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]

Sqrt[1 + x^2]/(1 - x)

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Rubi [A]  time = 0.0090941, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, 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*}

Mathematica [A]  time = 0.0069062, 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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Maple [A]  time = 0.003, size = 15, normalized size = 0.9 \begin{align*} -{\frac{1}{-1+x}\sqrt{{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.41986, size = 19, normalized size = 1.12 \begin{align*} -\frac{\sqrt{x^{2} + 1}}{x - 1} \end{align*}

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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Fricas [A]  time = 1.63456, size = 46, normalized size = 2.71 \begin{align*} -\frac{x + \sqrt{x^{2} + 1} - 1}{x - 1} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{\left (x - 1\right )^{2} \sqrt{x^{2} + 1}}\, dx \end{align*}

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

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]

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