3.924 \(\int \frac{1}{x^3 \sqrt{1+x^4}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{\sqrt{x^4+1}}{2 x^2} \]

[Out]

-Sqrt[1 + x^4]/(2*x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0026284, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {264} \[ -\frac{\sqrt{x^4+1}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(2*x^2)

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 \sqrt{1+x^4}} \, dx &=-\frac{\sqrt{1+x^4}}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0021295, size = 16, normalized size = 1. \[ -\frac{\sqrt{x^4+1}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[1 + x^4]),x]

[Out]

-Sqrt[1 + x^4]/(2*x^2)

________________________________________________________________________________________

Maple [A]  time = 0.038, size = 13, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{x}^{2}}\sqrt{{x}^{4}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.06402, size = 16, normalized size = 1. \begin{align*} -\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)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(x^4 + 1)/x^2

________________________________________________________________________________________

Fricas [A]  time = 1.4882, size = 43, normalized size = 2.69 \begin{align*} -\frac{x^{2} + \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)^(1/2),x, algorithm="fricas")

[Out]

-1/2*(x^2 + sqrt(x^4 + 1))/x^2

________________________________________________________________________________________

Sympy [A]  time = 0.578022, size = 12, normalized size = 0.75 \begin{align*} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1 + x**(-4))/2

________________________________________________________________________________________

Giac [A]  time = 1.22542, size = 12, normalized size = 0.75 \begin{align*} -\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)^(1/2),x, algorithm="giac")

[Out]

-1/2*sqrt(1/x^4 + 1)