3.2.56 \(\int \text {ArcTan}(\sqrt {x}) \, dx\) [156]

Optimal. Leaf size=22 \[ -\sqrt {x}+\text {ArcTan}\left (\sqrt {x}\right )+x \text {ArcTan}\left (\sqrt {x}\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, 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 = {4930, 52, 65, 209} \begin {gather*} x \text {ArcTan}\left (\sqrt {x}\right )+\text {ArcTan}\left (\sqrt {x}\right )-\sqrt {x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 52

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 65

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 209

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

Rule 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[b*c
*n*p, Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0
] && (EqQ[n, 1] || EqQ[p, 1])

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 )+\text {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 \begin {gather*} -\sqrt {x}+(1+x) \text {ArcTan}\left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[x] + (1 + x)*ArcTan[Sqrt[x]]

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Maple [A]
time = 0.01, size = 17, normalized size = 0.77

method	result	size
derivativedivides	\(\arctan \left (\sqrt {x}\right )+x \arctan \left (\sqrt {x}\right )-\sqrt {x}\)	\(17\)
default	\(\arctan \left (\sqrt {x}\right )+x \arctan \left (\sqrt {x}\right )-\sqrt {x}\)	\(17\)
meijerg	\(-\sqrt {x}+\frac {\left (3 x +3\right ) \arctan \left (\sqrt {x}\right )}{3}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(x^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.47, size = 16, normalized size = 0.73 \begin {gather*} x \arctan \left (\sqrt {x}\right ) - \sqrt {x} + \arctan \left (\sqrt {x}\right ) \end {gather*}

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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Fricas [A]
time = 1.21, size = 14, normalized size = 0.64 \begin {gather*} {\left (x + 1\right )} \arctan \left (\sqrt {x}\right ) - \sqrt {x} \end {gather*}

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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Sympy [A]
time = 0.69, size = 19, normalized size = 0.86 \begin {gather*} - \sqrt {x} + x \operatorname {atan}{\left (\sqrt {x} \right )} + \operatorname {atan}{\left (\sqrt {x} \right )} \end {gather*}

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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Giac [A]
time = 0.43, size = 16, normalized size = 0.73 \begin {gather*} x \arctan \left (\sqrt {x}\right ) - \sqrt {x} + \arctan \left (\sqrt {x}\right ) \end {gather*}

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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Mupad [B]
time = 0.07, size = 16, normalized size = 0.73 \begin {gather*} \mathrm {atan}\left (\sqrt {x}\right )+x\,\mathrm {atan}\left (\sqrt {x}\right )-\sqrt {x} \end {gather*}

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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