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

Optimal. Leaf size=22 \[ -\sqrt {x}+x \tan ^{-1}\left (\sqrt {x}\right )+\tan ^{-1}\left (\sqrt {x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5027, 50, 63, 203} \[ -\sqrt {x}+x \tan ^{-1}\left (\sqrt {x}\right )+\tan ^{-1}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[Sqrt[x]],x]

[Out]

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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])

Rule 5027

Int[ArcTan[(c_.)*(x_)^(n_)], x_Symbol] :> Simp[x*ArcTan[c*x^n], x] - Dist[c*n, Int[x^n/(1 + c^2*x^(2*n)), x],
x] /; FreeQ[{c, n}, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.82 \[ (x+1) \tan ^{-1}\left (\sqrt {x}\right )-\sqrt {x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[Sqrt[x]],x]

[Out]

-Sqrt[x] + (1 + x)*ArcTan[Sqrt[x]]

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fricas [A]  time = 0.44, size = 14, normalized size = 0.64 \[ {\left (x + 1\right )} \arctan \left (\sqrt {x}\right ) - \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*arctan(sqrt(x)) - sqrt(x)

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giac [A]  time = 0.02, size = 16, normalized size = 0.73 \[ x \arctan \left (\sqrt {x}\right ) - \sqrt {x} + \arctan \left (\sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 17, normalized size = 0.77 \[ x \arctan \left (\sqrt {x}\right )+\arctan \left (\sqrt {x}\right )-\sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.13, size = 16, normalized size = 0.73 \[ x \arctan \left (\sqrt {x}\right ) - \sqrt {x} + \arctan \left (\sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.06, size = 16, normalized size = 0.73 \[ \mathrm {atan}\left (\sqrt {x}\right )+x\,\mathrm {atan}\left (\sqrt {x}\right )-\sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atan(x^(1/2)),x)

[Out]

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

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sympy [A]  time = 1.52, size = 19, normalized size = 0.86 \[ - \sqrt {x} + x \operatorname {atan}{\left (\sqrt {x} \right )} + \operatorname {atan}{\left (\sqrt {x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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