3.163 \(\int \frac{\tan ^{-1}(\sqrt{x})}{x^{3/2}} \, dx\)

Optimal. Leaf size=22 \[ \log (x)-\log (x+1)-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0087961, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5033, 36, 29, 31} \[ \log (x)-\log (x+1)-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[Sqrt[x]]/x^(3/2),x]

[Out]

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

Rule 5033

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}} \, dx &=-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}}+\int \frac{1}{x (1+x)} \, dx\\ &=-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}}+\int \frac{1}{x} \, dx-\int \frac{1}{1+x} \, dx\\ &=-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}}+\log (x)-\log (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0109185, size = 22, normalized size = 1. \[ \log (x)-\log (x+1)-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[Sqrt[x]]/x^(3/2),x]

[Out]

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

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Maple [A]  time = 0.026, size = 19, normalized size = 0.9 \begin{align*} \ln \left ( x \right ) -\ln \left ( x+1 \right ) -2\,{\frac{\arctan \left ( \sqrt{x} \right ) }{\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(x^(1/2))/x^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.14953, size = 24, normalized size = 1.09 \begin{align*} -\frac{2 \, \arctan \left (\sqrt{x}\right )}{\sqrt{x}} - \log \left (x + 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x^(1/2))/x^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.18452, size = 78, normalized size = 3.55 \begin{align*} -\frac{x \log \left (x + 1\right ) - x \log \left (x\right ) + 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x^(1/2))/x^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.91852, size = 20, normalized size = 0.91 \begin{align*} \log{\left (x \right )} - \log{\left (x + 1 \right )} - \frac{2 \operatorname{atan}{\left (\sqrt{x} \right )}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(x**(1/2))/x**(3/2),x)

[Out]

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

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Giac [A]  time = 1.11018, size = 24, normalized size = 1.09 \begin{align*} -\frac{2 \, \arctan \left (\sqrt{x}\right )}{\sqrt{x}} - \log \left (x + 1\right ) + \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x^(1/2))/x^(3/2),x, algorithm="giac")

[Out]

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