}{45 x^{15}}\)	\(25\)
risch	\(\frac {23 x^{18}-19 x^{12}-x^{6}-3}{45 x^{15} \sqrt {x^{6}-1}}\)	\(30\)
gosper	\(\frac {\left (23 x^{12}+4 x^{6}+3\right ) \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{45 \sqrt {x^{6}-1}\, x^{15}}\)	\(45\)
meijerg	\(-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {-x^{6}+1}}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{3}}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (\frac {8}{3} x^{12}+\frac {4}{3} x^{6}+1\right ) \sqrt {-x^{6}+1}}{15 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{15}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^12+1)/x^16/(x^6-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/45*(x^6-1)^(1/2)*(23*x^12+4*x^6+3)/x^15

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maxima [A]  time = 0.47, size = 37, normalized size = 1.32 \begin {gather*} \frac {2 \, \sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {2 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} + \frac {{\left (x^{6} - 1\right )}^{\frac {5}{2}}}{15 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^12+1)/x^16/(x^6-1)^(1/2),x, algorithm="maxima")

[Out]

2/3*sqrt(x^6 - 1)/x^3 - 2/9*(x^6 - 1)^(3/2)/x^9 + 1/15*(x^6 - 1)^(5/2)/x^15

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mupad [B]  time = 0.32, size = 24, normalized size = 0.86 \begin {gather*} \frac {\sqrt {x^6-1}\,\left (23\,x^{12}+4\,x^6+3\right )}{45\,x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^12 + 1)/(x^16*(x^6 - 1)^(1/2)),x)

[Out]

((x^6 - 1)^(1/2)*(4*x^6 + 23*x^12 + 3))/(45*x^15)

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sympy [A]  time = 3.69, size = 65, normalized size = 2.32 \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 {2 \left (x^{6} - 1\right )^{\frac {3}{2}}}{3 x^{9}} + \frac {\left (x^{6} - 1\right )^{\frac {5}{2}}}{5 x^{15}} & \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**12+1)/x**16/(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 - 2*(x**6 - 1)**(3/2)/
(3*x**9) + (x**6 - 1)**(5/2)/(5*x**15), (x > -1) & (x < 1)))/3

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