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

Optimal. Leaf size=38 \[ \frac {7}{24} \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1} \left (7 x^6+2\right )}{24 x^{12}} \]

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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {446, 78, 51, 63, 203} \begin {gather*} \frac {7 \sqrt {x^6-1}}{24 x^6}+\frac {7}{24} \tan ^{-1}\left (\sqrt {x^6-1}\right )+\frac {\sqrt {x^6-1}}{12 x^{12}} \end {gather*}

Antiderivative was successfully verified.

[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 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 78

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] || LtQ
[p, n]))))

Rule 203

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

Rule 446

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*} \int \frac {1+x^6}{x^{13} \sqrt {-1+x^6}} \, dx &=\frac {1}{6} \operatorname {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} \operatorname {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} \operatorname {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} \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 {7 \sqrt {-1+x^6}}{24 x^6}+\frac {7}{24} \tan ^{-1}\left (\sqrt {-1+x^6}\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 50, normalized size = 1.32 \begin {gather*} \frac {1}{24} \sqrt {x^6-1} \left (\frac {7 \tanh ^{-1}\left (\sqrt {1-x^6}\right )}{\sqrt {1-x^6}}+\frac {7 x^6+2}{x^{12}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.04, size = 38, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6} \left (2+7 x^6\right )}{24 x^{12}}+\frac {7}{24} \tan ^{-1}\left (\sqrt {-1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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

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fricas [A]  time = 0.45, size = 34, normalized size = 0.89 \begin {gather*} \frac {7 \, x^{12} \arctan \left (\sqrt {x^{6} - 1}\right ) + {\left (7 \, x^{6} + 2\right )} \sqrt {x^{6} - 1}}{24 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.28, size = 35, normalized size = 0.92 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.38, size = 32, normalized size = 0.84

method	result	size
risch	\(\frac {7 x^{12}-5 x^{6}-2}{24 x^{12} \sqrt {x^{6}-1}}-\frac {7 \arcsin \left (\frac {1}{x^{3}}\right )}{24}\)	\(32\)
trager	\(\frac {\sqrt {x^{6}-1}\, \left (7 x^{6}+2\right )}{24 x^{12}}-\frac {7 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {x^{6}-1}}{x^{3}}\right )}{24}\)	\(50\)
meijerg	\(-\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (\frac {\sqrt {\pi }}{x^{6}}-\frac {\left (1-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (-4 x^{6}+8\right )}{8 x^{6}}+\frac {\sqrt {\pi }\, \sqrt {-x^{6}+1}}{x^{6}}+\ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right ) \sqrt {\pi }\right )}{6 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}+\frac {\sqrt {-\mathrm {signum}\left (x^{6}-1\right )}\, \left (-\frac {\sqrt {\pi }}{2 x^{12}}-\frac {\sqrt {\pi }}{2 x^{6}}+\frac {3 \left (\frac {7}{6}-2 \ln \relax (2)+6 \ln \relax (x )+i \pi \right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{12}+8 x^{6}+8\right )}{16 x^{12}}-\frac {\sqrt {\pi }\, \left (12 x^{6}+8\right ) \sqrt {-x^{6}+1}}{16 x^{12}}-\frac {3 \ln \left (\frac {1}{2}+\frac {\sqrt {-x^{6}+1}}{2}\right ) \sqrt {\pi }}{4}\right )}{6 \sqrt {\mathrm {signum}\left (x^{6}-1\right )}\, \sqrt {\pi }}\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(7*x^12-5*x^6-2)/x^12/(x^6-1)^(1/2)-7/24*arcsin(1/x^3)

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maxima [A]  time = 0.61, size = 60, normalized size = 1.58 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 0.58, size = 35, normalized size = 0.92 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 138.36, size = 63, normalized size = 1.66 \begin {gather*} - \frac {1 - \frac {1}{x^{6} - 1}}{24 \left (1 + \frac {1}{x^{6} - 1}\right )^{2} \sqrt {x^{6} - 1}} - \frac {7 \operatorname {atan}{\left (\frac {1}{\sqrt {x^{6} - 1}} \right )}}{24} + \frac {1}{3 \left (1 + \frac {1}{x^{6} - 1}\right ) \sqrt {x^{6} - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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