3.6.73 \(\int \frac {1+x^{12}}{x^{10} \sqrt {-1+x^6}} \, dx\)

Optimal. Leaf size=44 \[ \frac {\sqrt {x^6-1} \left (2 x^6+1\right )}{9 x^9}+\frac {2}{3} \tanh ^{-1}\left (\frac {x^3+1}{\sqrt {x^6-1}}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1489, 271, 264, 275, 217, 206} \begin {gather*} \frac {\sqrt {x^6-1}}{9 x^9}+\frac {2 \sqrt {x^6-1}}{9 x^3}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x^12)/(x^10*Sqrt[-1 + x^6]),x]

[Out]

Sqrt[-1 + x^6]/(9*x^9) + (2*Sqrt[-1 + x^6])/(9*x^3) + ArcTanh[x^3/Sqrt[-1 + x^6]]/3

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 1489

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

Rubi steps

\begin {align*} \int \frac {1+x^{12}}{x^{10} \sqrt {-1+x^6}} \, dx &=\int \left (\frac {1}{x^{10} \sqrt {-1+x^6}}+\frac {x^2}{\sqrt {-1+x^6}}\right ) \, dx\\ &=\int \frac {1}{x^{10} \sqrt {-1+x^6}} \, dx+\int \frac {x^2}{\sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{9 x^9}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^3\right )+\frac {2}{3} \int \frac {1}{x^4 \sqrt {-1+x^6}} \, dx\\ &=\frac {\sqrt {-1+x^6}}{9 x^9}+\frac {2 \sqrt {-1+x^6}}{9 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^3}{\sqrt {-1+x^6}}\right )\\ &=\frac {\sqrt {-1+x^6}}{9 x^9}+\frac {2 \sqrt {-1+x^6}}{9 x^3}+\frac {1}{3} \tanh ^{-1}\left (\frac {x^3}{\sqrt {-1+x^6}}\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 41, normalized size = 0.93 \begin {gather*} \frac {1}{9} \left (\frac {\sqrt {x^6-1} \left (2 x^6+1\right )}{x^9}+3 \tanh ^{-1}\left (\frac {x^3}{\sqrt {x^6-1}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^12)/(x^10*Sqrt[-1 + x^6]),x]

[Out]

((Sqrt[-1 + x^6]*(1 + 2*x^6))/x^9 + 3*ArcTanh[x^3/Sqrt[-1 + x^6]])/9

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IntegrateAlgebraic [A]  time = 0.31, size = 44, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{9 x^9}+\frac {2}{3} \tanh ^{-1}\left (\frac {1+x^3}{\sqrt {-1+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x^12)/(x^10*Sqrt[-1 + x^6]),x]

[Out]

(Sqrt[-1 + x^6]*(1 + 2*x^6))/(9*x^9) + (2*ArcTanh[(1 + x^3)/Sqrt[-1 + x^6]])/3

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fricas [A]  time = 0.43, size = 46, normalized size = 1.05 \begin {gather*} -\frac {3 \, x^{9} \log \left (-x^{3} + \sqrt {x^{6} - 1}\right ) - 2 \, x^{9} - {\left (2 \, 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^12+1)/x^10/(x^6-1)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.42, size = 59, normalized size = 1.34 \begin {gather*} -\frac {2 \, {\left (-\frac {1}{x^{6}} + 1\right )}^{\frac {3}{2}} - 6 \, \sqrt {-\frac {1}{x^{6}} + 1} - 3 \, \log \left (\sqrt {-\frac {1}{x^{6}} + 1} + 1\right ) + 3 \, \log \left (-\sqrt {-\frac {1}{x^{6}} + 1} + 1\right )}{18 \, \mathrm {sgn}\relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^12+1)/x^10/(x^6-1)^(1/2),x, algorithm="giac")

[Out]

-1/18*(2*(-1/x^6 + 1)^(3/2) - 6*sqrt(-1/x^6 + 1) - 3*log(sqrt(-1/x^6 + 1) + 1) + 3*log(-sqrt(-1/x^6 + 1) + 1))
/sgn(x)

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maple [A]  time = 0.27, size = 37, normalized size = 0.84

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}\, \left (2 x^{6}+1\right )}{9 x^{9}}-\frac {\ln \left (-x^{3}+\sqrt {x^{6}-1}\right )}{3}\)	\(37\)
risch	\(\frac {2 x^{12}-x^{6}-1}{9 x^{9} \sqrt {x^{6}-1}}+\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}\)	\(50\)
meijerg	\(\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}}-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (2 x^{6}+1\right ) \sqrt {-x^{6}+1}}{9 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, x^{9}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^12+1)/x^10/(x^6-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.45, size = 57, normalized size = 1.30 \begin {gather*} \frac {\sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} + \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {\sqrt {x^{6} - 1}}{x^{3}} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^12+1)/x^10/(x^6-1)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{12}+1}{x^{10}\,\sqrt {x^6-1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^12 + 1)/(x^10*(x^6 - 1)^(1/2)),x)

[Out]

int((x^12 + 1)/(x^10*(x^6 - 1)^(1/2)), x)

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sympy [A]  time = 2.62, size = 37, normalized size = 0.84 \begin {gather*} \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} + \frac {\operatorname {acosh}{\left (x^{3} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**12+1)/x**10/(x**6-1)**(1/2),x)

[Out]

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

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