\(\int \frac {-1+x^6}{x^{13} \sqrt {1+x^6}} \, dx\) [491]

Optimal result

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

[Out]

1/24*(-7*x^6+2)*(x^6+1)^(1/2)/x^12+7/24*arctanh((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, 213} \[ \int \frac {-1+x^6}{x^{13} \sqrt {1+x^6}} \, dx=\frac {7}{24} \text {arctanh}\left (\sqrt {x^6+1}\right )-\frac {7 \sqrt {x^6+1}}{24 x^6}+\frac {\sqrt {x^6+1}}{12 x^{12}} \]

[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*ArcTanh[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 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[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}{x^3 \sqrt {1+x}} \, dx,x,x^6\right ) \\ & = \frac {\sqrt {1+x^6}}{12 x^{12}}+\frac {7}{24} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+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}{48} \text {Subst}\left (\int \frac {1}{x \sqrt {1+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} \text {arctanh}\left (\sqrt {1+x^6}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^6}{x^{13} \sqrt {1+x^6}} \, dx=\frac {\left (2-7 x^6\right ) \sqrt {1+x^6}}{24 x^{12}}+\frac {7}{24} \text {arctanh}\left (\sqrt {1+x^6}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97

method	result	size
trager	\(-\frac {\left (7 x^{6}-2\right ) \sqrt {x^{6}+1}}{24 x^{12}}-\frac {7 \ln \left (\frac {\sqrt {x^{6}+1}-1}{x^{3}}\right )}{24}\)	\(37\)
pseudoelliptic	\(\frac {7 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{6}+1}}\right ) x^{12}-7 \sqrt {x^{6}+1}\, x^{6}+2 \sqrt {x^{6}+1}}{24 x^{12}}\)	\(41\)
risch	\(-\frac {7 x^{12}+5 x^{6}-2}{24 x^{12} \sqrt {x^{6}+1}}-\frac {7 \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )+\left (-2 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }\right )}{48 \sqrt {\pi }}\)	\(62\)
meijerg	\(\frac {\frac {\sqrt {\pi }\, \left (4 x^{6}+8\right )}{8 x^{6}}-\frac {\sqrt {\pi }\, \sqrt {x^{6}+1}}{x^{6}}+\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )-\frac {\left (1-2 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{x^{6}}}{6 \sqrt {\pi }}-\frac {\frac {\sqrt {\pi }\, \left (-7 x^{12}-8 x^{6}+8\right )}{16 x^{12}}-\frac {\sqrt {\pi }\, \left (-12 x^{6}+8\right ) \sqrt {x^{6}+1}}{16 x^{12}}-\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right )}{4}+\frac {3 \left (\frac {7}{6}-2 \ln \left (2\right )+6 \ln \left (x \right )\right ) \sqrt {\pi }}{8}-\frac {\sqrt {\pi }}{2 x^{12}}+\frac {\sqrt {\pi }}{2 x^{6}}}{6 \sqrt {\pi }}\)	\(173\)

[In]

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

[Out]

-1/24*(7*x^6-2)/x^12*(x^6+1)^(1/2)-7/24*ln(((x^6+1)^(1/2)-1)/x^3)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.37 \[ \int \frac {-1+x^6}{x^{13} \sqrt {1+x^6}} \, dx=\frac {7 \, x^{12} \log \left (\sqrt {x^{6} + 1} + 1\right ) - 7 \, x^{12} \log \left (\sqrt {x^{6} + 1} - 1\right ) - 2 \, {\left (7 \, x^{6} - 2\right )} \sqrt {x^{6} + 1}}{48 \, x^{12}} \]

[In]

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

[Out]

1/48*(7*x^12*log(sqrt(x^6 + 1) + 1) - 7*x^12*log(sqrt(x^6 + 1) - 1) - 2*(7*x^6 - 2)*sqrt(x^6 + 1))/x^12

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (34) = 68\).

Time = 33.63 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.29 \[ \int \frac {-1+x^6}{x^{13} \sqrt {1+x^6}} \, dx=- \frac {7 \log {\left (\sqrt {x^{6} + 1} - 1 \right )}}{48} + \frac {7 \log {\left (\sqrt {x^{6} + 1} + 1 \right )}}{48} - \frac {7}{48 \left (\sqrt {x^{6} + 1} + 1\right )} - \frac {1}{48 \left (\sqrt {x^{6} + 1} + 1\right )^{2}} - \frac {7}{48 \left (\sqrt {x^{6} + 1} - 1\right )} + \frac {1}{48 \left (\sqrt {x^{6} + 1} - 1\right )^{2}} \]

[In]

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

[Out]

-7*log(sqrt(x**6 + 1) - 1)/48 + 7*log(sqrt(x**6 + 1) + 1)/48 - 7/(48*(sqrt(x**6 + 1) + 1)) - 1/(48*(sqrt(x**6
+ 1) + 1)**2) - 7/(48*(sqrt(x**6 + 1) - 1)) + 1/(48*(sqrt(x**6 + 1) - 1)**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (30) = 60\).

Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.00 \[ \int \frac {-1+x^6}{x^{13} \sqrt {1+x^6}} \, dx=\frac {3 \, {\left (x^{6} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {x^{6} + 1}}{24 \, {\left (2 \, x^{6} - {\left (x^{6} + 1\right )}^{2} + 1\right )}} - \frac {\sqrt {x^{6} + 1}}{6 \, x^{6}} + \frac {7}{48} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {7}{48} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]

[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 + 7/48*log(sqrt(x
^6 + 1) + 1) - 7/48*log(sqrt(x^6 + 1) - 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.29 \[ \int \frac {-1+x^6}{x^{13} \sqrt {1+x^6}} \, dx=-\frac {7 \, {\left (x^{6} + 1\right )}^{\frac {3}{2}} - 9 \, \sqrt {x^{6} + 1}}{24 \, x^{12}} + \frac {7}{48} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {7}{48} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \]

[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 + 7/48*log(sqrt(x^6 + 1) + 1) - 7/48*log(sqrt(x^6 + 1) - 1)

Mupad [B] (verification not implemented)

Time = 5.55 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.24 \[ \int \frac {-1+x^6}{x^{13} \sqrt {1+x^6}} \, dx=\frac {7\,\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{24}-\frac {\sqrt {x^6+1}}{6\,x^6}+\frac {5\,\sqrt {x^6+1}}{24\,x^{12}}-\frac {{\left (x^6+1\right )}^{3/2}}{8\,x^{12}} \]

[In]

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

[Out]

(7*atanh((x^6 + 1)^(1/2)))/24 - (x^6 + 1)^(1/2)/(6*x^6) + (5*(x^6 + 1)^(1/2))/(24*x^12) - (x^6 + 1)^(3/2)/(8*x
^12)