\(\int \frac {\arctan (\sqrt {x})}{\sqrt {x}} \, dx\) [162]

Optimal result

Integrand size = 12, antiderivative size = 20 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \sqrt {x} \arctan \left (\sqrt {x}\right )-\log (1+x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4946, 31} \[ \int \frac {\arctan \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \sqrt {x} \arctan \left (\sqrt {x}\right )-\log (x+1) \]

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 4946

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

Rubi steps \begin{align*} \text {integral}& = 2 \sqrt {x} \arctan \left (\sqrt {x}\right )-\int \frac {1}{1+x} \, dx \\ & = 2 \sqrt {x} \arctan \left (\sqrt {x}\right )-\log (1+x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \sqrt {x} \arctan \left (\sqrt {x}\right )-\log (1+x) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(-\ln \left (x +1\right )+2 \sqrt {x}\, \arctan \left (\sqrt {x}\right )\)	\(17\)
default	\(-\ln \left (x +1\right )+2 \sqrt {x}\, \arctan \left (\sqrt {x}\right )\)	\(17\)
meijerg	\(-\ln \left (x +1\right )+2 \sqrt {x}\, \arctan \left (\sqrt {x}\right )\)	\(17\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \, \sqrt {x} \arctan \left (\sqrt {x}\right ) - \log \left (x + 1\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \sqrt {x} \operatorname {atan}{\left (\sqrt {x} \right )} - \log {\left (x + 1 \right )} \]

[In]

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

[Out]

2*sqrt(x)*atan(sqrt(x)) - log(x + 1)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \, \sqrt {x} \arctan \left (\sqrt {x}\right ) - \log \left (x + 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2 \, \sqrt {x} \arctan \left (\sqrt {x}\right ) - \log \left (x + 1\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {\arctan \left (\sqrt {x}\right )}{\sqrt {x}} \, dx=2\,\sqrt {x}\,\mathrm {atan}\left (\sqrt {x}\right )-\ln \left (x+1\right ) \]

[In]

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

[Out]

2*x^(1/2)*atan(x^(1/2)) - log(x + 1)