3.122 \(\int \tan ^{-1}(\sqrt{1+x}) \, dx\)

Optimal. Leaf size=30 \[ -\sqrt{x+1}+x \tan ^{-1}\left (\sqrt{x+1}\right )+2 \tan ^{-1}\left (\sqrt{x+1}\right ) \]

[Out]

-Sqrt[1 + x] + 2*ArcTan[Sqrt[1 + x]] + x*ArcTan[Sqrt[1 + x]]

________________________________________________________________________________________

Rubi [A]  time = 0.0110627, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5203, 80, 63, 203} \[ -\sqrt{x+1}+x \tan ^{-1}\left (\sqrt{x+1}\right )+2 \tan ^{-1}\left (\sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[Sqrt[1 + x]],x]

[Out]

-Sqrt[1 + x] + 2*ArcTan[Sqrt[1 + x]] + x*ArcTan[Sqrt[1 + x]]

Rule 5203

Int[ArcTan[u_], x_Symbol] :> Simp[x*ArcTan[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(1 + u^2), x], x] /; Inv
erseFunctionFreeQ[u, x]

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.009583, size = 22, normalized size = 0.73 \[ (x+2) \tan ^{-1}\left (\sqrt{x+1}\right )-\sqrt{x+1} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[Sqrt[1 + x]],x]

[Out]

-Sqrt[1 + x] + (2 + x)*ArcTan[Sqrt[1 + x]]

________________________________________________________________________________________

Maple [A]  time = 0.038, size = 25, normalized size = 0.8 \begin{align*} \left ( x+1 \right ) \arctan \left ( \sqrt{x+1} \right ) -\sqrt{x+1}+\arctan \left ( \sqrt{x+1} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.43828, size = 32, normalized size = 1.07 \begin{align*}{\left (x + 1\right )} \arctan \left (\sqrt{x + 1}\right ) - \sqrt{x + 1} + \arctan \left (\sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*arctan(sqrt(x + 1)) - sqrt(x + 1) + arctan(sqrt(x + 1))

________________________________________________________________________________________

Fricas [A]  time = 2.39461, size = 58, normalized size = 1.93 \begin{align*}{\left (x + 2\right )} \arctan \left (\sqrt{x + 1}\right ) - \sqrt{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 2)*arctan(sqrt(x + 1)) - sqrt(x + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.255563, size = 26, normalized size = 0.87 \begin{align*} x \operatorname{atan}{\left (\sqrt{x + 1} \right )} - \sqrt{x + 1} + 2 \operatorname{atan}{\left (\sqrt{x + 1} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*atan(sqrt(x + 1)) - sqrt(x + 1) + 2*atan(sqrt(x + 1))

________________________________________________________________________________________

Giac [A]  time = 1.1122, size = 32, normalized size = 1.07 \begin{align*}{\left (x + 1\right )} \arctan \left (\sqrt{x + 1}\right ) - \sqrt{x + 1} + \arctan \left (\sqrt{x + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*arctan(sqrt(x + 1)) - sqrt(x + 1) + arctan(sqrt(x + 1))