3.941 \(\int \frac{1}{x^5 (1+x^4)^{3/2}} \, dx\)

Optimal. Leaf size=44 \[ -\frac{1}{4 x^4 \sqrt{x^4+1}}-\frac{3}{4 \sqrt{x^4+1}}+\frac{3}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right ) \]

[Out]

-3/(4*Sqrt[1 + x^4]) - 1/(4*x^4*Sqrt[1 + x^4]) + (3*ArcTanh[Sqrt[1 + x^4]])/4

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Rubi [A]  time = 0.0153616, antiderivative size = 47, normalized size of antiderivative = 1.07, 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} \[ -\frac{3 \sqrt{x^4+1}}{4 x^4}+\frac{1}{2 x^4 \sqrt{x^4+1}}+\frac{3}{4} \tanh ^{-1}\left (\sqrt{x^4+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(1 + x^4)^(3/2)),x]

[Out]

1/(2*x^4*Sqrt[1 + x^4]) - (3*Sqrt[1 + x^4])/(4*x^4) + (3*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 \left (1+x^4\right )^{3/2}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 (1+x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac{1}{2 x^4 \sqrt{1+x^4}}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x}} \, dx,x,x^4\right )\\ &=\frac{1}{2 x^4 \sqrt{1+x^4}}-\frac{3 \sqrt{1+x^4}}{4 x^4}-\frac{3}{8} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,x^4\right )\\ &=\frac{1}{2 x^4 \sqrt{1+x^4}}-\frac{3 \sqrt{1+x^4}}{4 x^4}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+x^4}\right )\\ &=\frac{1}{2 x^4 \sqrt{1+x^4}}-\frac{3 \sqrt{1+x^4}}{4 x^4}+\frac{3}{4} \tanh ^{-1}\left (\sqrt{1+x^4}\right )\\ \end{align*}

Mathematica [C]  time = 0.0050258, size = 26, normalized size = 0.59 \[ -\frac{\, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};x^4+1\right )}{2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(1 + x^4)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.011, size = 33, normalized size = 0.8 \begin{align*} -{\frac{1}{4\,{x}^{4}}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{3}{4}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{3}{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)^(3/2),x)

[Out]

-1/4/x^4/(x^4+1)^(1/2)-3/4/(x^4+1)^(1/2)+3/4*arctanh(1/(x^4+1)^(1/2))

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Maxima [A]  time = 1.01276, size = 72, normalized size = 1.64 \begin{align*} -\frac{3 \, x^{4} + 1}{4 \,{\left ({\left (x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{x^{4} + 1}\right )}} + \frac{3}{8} \, \log \left (\sqrt{x^{4} + 1} + 1\right ) - \frac{3}{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)^(3/2),x, algorithm="maxima")

[Out]

-1/4*(3*x^4 + 1)/((x^4 + 1)^(3/2) - sqrt(x^4 + 1)) + 3/8*log(sqrt(x^4 + 1) + 1) - 3/8*log(sqrt(x^4 + 1) - 1)

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Fricas [B]  time = 1.49287, size = 169, normalized size = 3.84 \begin{align*} \frac{3 \,{\left (x^{8} + x^{4}\right )} \log \left (\sqrt{x^{4} + 1} + 1\right ) - 3 \,{\left (x^{8} + x^{4}\right )} \log \left (\sqrt{x^{4} + 1} - 1\right ) - 2 \,{\left (3 \, x^{4} + 1\right )} \sqrt{x^{4} + 1}}{8 \,{\left (x^{8} + x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^4+1)^(3/2),x, algorithm="fricas")

[Out]

1/8*(3*(x^8 + x^4)*log(sqrt(x^4 + 1) + 1) - 3*(x^8 + x^4)*log(sqrt(x^4 + 1) - 1) - 2*(3*x^4 + 1)*sqrt(x^4 + 1)
)/(x^8 + x^4)

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Sympy [A]  time = 2.81343, size = 42, normalized size = 0.95 \begin{align*} \frac{3 \operatorname{asinh}{\left (\frac{1}{x^{2}} \right )}}{4} - \frac{3}{4 x^{2} \sqrt{1 + \frac{1}{x^{4}}}} - \frac{1}{4 x^{6} \sqrt{1 + \frac{1}{x^{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(x**4+1)**(3/2),x)

[Out]

3*asinh(x**(-2))/4 - 3/(4*x**2*sqrt(1 + x**(-4))) - 1/(4*x**6*sqrt(1 + x**(-4)))

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Giac [A]  time = 1.14099, size = 72, normalized size = 1.64 \begin{align*} -\frac{3 \, x^{4} + 1}{4 \,{\left ({\left (x^{4} + 1\right )}^{\frac{3}{2}} - \sqrt{x^{4} + 1}\right )}} + \frac{3}{8} \, \log \left (\sqrt{x^{4} + 1} + 1\right ) - \frac{3}{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)^(3/2),x, algorithm="giac")

[Out]

-1/4*(3*x^4 + 1)/((x^4 + 1)^(3/2) - sqrt(x^4 + 1)) + 3/8*log(sqrt(x^4 + 1) + 1) - 3/8*log(sqrt(x^4 + 1) - 1)