3.212 \(\int \frac{1+\sqrt{1+x}}{-1+\sqrt{1+x}} \, dx\)

Optimal. Leaf size=25 \[ x+4 \sqrt{x+1}+4 \log \left (1-\sqrt{x+1}\right ) \]

[Out]

x + 4*Sqrt[1 + x] + 4*Log[1 - Sqrt[1 + x]]

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Rubi [A]  time = 0.0173864, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {431, 376, 77} \[ x+4 \sqrt{x+1}+4 \log \left (1-\sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x + 4*Sqrt[1 + x] + 4*Log[1 - Sqrt[1 + x]]

Rule 431

Int[((a_.) + (b_.)*(u_)^(n_))^(p_.)*((c_.) + (d_.)*(u_)^(n_))^(q_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1],
Subst[Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x, u], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && LinearQ[u, x] && N
eQ[u, x]

Rule 376

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, Dis
t[g, Subst[Int[x^(g - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}
, x] && NeQ[b*c - a*d, 0] && FractionQ[n]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0104995, size = 24, normalized size = 0.96 \[ x+4 \left (\sqrt{x+1}+\log \left (1-\sqrt{x+1}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x + 4*(Sqrt[1 + x] + Log[1 - Sqrt[1 + x]])

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Maple [A]  time = 0.002, size = 21, normalized size = 0.8 \begin{align*} 1+x+4\,\sqrt{1+x}+4\,\ln \left ( -1+\sqrt{1+x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.941341, size = 27, normalized size = 1.08 \begin{align*} x + 4 \, \sqrt{x + 1} + 4 \, \log \left (\sqrt{x + 1} - 1\right ) + 1 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 4*sqrt(x + 1) + 4*log(sqrt(x + 1) - 1) + 1

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Fricas [A]  time = 1.71695, size = 59, normalized size = 2.36 \begin{align*} x + 4 \, \sqrt{x + 1} + 4 \, \log \left (\sqrt{x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 4*sqrt(x + 1) + 4*log(sqrt(x + 1) - 1)

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Sympy [A]  time = 0.139086, size = 20, normalized size = 0.8 \begin{align*} x + 4 \sqrt{x + 1} + 4 \log{\left (\sqrt{x + 1} - 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 4*sqrt(x + 1) + 4*log(sqrt(x + 1) - 1)

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Giac [A]  time = 1.07094, size = 28, normalized size = 1.12 \begin{align*} x + 4 \, \sqrt{x + 1} + 4 \, \log \left ({\left | \sqrt{x + 1} - 1 \right |}\right ) + 1 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + 4*sqrt(x + 1) + 4*log(abs(sqrt(x + 1) - 1)) + 1