3.2.63 \(\int \frac {\text {ArcTan}(\sqrt {x})}{x^{3/2}} \, dx\) [163]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {4946, 36, 29, 31} \begin {gather*} -\frac {2 \text {ArcTan}\left (\sqrt {x}\right )}{\sqrt {x}}+\log (x)-\log (x+1) \end {gather*}

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 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]

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 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 )}{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*}

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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} -\frac {2 \text {ArcTan}\left (\sqrt {x}\right )}{\sqrt {x}}+\log (x)-\log (1+x) \end {gather*}

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 = 0.02, size = 19, normalized size = 0.86

method	result	size
derivativedivides	\(\ln \left (x \right )-\ln \left (1+x \right )-\frac {2 \arctan \left (\sqrt {x}\right )}{\sqrt {x}}\)	\(19\)
default	\(\ln \left (x \right )-\ln \left (1+x \right )-\frac {2 \arctan \left (\sqrt {x}\right )}{\sqrt {x}}\)	\(19\)
meijerg	\(\ln \left (x \right )-\ln \left (1+x \right )-\frac {2 \arctan \left (\sqrt {x}\right )}{\sqrt {x}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(x^(1/2))/x^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 18, normalized size = 0.82 \begin {gather*} -\frac {2 \, \arctan \left (\sqrt {x}\right )}{\sqrt {x}} - \log \left (x + 1\right ) + \log \left (x\right ) \end {gather*}

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 = 5.14, size = 26, normalized size = 1.18 \begin {gather*} -\frac {x \log \left (x + 1\right ) - x \log \left (x\right ) + 2 \, \sqrt {x} \arctan \left (\sqrt {x}\right )}{x} \end {gather*}

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 = 0.38, size = 20, normalized size = 0.91 \begin {gather*} \log {\left (x \right )} - \log {\left (x + 1 \right )} - \frac {2 \operatorname {atan}{\left (\sqrt {x} \right )}}{\sqrt {x}} \end {gather*}

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 = 0.45, size = 18, normalized size = 0.82 \begin {gather*} -\frac {2 \, \arctan \left (\sqrt {x}\right )}{\sqrt {x}} - \log \left (x + 1\right ) + \log \left (x\right ) \end {gather*}

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)

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Mupad [B]
time = 0.36, size = 22, normalized size = 1.00 \begin {gather*} 2\,\ln \left (\sqrt {x}\right )-\ln \left (x+1\right )-\frac {2\,\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^(3/2),x)

[Out]

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

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