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

Optimal. Leaf size=17 \[ \frac {\sqrt {x^2+1}}{1-x} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {803} \[ \frac {\sqrt {x^2+1}}{1-x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + x^2]/(1 - x)

Rule 803

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] /; FreeQ[{a, c, d, e, f, g, m, p}, x]
&& NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0] && EqQ[c*d*f + a*e*g, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.94 \[ -\frac {\sqrt {x^2+1}}{x-1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 + x^2]/(-1 + x))

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fricas [A]  time = 0.40, size = 17, normalized size = 1.00 \[ -\frac {x + \sqrt {x^{2} + 1} - 1}{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + sqrt(x^2 + 1) - 1)/(x - 1)

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giac [B]  time = 1.20, size = 35, normalized size = 2.06 \[ -\frac {\sqrt {\frac {2}{x - 1} + \frac {2}{{\left (x - 1\right )}^{2}} + 1}}{\mathrm {sgn}\left (\frac {1}{x - 1}\right )} + \mathrm {sgn}\left (\frac {1}{x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2/(x - 1) + 2/(x - 1)^2 + 1)/sgn(1/(x - 1)) + sgn(1/(x - 1))

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maple [A]  time = 0.00, size = 15, normalized size = 0.88 \[ -\frac {\sqrt {x^{2}+1}}{x -1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.95, size = 14, normalized size = 0.82 \[ -\frac {\sqrt {x^{2} + 1}}{x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x^2 + 1)/(x - 1)

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mupad [B]  time = 0.15, size = 14, normalized size = 0.82 \[ -\frac {\sqrt {x^2+1}}{x-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x^2 + 1)^(1/2)/(x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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