3.1523 \(\int \frac{1}{x^{17} \sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=47 \[ \frac{3 \sqrt{x^8+1}}{32 x^8}-\frac{\sqrt{x^8+1}}{16 x^{16}}-\frac{3}{32} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

[Out]

-Sqrt[1 + x^8]/(16*x^16) + (3*Sqrt[1 + x^8])/(32*x^8) - (3*ArcTanh[Sqrt[1 + x^8]])/32

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Rubi [A]  time = 0.0163496, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 207} \[ \frac{3 \sqrt{x^8+1}}{32 x^8}-\frac{\sqrt{x^8+1}}{16 x^{16}}-\frac{3}{32} \tanh ^{-1}\left (\sqrt{x^8+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^17*Sqrt[1 + x^8]),x]

[Out]

-Sqrt[1 + x^8]/(16*x^16) + (3*Sqrt[1 + x^8])/(32*x^8) - (3*ArcTanh[Sqrt[1 + x^8]])/32

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]]

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])

Rubi steps

\begin{align*} \int \frac{1}{x^{17} \sqrt{1+x^8}} \, dx &=\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1+x}} \, dx,x,x^8\right )\\ &=-\frac{\sqrt{1+x^8}}{16 x^{16}}-\frac{3}{32} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x}} \, dx,x,x^8\right )\\ &=-\frac{\sqrt{1+x^8}}{16 x^{16}}+\frac{3 \sqrt{1+x^8}}{32 x^8}+\frac{3}{64} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,x^8\right )\\ &=-\frac{\sqrt{1+x^8}}{16 x^{16}}+\frac{3 \sqrt{1+x^8}}{32 x^8}+\frac{3}{32} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+x^8}\right )\\ &=-\frac{\sqrt{1+x^8}}{16 x^{16}}+\frac{3 \sqrt{1+x^8}}{32 x^8}-\frac{3}{32} \tanh ^{-1}\left (\sqrt{1+x^8}\right )\\ \end{align*}

Mathematica [C]  time = 0.0046953, size = 26, normalized size = 0.55 \[ -\frac{1}{4} \sqrt{x^8+1} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};x^8+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^17*Sqrt[1 + x^8]),x]

[Out]

-(Sqrt[1 + x^8]*Hypergeometric2F1[1/2, 3, 3/2, 1 + x^8])/4

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Maple [A]  time = 0.02, size = 42, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{16}+{x}^{8}-2}{32\,{x}^{16}}{\frac{1}{\sqrt{{x}^{8}+1}}}}+{\frac{3}{32}\ln \left ({ \left ( \sqrt{{x}^{8}+1}-1 \right ){\frac{1}{\sqrt{{x}^{8}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^17/(x^8+1)^(1/2),x)

[Out]

1/32*(3*x^16+x^8-2)/x^16/(x^8+1)^(1/2)+3/32*ln(((x^8+1)^(1/2)-1)/(x^8)^(1/2))

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Maxima [A]  time = 0.964054, size = 86, normalized size = 1.83 \begin{align*} -\frac{3 \,{\left (x^{8} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{x^{8} + 1}}{32 \,{\left (2 \, x^{8} -{\left (x^{8} + 1\right )}^{2} + 1\right )}} - \frac{3}{64} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{3}{64} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^17/(x^8+1)^(1/2),x, algorithm="maxima")

[Out]

-1/32*(3*(x^8 + 1)^(3/2) - 5*sqrt(x^8 + 1))/(2*x^8 - (x^8 + 1)^2 + 1) - 3/64*log(sqrt(x^8 + 1) + 1) + 3/64*log
(sqrt(x^8 + 1) - 1)

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Fricas [A]  time = 1.32907, size = 143, normalized size = 3.04 \begin{align*} -\frac{3 \, x^{16} \log \left (\sqrt{x^{8} + 1} + 1\right ) - 3 \, x^{16} \log \left (\sqrt{x^{8} + 1} - 1\right ) - 2 \,{\left (3 \, x^{8} - 2\right )} \sqrt{x^{8} + 1}}{64 \, x^{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^17/(x^8+1)^(1/2),x, algorithm="fricas")

[Out]

-1/64*(3*x^16*log(sqrt(x^8 + 1) + 1) - 3*x^16*log(sqrt(x^8 + 1) - 1) - 2*(3*x^8 - 2)*sqrt(x^8 + 1))/x^16

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Sympy [A]  time = 6.25812, size = 60, normalized size = 1.28 \begin{align*} - \frac{3 \operatorname{asinh}{\left (\frac{1}{x^{4}} \right )}}{32} + \frac{3}{32 x^{4} \sqrt{1 + \frac{1}{x^{8}}}} + \frac{1}{32 x^{12} \sqrt{1 + \frac{1}{x^{8}}}} - \frac{1}{16 x^{20} \sqrt{1 + \frac{1}{x^{8}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**17/(x**8+1)**(1/2),x)

[Out]

-3*asinh(x**(-4))/32 + 3/(32*x**4*sqrt(1 + x**(-8))) + 1/(32*x**12*sqrt(1 + x**(-8))) - 1/(16*x**20*sqrt(1 + x
**(-8)))

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Giac [A]  time = 1.19435, size = 66, normalized size = 1.4 \begin{align*} \frac{3 \,{\left (x^{8} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{x^{8} + 1}}{32 \, x^{16}} - \frac{3}{64} \, \log \left (\sqrt{x^{8} + 1} + 1\right ) + \frac{3}{64} \, \log \left (\sqrt{x^{8} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^17/(x^8+1)^(1/2),x, algorithm="giac")

[Out]

1/32*(3*(x^8 + 1)^(3/2) - 5*sqrt(x^8 + 1))/x^16 - 3/64*log(sqrt(x^8 + 1) + 1) + 3/64*log(sqrt(x^8 + 1) - 1)