3.6.55 \(\int \frac {-1+x^6}{x^{19} \sqrt {1+x^6}} \, dx\)

Optimal. Leaf size=43 \[ \frac {\sqrt {x^6+1} \left (33 x^{12}-22 x^6+8\right )}{144 x^{18}}-\frac {11}{48} \tanh ^{-1}\left (\sqrt {x^6+1}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.47, number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {446, 78, 51, 63, 207} \begin {gather*} \frac {11 \sqrt {x^6+1}}{48 x^6}-\frac {11}{48} \tanh ^{-1}\left (\sqrt {x^6+1}\right )+\frac {\sqrt {x^6+1}}{18 x^{18}}-\frac {11 \sqrt {x^6+1}}{72 x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^6)/(x^19*Sqrt[1 + x^6]),x]

[Out]

Sqrt[1 + x^6]/(18*x^18) - (11*Sqrt[1 + x^6])/(72*x^12) + (11*Sqrt[1 + x^6])/(48*x^6) - (11*ArcTanh[Sqrt[1 + x^
6]])/48

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 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 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^{19} \sqrt {1+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {-1+x}{x^4 \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{18 x^{18}}+\frac {11}{36} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{18 x^{18}}-\frac {11 \sqrt {1+x^6}}{72 x^{12}}-\frac {11}{48} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{18 x^{18}}-\frac {11 \sqrt {1+x^6}}{72 x^{12}}+\frac {11 \sqrt {1+x^6}}{48 x^6}+\frac {11}{96} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^6\right )\\ &=\frac {\sqrt {1+x^6}}{18 x^{18}}-\frac {11 \sqrt {1+x^6}}{72 x^{12}}+\frac {11 \sqrt {1+x^6}}{48 x^6}+\frac {11}{48} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^6}\right )\\ &=\frac {\sqrt {1+x^6}}{18 x^{18}}-\frac {11 \sqrt {1+x^6}}{72 x^{12}}+\frac {11 \sqrt {1+x^6}}{48 x^6}-\frac {11}{48} \tanh ^{-1}\left (\sqrt {1+x^6}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 36, normalized size = 0.84 \begin {gather*} \frac {\sqrt {x^6+1} \left (1-11 x^{18} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};x^6+1\right )\right )}{18 x^{18}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^6)/(x^19*Sqrt[1 + x^6]),x]

[Out]

(Sqrt[1 + x^6]*(1 - 11*x^18*Hypergeometric2F1[1/2, 3, 3/2, 1 + x^6]))/(18*x^18)

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IntegrateAlgebraic [A]  time = 0.06, size = 43, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^6} \left (8-22 x^6+33 x^{12}\right )}{144 x^{18}}-\frac {11}{48} \tanh ^{-1}\left (\sqrt {1+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^6)/(x^19*Sqrt[1 + x^6]),x]

[Out]

(Sqrt[1 + x^6]*(8 - 22*x^6 + 33*x^12))/(144*x^18) - (11*ArcTanh[Sqrt[1 + x^6]])/48

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fricas [A]  time = 0.45, size = 57, normalized size = 1.33 \begin {gather*} -\frac {33 \, x^{18} \log \left (\sqrt {x^{6} + 1} + 1\right ) - 33 \, x^{18} \log \left (\sqrt {x^{6} + 1} - 1\right ) - 2 \, {\left (33 \, x^{12} - 22 \, x^{6} + 8\right )} \sqrt {x^{6} + 1}}{288 \, x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)/x^19/(x^6+1)^(1/2),x, algorithm="fricas")

[Out]

-1/288*(33*x^18*log(sqrt(x^6 + 1) + 1) - 33*x^18*log(sqrt(x^6 + 1) - 1) - 2*(33*x^12 - 22*x^6 + 8)*sqrt(x^6 +
1))/x^18

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giac [A]  time = 0.32, size = 58, normalized size = 1.35 \begin {gather*} \frac {33 \, {\left (x^{6} + 1\right )}^{\frac {5}{2}} - 88 \, {\left (x^{6} + 1\right )}^{\frac {3}{2}} + 63 \, \sqrt {x^{6} + 1}}{144 \, x^{18}} - \frac {11}{96} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {11}{96} \, \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)/x^19/(x^6+1)^(1/2),x, algorithm="giac")

[Out]

1/144*(33*(x^6 + 1)^(5/2) - 88*(x^6 + 1)^(3/2) + 63*sqrt(x^6 + 1))/x^18 - 11/96*log(sqrt(x^6 + 1) + 1) + 11/96
*log(sqrt(x^6 + 1) - 1)

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maple [A]  time = 0.26, size = 42, normalized size = 0.98

method	result	size
trager	\(\frac {\sqrt {x^{6}+1}\, \left (33 x^{12}-22 x^{6}+8\right )}{144 x^{18}}+\frac {11 \ln \left (\frac {\sqrt {x^{6}+1}-1}{x^{3}}\right )}{48}\)	\(42\)
risch	\(\frac {33 x^{18}+11 x^{12}-14 x^{6}+8}{144 x^{18} \sqrt {x^{6}+1}}+\frac {11 \ln \left (\frac {\sqrt {x^{6}+1}-1}{\sqrt {x^{6}}}\right )}{48}\)	\(49\)
meijerg	\(\frac {-\frac {\sqrt {\pi }}{2 x^{12}}+\frac {\sqrt {\pi }}{2 x^{6}}+\frac {3 \left (\frac {7}{6}-2 \ln \relax (2)+6 \ln \relax (x )\right ) \sqrt {\pi }}{8}+\frac {\sqrt {\pi }\, \left (-7 x^{12}-8 x^{6}+8\right )}{16 x^{12}}-\frac {\sqrt {\pi }\, \left (-12 x^{6}+8\right ) \sqrt {x^{6}+1}}{16 x^{12}}-\frac {3 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right ) \sqrt {\pi }}{4}}{6 \sqrt {\pi }}-\frac {-\frac {\sqrt {\pi }}{3 x^{18}}+\frac {\sqrt {\pi }}{4 x^{12}}-\frac {3 \sqrt {\pi }}{8 x^{6}}-\frac {5 \left (\frac {37}{30}-2 \ln \relax (2)+6 \ln \relax (x )\right ) \sqrt {\pi }}{16}+\frac {\sqrt {\pi }\, \left (148 x^{18}+144 x^{12}-96 x^{6}+128\right )}{384 x^{18}}-\frac {\sqrt {\pi }\, \left (240 x^{12}-160 x^{6}+128\right ) \sqrt {x^{6}+1}}{384 x^{18}}+\frac {5 \ln \left (\frac {1}{2}+\frac {\sqrt {x^{6}+1}}{2}\right ) \sqrt {\pi }}{8}}{6 \sqrt {\pi }}\)	\(212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-1)/x^19/(x^6+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/144*(x^6+1)^(1/2)*(33*x^12-22*x^6+8)/x^18+11/48*ln(((x^6+1)^(1/2)-1)/x^3)

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maxima [B]  time = 0.49, size = 119, normalized size = 2.77 \begin {gather*} \frac {15 \, {\left (x^{6} + 1\right )}^{\frac {5}{2}} - 40 \, {\left (x^{6} + 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {x^{6} + 1}}{144 \, {\left (3 \, x^{6} + {\left (x^{6} + 1\right )}^{3} - 3 \, {\left (x^{6} + 1\right )}^{2} + 2\right )}} - \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 {11}{96} \, \log \left (\sqrt {x^{6} + 1} + 1\right ) + \frac {11}{96} \, \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)/x^19/(x^6+1)^(1/2),x, algorithm="maxima")

[Out]

1/144*(15*(x^6 + 1)^(5/2) - 40*(x^6 + 1)^(3/2) + 33*sqrt(x^6 + 1))/(3*x^6 + (x^6 + 1)^3 - 3*(x^6 + 1)^2 + 2) -
1/24*(3*(x^6 + 1)^(3/2) - 5*sqrt(x^6 + 1))/(2*x^6 - (x^6 + 1)^2 + 1) - 11/96*log(sqrt(x^6 + 1) + 1) + 11/96*l
og(sqrt(x^6 + 1) - 1)

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mupad [B]  time = 1.01, size = 85, normalized size = 1.98 \begin {gather*} \frac {\frac {5\,\sqrt {x^6+1}}{24}-\frac {{\left (x^6+1\right )}^{3/2}}{8}}{2\,x^6-{\left (x^6+1\right )}^2+1}-\frac {11\,\mathrm {atanh}\left (\sqrt {x^6+1}\right )}{48}+\frac {11\,\sqrt {x^6+1}}{48\,x^{18}}-\frac {5\,{\left (x^6+1\right )}^{3/2}}{18\,x^{18}}+\frac {5\,{\left (x^6+1\right )}^{5/2}}{48\,x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6 - 1)/(x^19*(x^6 + 1)^(1/2)),x)

[Out]

((5*(x^6 + 1)^(1/2))/24 - (x^6 + 1)^(3/2)/8)/(2*x^6 - (x^6 + 1)^2 + 1) - (11*atanh((x^6 + 1)^(1/2)))/48 + (11*
(x^6 + 1)^(1/2))/(48*x^18) - (5*(x^6 + 1)^(3/2))/(18*x^18) + (5*(x^6 + 1)^(5/2))/(48*x^18)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-1)/x**19/(x**6+1)**(1/2),x)

[Out]

Timed out

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