3.2.25 \(\int \frac {-1+x^2}{(1+x^2) \sqrt {x+x^3}} \, dx\)

Optimal. Leaf size=18 \[ -\frac {2 \sqrt {x^3+x}}{x^2+1} \]

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Rubi [A]  time = 0.07, antiderivative size = 12, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2056, 449} \begin {gather*} -\frac {2 x}{\sqrt {x^3+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^2)/((1 + x^2)*Sqrt[x + x^3]),x]

[Out]

(-2*x)/Sqrt[x + x^3]

Rule 449

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

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 12, normalized size = 0.67 \begin {gather*} -\frac {2 x}{\sqrt {x^3+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^2)/((1 + x^2)*Sqrt[x + x^3]),x]

[Out]

(-2*x)/Sqrt[x + x^3]

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IntegrateAlgebraic [A]  time = 0.11, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {x+x^3}}{1+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^2)/((1 + x^2)*Sqrt[x + x^3]),x]

[Out]

(-2*Sqrt[x + x^3])/(1 + x^2)

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fricas [A]  time = 0.47, size = 16, normalized size = 0.89 \begin {gather*} -\frac {2 \, \sqrt {x^{3} + x}}{x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(x^3 + x)/(x^2 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^2 - 1)/(sqrt(x^3 + x)*(x^2 + 1)), x)

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maple [A]  time = 0.08, size = 11, normalized size = 0.61

method	result	size
gosper	\(-\frac {2 x}{\sqrt {x^{3}+x}}\)	\(11\)
default	\(-\frac {2 x}{\sqrt {\left (x^{2}+1\right ) x}}\)	\(13\)
risch	\(-\frac {2 x}{\sqrt {\left (x^{2}+1\right ) x}}\)	\(13\)
elliptic	\(-\frac {2 x}{\sqrt {\left (x^{2}+1\right ) x}}\)	\(13\)
trager	\(-\frac {2 \sqrt {x^{3}+x}}{x^{2}+1}\)	\(17\)
meijerg	\(\frac {2 \hypergeom \left (\left [\frac {5}{4}, \frac {3}{2}\right ], \left [\frac {9}{4}\right ], -x^{2}\right ) x^{\frac {5}{2}}}{5}-2 \hypergeom \left (\left [\frac {1}{4}, \frac {3}{2}\right ], \left [\frac {5}{4}\right ], -x^{2}\right ) \sqrt {x}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-1)/(x^2+1)/(x^3+x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^2 - 1)/(sqrt(x^3 + x)*(x^2 + 1)), x)

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mupad [B]  time = 0.05, size = 10, normalized size = 0.56 \begin {gather*} -\frac {2\,x}{\sqrt {x^3+x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 1)/((x^2 + 1)*(x + x^3)^(1/2)),x)

[Out]

-(2*x)/(x + x^3)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt {x \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((x**2-1)/(x**2+1)/(x**3+x)**(1/2),x)

[Out]

Integral((x - 1)*(x + 1)/(sqrt(x*(x**2 + 1))*(x**2 + 1)), x)

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