3.4.32 \(\int \frac {\tan ^{-1}(\sqrt {x})}{\sqrt {x}} \, dx\) [332]

Optimal. Leaf size=20 \[ 2 \sqrt {x} \tan ^{-1}\left (\sqrt {x}\right )-\log (1+x) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4946, 31} \begin {gather*} 2 \sqrt {x} \text {ArcTan}\left (\sqrt {x}\right )-\log (x+1) \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} 2 \sqrt {x} \tan ^{-1}\left (\sqrt {x}\right )-\log (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.01, size = 17, normalized size = 0.85

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.58, size = 16, normalized size = 0.80 \begin {gather*} 2 \, \sqrt {x} \arctan \left (\sqrt {x}\right ) - \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.60, size = 16, normalized size = 0.80 \begin {gather*} 2 \, \sqrt {x} \arctan \left (\sqrt {x}\right ) - \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.13, size = 17, normalized size = 0.85 \begin {gather*} 2 \sqrt {x} \operatorname {atan}{\left (\sqrt {x} \right )} - \log {\left (x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*atan(sqrt(x)) - log(x + 1)

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Giac [A]
time = 1.07, size = 16, normalized size = 0.80 \begin {gather*} 2 \, \sqrt {x} \arctan \left (\sqrt {x}\right ) - \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.25, size = 16, normalized size = 0.80 \begin {gather*} 2\,\sqrt {x}\,\mathrm {atan}\left (\sqrt {x}\right )-\ln \left (x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^(1/2)*atan(x^(1/2)) - log(x + 1)

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