3.5.92 \(\int \frac {\sqrt {1+x^6}}{x^{13}} \, dx\)

Optimal. Leaf size=38 \[ \frac {1}{24} \tanh ^{-1}\left (\sqrt {x^6+1}\right )+\frac {\sqrt {x^6+1} \left (-x^6-2\right )}{24 x^{12}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {266, 47, 51, 63, 207} \begin {gather*} -\frac {\sqrt {x^6+1}}{24 x^6}+\frac {1}{24} \tanh ^{-1}\left (\sqrt {x^6+1}\right )-\frac {\sqrt {x^6+1}}{12 x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^6]/x^13,x]

[Out]

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

Rule 47

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

Rule 51

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 207

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

Rule 266

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

Rubi steps

\begin {align*} \int \frac {\sqrt {1+x^6}}{x^{13}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^3} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {1+x^6}}{12 x^{12}}+\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {1+x^6}}{12 x^{12}}-\frac {\sqrt {1+x^6}}{24 x^6}-\frac {1}{48} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right )\\ &=-\frac {\sqrt {1+x^6}}{12 x^{12}}-\frac {\sqrt {1+x^6}}{24 x^6}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^6}\right )\\ &=-\frac {\sqrt {1+x^6}}{12 x^{12}}-\frac {\sqrt {1+x^6}}{24 x^6}+\frac {1}{24} \tanh ^{-1}\left (\sqrt {1+x^6}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.01, size = 26, normalized size = 0.68 \begin {gather*} -\frac {1}{9} \left (x^6+1\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};x^6+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x^6]/x^13,x]

[Out]

-1/9*((1 + x^6)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + x^6])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.05, size = 38, normalized size = 1.00 \begin {gather*} \frac {\left (-2-x^6\right ) \sqrt {1+x^6}}{24 x^{12}}+\frac {1}{24} \tanh ^{-1}\left (\sqrt {1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 + x^6]/x^13,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 49, normalized size = 1.29 \begin {gather*} \frac {x^{12} \log \left (\sqrt {x^{6} + 1} + 1\right ) - x^{12} \log \left (\sqrt {x^{6} + 1} - 1\right ) - 2 \, {\left (x^{6} + 2\right )} \sqrt {x^{6} + 1}}{48 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.42, size = 77, normalized size = 2.03 \begin {gather*} -\frac {\sqrt {x^{6} + 1} + \frac {1}{\sqrt {x^{6} + 1}}}{24 \, {\left ({\left (\sqrt {x^{6} + 1} + \frac {1}{\sqrt {x^{6} + 1}}\right )}^{2} - 4\right )}} + \frac {1}{96} \, \log \left (\sqrt {x^{6} + 1} + \frac {1}{\sqrt {x^{6} + 1}} + 2\right ) - \frac {1}{96} \, \log \left (\sqrt {x^{6} + 1} + \frac {1}{\sqrt {x^{6} + 1}} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(sqrt(x^6 + 1) + 1/sqrt(x^6 + 1))/((sqrt(x^6 + 1) + 1/sqrt(x^6 + 1))^2 - 4) + 1/96*log(sqrt(x^6 + 1) + 1
/sqrt(x^6 + 1) + 2) - 1/96*log(sqrt(x^6 + 1) + 1/sqrt(x^6 + 1) - 2)

________________________________________________________________________________________

maple [A]  time = 0.23, size = 35, normalized size = 0.92

method	result	size
trager	\(-\frac {\left (x^{6}+2\right ) \sqrt {x^{6}+1}}{24 x^{12}}+\frac {\ln \left (\frac {\sqrt {x^{6}+1}+1}{x^{3}}\right )}{24}\)	\(35\)
risch	\(-\frac {x^{12}+3 x^{6}+2}{24 x^{12} \sqrt {x^{6}+1}}-\frac {\ln \left (\frac {\sqrt {x^{6}+1}-1}{\sqrt {x^{6}}}\right )}{24}\)	\(42\)
meijerg	\(-\frac {\frac {\sqrt {\pi }}{x^{12}}+\frac {\sqrt {\pi }}{x^{6}}+\frac {\left (\frac {1}{2}-2 \ln \relax (2)+6 \ln \relax (x )\right ) \sqrt {\pi }}{4}-\frac {\sqrt {\pi }\, \left (x^{12}+8 x^{6}+8\right )}{8 x^{12}}+\frac {\sqrt {\pi }\, \left (4 x^{6}+8\right ) \sqrt {x^{6}+1}}{8 x^{12}}-\frac {\ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right ) \sqrt {\pi }}{2}}{12 \sqrt {\pi }}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.36, size = 60, normalized size = 1.58 \begin {gather*} \frac {{\left (x^{6} + 1\right )}^{\frac {3}{2}} + \sqrt {x^{6} + 1}}{24 \, {\left (2 \, x^{6} - {\left (x^{6} + 1\right )}^{2} + 1\right )}} + \frac {1}{48} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) - \frac {1}{48} \, \log \left (\sqrt {x^{6} + 1} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((x^6 + 1)^(3/2) + sqrt(x^6 + 1))/(2*x^6 - (x^6 + 1)^2 + 1) + 1/48*log(sqrt(x^6 + 1) + 1) - 1/48*log(sqrt
(x^6 + 1) - 1)

________________________________________________________________________________________

mupad [B]  time = 0.45, size = 49, normalized size = 1.29 \begin {gather*} \frac {\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{24}+\frac {\frac {\sqrt {x^6+1}}{24}+\frac {{\left (x^6+1\right )}^{3/2}}{24}}{2\,x^6-{\left (x^6+1\right )}^2+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 + 1)^(1/2)/x^13,x)

[Out]

atanh((x^6 + 1)^(1/2))/24 + ((x^6 + 1)^(1/2)/24 + (x^6 + 1)^(3/2)/24)/(2*x^6 - (x^6 + 1)^2 + 1)

________________________________________________________________________________________

sympy [A]  time = 2.64, size = 58, normalized size = 1.53 \begin {gather*} \frac {\operatorname {asinh}{\left (\frac {1}{x^{3}} \right )}}{24} - \frac {1}{24 x^{3} \sqrt {1 + \frac {1}{x^{6}}}} - \frac {1}{8 x^{9} \sqrt {1 + \frac {1}{x^{6}}}} - \frac {1}{12 x^{15} \sqrt {1 + \frac {1}{x^{6}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asinh(x**(-3))/24 - 1/(24*x**3*sqrt(1 + x**(-6))) - 1/(8*x**9*sqrt(1 + x**(-6))) - 1/(12*x**15*sqrt(1 + x**(-6
)))

________________________________________________________________________________________