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

Optimal. Leaf size=45 \[ \frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.0307211, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {827, 1165, 628} \[ \frac{\log \left (x+\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}}-\frac{\log \left (x-\sqrt{2} \sqrt{x}+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)/(Sqrt[x]*(1 + x^2)),x]

[Out]

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

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [C]  time = 0.0352161, size = 99, normalized size = 2.2 \[ \frac{1}{12} \left (3 \sqrt{2} \left (-\log \left (x-\sqrt{2} \sqrt{x}+1\right )+\log \left (x+\sqrt{2} \sqrt{x}+1\right )-2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+2 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right )-8 x^{3/2} \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*x^(3/2)*Hypergeometric2F1[3/4, 1, 7/4, -x^2] + 3*Sqrt[2]*(-2*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 2*ArcTan[1 + Sq
rt[2]*Sqrt[x]] - Log[1 - Sqrt[2]*Sqrt[x] + x] + Log[1 + Sqrt[2]*Sqrt[x] + x]))/12

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Maple [A]  time = 0.006, size = 62, normalized size = 1.4 \begin{align*}{\frac{\sqrt{2}}{4}\ln \left ({ \left ( 1+x+\sqrt{2}\sqrt{x} \right ) \left ( 1+x-\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{4}\ln \left ({ \left ( 1+x-\sqrt{2}\sqrt{x} \right ) \left ( 1+x+\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.53666, size = 46, normalized size = 1.02 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{1}{2} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 1/2*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)

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Fricas [A]  time = 1.66104, size = 97, normalized size = 2.16 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (\frac{2 \, \sqrt{2}{\left (x + 1\right )} \sqrt{x} + x^{2} + 4 \, x + 1}{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.26455, size = 49, normalized size = 1.09 \begin{align*} - \frac{\sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{2} + \frac{\sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/2 + sqrt(2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/2

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Giac [A]  time = 1.31005, size = 46, normalized size = 1.02 \begin{align*} \frac{1}{2} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{1}{2} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 1/2*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1)