3.3.24 \(\int \frac {1}{(1+x^2) \sqrt [4]{x^2+x^4}} \, dx\)

Optimal. Leaf size=23 \[ \frac {2 \left (x^4+x^2\right )^{3/4}}{x \left (x^2+1\right )} \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.61, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1146, 264} \begin {gather*} \frac {2 x}{\sqrt [4]{x^4+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 + x^2)*(x^2 + x^4)^(1/4)),x]

[Out]

(2*x)/(x^2 + x^4)^(1/4)

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]

Rule 1146

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(b*x^2 + c*x^4)^FracPart
[p]/(x^(2*FracPart[p])*(b + c*x^2)^FracPart[p]), Int[x^(2*p)*(d + e*x^2)^q*(b + c*x^2)^p, x], x] /; FreeQ[{b,
c, d, e, p, q}, x] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{\left (1+x^2\right ) \sqrt [4]{x^2+x^4}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1+x^2\right )^{5/4}} \, dx}{\sqrt [4]{x^2+x^4}}\\ &=\frac {2 x}{\sqrt [4]{x^2+x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.61 \begin {gather*} \frac {2 x}{\sqrt [4]{x^4+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 + x^2)*(x^2 + x^4)^(1/4)),x]

[Out]

(2*x)/(x^2 + x^4)^(1/4)

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IntegrateAlgebraic [A]  time = 0.09, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^2+x^4\right )^{3/4}}{x \left (1+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((1 + x^2)*(x^2 + x^4)^(1/4)),x]

[Out]

(2*(x^2 + x^4)^(3/4))/(x*(1 + x^2))

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fricas [A]  time = 0.45, size = 18, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x^{3} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(x^4 + x^2)^(3/4)/(x^3 + x)

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giac [A]  time = 0.23, size = 9, normalized size = 0.39 \begin {gather*} \frac {2}{{\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(1/x^2 + 1)^(1/4)

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maple [A]  time = 0.06, size = 13, normalized size = 0.57

method	result	size
gosper	\(\frac {2 x}{\left (x^{4}+x^{2}\right )^{\frac {1}{4}}}\)	\(13\)
meijerg	\(\frac {2 \sqrt {x}}{\left (x^{2}+1\right )^{\frac {1}{4}}}\)	\(13\)
risch	\(\frac {2 x}{\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\)	\(15\)
trager	\(\frac {2 \left (x^{4}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{2}+1\right )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+1)/(x^4+x^2)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {2 \, {\left (x^{3} + x\right )}}{3 \, {\left (x^{\frac {5}{2}} + \sqrt {x}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}} + \int \frac {4 \, {\left (x^{2} + 1\right )}^{\frac {3}{4}}}{3 \, {\left (x^{\frac {9}{2}} + 2 \, x^{\frac {5}{2}} + \sqrt {x}\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(x^3 + x)/((x^(5/2) + sqrt(x))*(x^2 + 1)^(1/4)) + integrate(4/3*(x^2 + 1)^(3/4)/(x^(9/2) + 2*x^(5/2) + sq
rt(x)), x)

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mupad [B]  time = 0.12, size = 21, normalized size = 0.91 \begin {gather*} \frac {2\,{\left (x^4+x^2\right )}^{3/4}}{x\,\left (x^2+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(x^2 + x^4)^(3/4))/(x*(x^2 + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((x**2*(x**2 + 1))**(1/4)*(x**2 + 1)), x)

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