\(\int \frac {1+x^6}{x^{10} \sqrt {-1+x^6}} \, dx\) [234]

Optimal result

Integrand size = 18, antiderivative size = 23 \[ \int \frac {1+x^6}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6} \left (1+5 x^6\right )}{9 x^9} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {464, 270} \[ \int \frac {1+x^6}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {x^6-1}}{9 x^9}+\frac {5 \sqrt {x^6-1}}{9 x^3} \]

[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 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 464

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^6}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6} \left (1+5 x^6\right )}{9 x^9} \]

[In]

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

[Out]

(Sqrt[-1 + x^6]*(1 + 5*x^6))/(9*x^9)

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {1+x^6}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {5 \, x^{9} + {\left (5 \, x^{6} + 1\right )} \sqrt {x^{6} - 1}}{9 \, x^{9}} \]

[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

Sympy [A] (verification not implemented)

Time = 1.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43 \[ \int \frac {1+x^6}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {\begin {cases} \frac {\sqrt {x^{6} - 1}}{x^{3}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} + \frac {\begin {cases} \frac {\sqrt {x^{6} - 1}}{x^{3}} - \frac {\left (x^{6} - 1\right )^{\frac {3}{2}}}{3 x^{9}} & \text {for}\: x^{3} > -1 \wedge x^{3} < 1 \end {cases}}{3} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1+x^6}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {2 \, \sqrt {x^{6} - 1}}{3 \, x^{3}} - \frac {{\left (x^{6} - 1\right )}^{\frac {3}{2}}}{9 \, x^{9}} \]

[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

Giac [F(-2)]

Exception generated. \[ \int \frac {1+x^6}{x^{10} \sqrt {-1+x^6}} \, dx=\text {Exception raised: TypeError} \]

[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,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 5.61 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1+x^6}{x^{10} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {x^6-1}+5\,x^6\,\sqrt {x^6-1}}{9\,x^9} \]

[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)