3.921 \(\int \frac{1}{x^5 \sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right )-\frac{\sqrt{x^4+1}}{4 x^4} \]

[Out]

-Sqrt[1 + x^4]/(4*x^4) + ArcTanh[Sqrt[1 + x^4]]/4

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Rubi [A]  time = 0.0108621, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 207} \[ \frac{1}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right )-\frac{\sqrt{x^4+1}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(4*x^4) + ArcTanh[Sqrt[1 + x^4]]/4

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^5 \sqrt{1+x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x}} \, dx,x,x^4\right )\\ &=-\frac{\sqrt{1+x^4}}{4 x^4}-\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,x^4\right )\\ &=-\frac{\sqrt{1+x^4}}{4 x^4}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+x^4}\right )\\ &=-\frac{\sqrt{1+x^4}}{4 x^4}+\frac{1}{4} \tanh ^{-1}\left (\sqrt{1+x^4}\right )\\ \end{align*}

Mathematica [A]  time = 0.007464, size = 31, normalized size = 1. \[ \frac{1}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right )-\frac{\sqrt{x^4+1}}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(4*x^4) + ArcTanh[Sqrt[1 + x^4]]/4

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Maple [A]  time = 0.045, size = 24, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{4}}\sqrt{{x}^{4}+1}}+{\frac{1}{4}{\it Artanh} \left ({\frac{1}{\sqrt{{x}^{4}+1}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(x^4+1)^(1/2),x)

[Out]

-1/4*(x^4+1)^(1/2)/x^4+1/4*arctanh(1/(x^4+1)^(1/2))

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Maxima [A]  time = 0.983707, size = 50, normalized size = 1.61 \begin{align*} -\frac{\sqrt{x^{4} + 1}}{4 \, x^{4}} + \frac{1}{8} \, \log \left (\sqrt{x^{4} + 1} + 1\right ) - \frac{1}{8} \, \log \left (\sqrt{x^{4} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-1/4*sqrt(x^4 + 1)/x^4 + 1/8*log(sqrt(x^4 + 1) + 1) - 1/8*log(sqrt(x^4 + 1) - 1)

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Fricas [A]  time = 1.49822, size = 115, normalized size = 3.71 \begin{align*} \frac{x^{4} \log \left (\sqrt{x^{4} + 1} + 1\right ) - x^{4} \log \left (\sqrt{x^{4} + 1} - 1\right ) - 2 \, \sqrt{x^{4} + 1}}{8 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^4+1)^(1/2),x, algorithm="fricas")

[Out]

1/8*(x^4*log(sqrt(x^4 + 1) + 1) - x^4*log(sqrt(x^4 + 1) - 1) - 2*sqrt(x^4 + 1))/x^4

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Sympy [A]  time = 2.42289, size = 22, normalized size = 0.71 \begin{align*} \frac{\operatorname{asinh}{\left (\frac{1}{x^{2}} \right )}}{4} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{4 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(x**4+1)**(1/2),x)

[Out]

asinh(x**(-2))/4 - sqrt(1 + x**(-4))/(4*x**2)

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Giac [A]  time = 1.16324, size = 50, normalized size = 1.61 \begin{align*} -\frac{\sqrt{x^{4} + 1}}{4 \, x^{4}} + \frac{1}{8} \, \log \left (\sqrt{x^{4} + 1} + 1\right ) - \frac{1}{8} \, \log \left (\sqrt{x^{4} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^4+1)^(1/2),x, algorithm="giac")

[Out]

-1/4*sqrt(x^4 + 1)/x^4 + 1/8*log(sqrt(x^4 + 1) + 1) - 1/8*log(sqrt(x^4 + 1) - 1)