3.164 \(\int \frac{\tan ^{-1}(\sqrt{x})}{x^{5/2}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{3 x^{3/2}}-\frac{1}{3 x}-\frac{\log (x)}{3}+\frac{1}{3} \log (x+1) \]

[Out]

-1/(3*x) - (2*ArcTan[Sqrt[x]])/(3*x^(3/2)) - Log[x]/3 + Log[1 + x]/3

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Rubi [A]  time = 0.0137589, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5033, 44} \[ -\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{3 x^{3/2}}-\frac{1}{3 x}-\frac{\log (x)}{3}+\frac{1}{3} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[Sqrt[x]]/x^(5/2),x]

[Out]

-1/(3*x) - (2*ArcTan[Sqrt[x]])/(3*x^(3/2)) - Log[x]/3 + Log[1 + x]/3

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 44

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

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}\left (\sqrt{x}\right )}{x^{5/2}} \, dx &=-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{3 x^{3/2}}+\frac{1}{3} \int \frac{1}{x^2 (1+x)} \, dx\\ &=-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{3 x^{3/2}}+\frac{1}{3} \int \left (\frac{1}{x^2}-\frac{1}{x}+\frac{1}{1+x}\right ) \, dx\\ &=-\frac{1}{3 x}-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{3 x^{3/2}}-\frac{\log (x)}{3}+\frac{1}{3} \log (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0179356, size = 31, normalized size = 0.84 \[ \frac{1}{3} \left (-\frac{2 \tan ^{-1}\left (\sqrt{x}\right )}{x^{3/2}}-\frac{1}{x}-\log (x)+\log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[Sqrt[x]]/x^(5/2),x]

[Out]

(-x^(-1) - (2*ArcTan[Sqrt[x]])/x^(3/2) - Log[x] + Log[1 + x])/3

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Maple [A]  time = 0.029, size = 26, normalized size = 0.7 \begin{align*} -{\frac{1}{3\,x}}-{\frac{2}{3}\arctan \left ( \sqrt{x} \right ){x}^{-{\frac{3}{2}}}}-{\frac{\ln \left ( x \right ) }{3}}+{\frac{\ln \left ( x+1 \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(x^(1/2))/x^(5/2),x)

[Out]

-1/3/x-2/3*arctan(x^(1/2))/x^(3/2)-1/3*ln(x)+1/3*ln(x+1)

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Maxima [A]  time = 0.995832, size = 34, normalized size = 0.92 \begin{align*} -\frac{2 \, \arctan \left (\sqrt{x}\right )}{3 \, x^{\frac{3}{2}}} - \frac{1}{3 \, x} + \frac{1}{3} \, \log \left (x + 1\right ) - \frac{1}{3} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x^(1/2))/x^(5/2),x, algorithm="maxima")

[Out]

-2/3*arctan(sqrt(x))/x^(3/2) - 1/3/x + 1/3*log(x + 1) - 1/3*log(x)

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Fricas [A]  time = 2.19221, size = 96, normalized size = 2.59 \begin{align*} \frac{x^{2} \log \left (x + 1\right ) - x^{2} \log \left (x\right ) - 2 \, \sqrt{x} \arctan \left (\sqrt{x}\right ) - x}{3 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x^(1/2))/x^(5/2),x, algorithm="fricas")

[Out]

1/3*(x^2*log(x + 1) - x^2*log(x) - 2*sqrt(x)*arctan(sqrt(x)) - x)/x^2

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Sympy [B]  time = 8.49852, size = 143, normalized size = 3.86 \begin{align*} - \frac{2 x^{\frac{3}{2}} \operatorname{atan}{\left (\sqrt{x} \right )}}{3 x^{3} + 3 x^{2}} - \frac{2 \sqrt{x} \operatorname{atan}{\left (\sqrt{x} \right )}}{3 x^{3} + 3 x^{2}} - \frac{x^{3} \log{\left (x \right )}}{3 x^{3} + 3 x^{2}} + \frac{x^{3} \log{\left (x + 1 \right )}}{3 x^{3} + 3 x^{2}} + \frac{x^{3}}{3 x^{3} + 3 x^{2}} - \frac{x^{2} \log{\left (x \right )}}{3 x^{3} + 3 x^{2}} + \frac{x^{2} \log{\left (x + 1 \right )}}{3 x^{3} + 3 x^{2}} - \frac{x}{3 x^{3} + 3 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(x**(1/2))/x**(5/2),x)

[Out]

-2*x**(3/2)*atan(sqrt(x))/(3*x**3 + 3*x**2) - 2*sqrt(x)*atan(sqrt(x))/(3*x**3 + 3*x**2) - x**3*log(x)/(3*x**3
+ 3*x**2) + x**3*log(x + 1)/(3*x**3 + 3*x**2) + x**3/(3*x**3 + 3*x**2) - x**2*log(x)/(3*x**3 + 3*x**2) + x**2*
log(x + 1)/(3*x**3 + 3*x**2) - x/(3*x**3 + 3*x**2)

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Giac [A]  time = 1.15805, size = 38, normalized size = 1.03 \begin{align*} \frac{x - 1}{3 \, x} - \frac{2 \, \arctan \left (\sqrt{x}\right )}{3 \, x^{\frac{3}{2}}} + \frac{1}{3} \, \log \left (x + 1\right ) - \frac{1}{3} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(x^(1/2))/x^(5/2),x, algorithm="giac")

[Out]

1/3*(x - 1)/x - 2/3*arctan(sqrt(x))/x^(3/2) + 1/3*log(x + 1) - 1/3*log(x)