\(\int \frac {1+x^6}{x^{13} \sqrt {-1+x^6}} \, dx\) [493]

Optimal result

Integrand size = 18, antiderivative size = 38 \[ \int \frac {1+x^6}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6} \left (2+7 x^6\right )}{24 x^{12}}+\frac {7}{24} \arctan \left (\sqrt {-1+x^6}\right ) \]

[Out]

1/24*(x^6-1)^(1/2)*(7*x^6+2)/x^12+7/24*arctan((x^6-1)^(1/2))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {457, 79, 44, 65, 209} \[ \int \frac {1+x^6}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {7}{24} \arctan \left (\sqrt {x^6-1}\right )+\frac {7 \sqrt {x^6-1}}{24 x^6}+\frac {\sqrt {x^6-1}}{12 x^{12}} \]

[In]

Int[(1 + x^6)/(x^13*Sqrt[-1 + x^6]),x]

[Out]

Sqrt[-1 + x^6]/(12*x^12) + (7*Sqrt[-1 + x^6])/(24*x^6) + (7*ArcTan[Sqrt[-1 + x^6]])/24

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 209

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

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \text {Subst}\left (\int \frac {1+x}{\sqrt {-1+x} x^3} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {7}{24} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {7 \sqrt {-1+x^6}}{24 x^6}+\frac {7}{48} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {7 \sqrt {-1+x^6}}{24 x^6}+\frac {7}{24} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+x^6}\right ) \\ & = \frac {\sqrt {-1+x^6}}{12 x^{12}}+\frac {7 \sqrt {-1+x^6}}{24 x^6}+\frac {7}{24} \arctan \left (\sqrt {-1+x^6}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^6}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {\sqrt {-1+x^6} \left (2+7 x^6\right )}{24 x^{12}}+\frac {7}{24} \arctan \left (\sqrt {-1+x^6}\right ) \]

[In]

Integrate[(1 + x^6)/(x^13*Sqrt[-1 + x^6]),x]

[Out]

(Sqrt[-1 + x^6]*(2 + 7*x^6))/(24*x^12) + (7*ArcTan[Sqrt[-1 + x^6]])/24

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08

method	result	size
pseudoelliptic	\(\frac {-7 \arctan \left (\frac {1}{\sqrt {x^{6}-1}}\right ) x^{12}+7 \sqrt {x^{6}-1}\, x^{6}+2 \sqrt {x^{6}-1}}{24 x^{12}}\)	\(41\)
trager	\(\frac {\sqrt {x^{6}-1}\, \left (7 x^{6}+2\right )}{24 x^{12}}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{24}\)	\(48\)
risch	\(\frac {7 x^{12}-5 x^{6}-2}{24 x^{12} \sqrt {x^{6}-1}}+\frac {7 \sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }\right )}{48 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\)	\(86\)
meijerg	\(-\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (-\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right )}{8 x^{6}}+\frac {\sqrt {\pi }\, \sqrt {-x^{6}+1}}{x^{6}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{2}+\frac {\sqrt {\pi }}{x^{6}}\right )}{6 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}+\frac {\sqrt {-\operatorname {signum}\left (x^{6}-1\right )}\, \left (\frac {\sqrt {\pi }\, \left (-7 x^{12}+8 x^{6}+8\right )}{16 x^{12}}-\frac {\sqrt {\pi }\, \left (12 x^{6}+8\right ) \sqrt {-x^{6}+1}}{16 x^{12}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right )}{4}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+6 \ln \left (x \right )+i \pi \right ) \sqrt {\pi }}{8}-\frac {\sqrt {\pi }}{2 x^{12}}-\frac {\sqrt {\pi }}{2 x^{6}}\right )}{6 \sqrt {\pi }\, \sqrt {\operatorname {signum}\left (x^{6}-1\right )}}\)	\(223\)

[In]

int((x^6+1)/x^13/(x^6-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/24*(-7*arctan(1/(x^6-1)^(1/2))*x^12+7*(x^6-1)^(1/2)*x^6+2*(x^6-1)^(1/2))/x^12

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {1+x^6}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {7 \, x^{12} \arctan \left (\sqrt {x^{6} - 1}\right ) + {\left (7 \, x^{6} + 2\right )} \sqrt {x^{6} - 1}}{24 \, x^{12}} \]

[In]

integrate((x^6+1)/x^13/(x^6-1)^(1/2),x, algorithm="fricas")

[Out]

1/24*(7*x^12*arctan(sqrt(x^6 - 1)) + (7*x^6 + 2)*sqrt(x^6 - 1))/x^12

Sympy [A] (verification not implemented)

Time = 24.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.16 \[ \int \frac {1+x^6}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {7 \operatorname {atan}{\left (\sqrt {x^{6} - 1} \right )}}{24} + \frac {\sqrt {x^{6} - 1}}{3 x^{6}} + \frac {\left (2 - x^{6}\right ) \sqrt {x^{6} - 1}}{24 x^{12}} \]

[In]

integrate((x**6+1)/x**13/(x**6-1)**(1/2),x)

[Out]

7*atan(sqrt(x**6 - 1))/24 + sqrt(x**6 - 1)/(3*x**6) + (2 - x**6)*sqrt(x**6 - 1)/(24*x**12)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.58 \[ \int \frac {1+x^6}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {3 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} + 5 \, \sqrt {x^{6} - 1}}{24 \, {\left (2 \, x^{6} + {\left (x^{6} - 1\right )}^{2} - 1\right )}} + \frac {\sqrt {x^{6} - 1}}{6 \, x^{6}} + \frac {7}{24} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]

[In]

integrate((x^6+1)/x^13/(x^6-1)^(1/2),x, algorithm="maxima")

[Out]

1/24*(3*(x^6 - 1)^(3/2) + 5*sqrt(x^6 - 1))/(2*x^6 + (x^6 - 1)^2 - 1) + 1/6*sqrt(x^6 - 1)/x^6 + 7/24*arctan(sqr
t(x^6 - 1))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {1+x^6}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {7 \, {\left (x^{6} - 1\right )}^{\frac {3}{2}} + 9 \, \sqrt {x^{6} - 1}}{24 \, x^{12}} + \frac {7}{24} \, \arctan \left (\sqrt {x^{6} - 1}\right ) \]

[In]

integrate((x^6+1)/x^13/(x^6-1)^(1/2),x, algorithm="giac")

[Out]

1/24*(7*(x^6 - 1)^(3/2) + 9*sqrt(x^6 - 1))/x^12 + 7/24*arctan(sqrt(x^6 - 1))

Mupad [B] (verification not implemented)

Time = 5.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92 \[ \int \frac {1+x^6}{x^{13} \sqrt {-1+x^6}} \, dx=\frac {7\,\mathrm {atan}\left (\sqrt {x^6-1}\right )}{24}+\frac {7\,\sqrt {x^6-1}}{24\,x^6}+\frac {\sqrt {x^6-1}}{12\,x^{12}} \]

[In]

int((x^6 + 1)/(x^13*(x^6 - 1)^(1/2)),x)

[Out]

(7*atan((x^6 - 1)^(1/2)))/24 + (7*(x^6 - 1)^(1/2))/(24*x^6) + (x^6 - 1)^(1/2)/(12*x^12)