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

Optimal. Leaf size=31 \[ -\frac{x^2}{\sqrt{x^4+1}}-\frac{1}{2 \sqrt{x^4+1} x^2} \]

[Out]

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

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Rubi [A]  time = 0.0059413, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {271, 264} \[ -\frac{x^2}{\sqrt{x^4+1}}-\frac{1}{2 \sqrt{x^4+1} x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(1 + x^4)^(3/2)),x]

[Out]

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

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (1+x^4\right )^{3/2}} \, dx &=-\frac{1}{2 x^2 \sqrt{1+x^4}}-2 \int \frac{x}{\left (1+x^4\right )^{3/2}} \, dx\\ &=-\frac{1}{2 x^2 \sqrt{1+x^4}}-\frac{x^2}{\sqrt{1+x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0043246, size = 23, normalized size = 0.74 \[ -\frac{2 x^4+1}{2 x^2 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(1 + x^4)^(3/2)),x]

[Out]

-(1 + 2*x^4)/(2*x^2*Sqrt[1 + x^4])

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Maple [A]  time = 0.003, size = 20, normalized size = 0.7 \begin{align*} -{\frac{2\,{x}^{4}+1}{2\,{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.05645, size = 34, normalized size = 1.1 \begin{align*} -\frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} - \frac{\sqrt{x^{4} + 1}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.58834, size = 84, normalized size = 2.71 \begin{align*} -\frac{2 \, x^{6} + 2 \, x^{2} +{\left (2 \, x^{4} + 1\right )} \sqrt{x^{4} + 1}}{2 \,{\left (x^{6} + x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.914509, size = 42, normalized size = 1.35 \begin{align*} - \frac{2 x^{4} \sqrt{x^{4} + 1}}{2 x^{6} + 2 x^{2}} - \frac{\sqrt{x^{4} + 1}}{2 x^{6} + 2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18925, size = 30, normalized size = 0.97 \begin{align*} -\frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} - \frac{1}{2} \, \sqrt{\frac{1}{x^{4}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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