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

Optimal. Leaf size=14 \[ -\frac {2 x}{\sqrt {x \left (x^2+1\right )}} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6719, 449} \[ -\frac {2 x}{\sqrt {x \left (x^2+1\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/Sqrt[x*(1 + x^2)]

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 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 12, normalized size = 0.86 \[ -\frac {2 x}{\sqrt {x^3+x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/Sqrt[x + x^3]

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fricas [A]  time = 0.42, size = 16, normalized size = 1.14 \[ -\frac {2 \, \sqrt {x^{3} + x}}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)/(x^2+1)/(x*(x^2+1))^(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 \[ \int \frac {x^{2} - 1}{\sqrt {{\left (x^{2} + 1\right )} x} {\left (x^{2} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 13, normalized size = 0.93 \[ -\frac {2 x}{\sqrt {\left (x^{2}+1\right ) x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.38, size = 138, normalized size = 9.86 \[ -\frac {2\,x}{\sqrt {x^3+x}}-\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {\frac {1}{2}+\frac {x\,1{}\mathrm {i}}{2}}\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {1-x\,1{}\mathrm {i}}\right )\middle |\frac {1}{2}\right )\,\sqrt {x\,1{}\mathrm {i}}\,2{}\mathrm {i}}{\sqrt {x^3+x}}+\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {\frac {1}{2}+\frac {x\,1{}\mathrm {i}}{2}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {1-x\,1{}\mathrm {i}}\right )\middle |\frac {1}{2}\right )\,\sqrt {x\,1{}\mathrm {i}}\,2{}\mathrm {i}}{\sqrt {x^3+x}}-\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {1+x\,1{}\mathrm {i}}\,\sqrt {-x\,1{}\mathrm {i}}\,\mathrm {E}\left (\mathrm {asin}\left (\sqrt {-x\,1{}\mathrm {i}}\right )\middle |-1\right )\,1{}\mathrm {i}}{\sqrt {x^3+x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((1 - x*1i)^(1/2)*((x*1i)/2 + 1/2)^(1/2)*ellipticF(asin((1 - x*1i)^(1/2)), 1/2)*(x*1i)^(1/2)*2i)/(x + x^3)^(1/
2) - ((1 - x*1i)^(1/2)*((x*1i)/2 + 1/2)^(1/2)*ellipticE(asin((1 - x*1i)^(1/2)), 1/2)*(x*1i)^(1/2)*2i)/(x + x^3
)^(1/2) - (2*x)/(x + x^3)^(1/2) - ((1 - x*1i)^(1/2)*(x*1i + 1)^(1/2)*(-x*1i)^(1/2)*ellipticE(asin((-x*1i)^(1/2
)), -1)*1i)/(x + x^3)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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