3.1.94 \(\int \frac {\cot ^{-1}(\sqrt {x})}{x^{3/2}} \, dx\) [94]

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

[Out]

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

________________________________________________________________________________________

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 = {4947, 36, 29, 31} \begin {gather*} -\log (x)+\log (x+1)-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*ArcCot[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 4947

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^
n])^p/(m + 1)), x] + Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCot[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 {\cot ^{-1}\left (\sqrt {x}\right )}{x^{3/2}} \, dx &=-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}-\int \frac {1}{x (1+x)} \, dx\\ &=-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}-\int \frac {1}{x} \, dx+\int \frac {1}{1+x} \, dx\\ &=-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}}-\log (x)+\log (1+x)\\ \end {align*}

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 19, normalized size = 0.86

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 18, normalized size = 0.82 \begin {gather*} -\frac {2 \, \operatorname {arccot}\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(arccot(x^(1/2))/x^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 3.15, size = 25, normalized size = 1.14 \begin {gather*} \frac {x \log \left (x + 1\right ) - x \log \left (x\right ) - 2 \, \sqrt {x} \operatorname {arccot}\left (\sqrt {x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.40, size = 20, normalized size = 0.91 \begin {gather*} - \log {\left (x \right )} + \log {\left (x + 1 \right )} - \frac {2 \operatorname {acot}{\left (\sqrt {x} \right )}}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.40, size = 16, normalized size = 0.73 \begin {gather*} -\frac {2 \, \arctan \left (\frac {1}{\sqrt {x}}\right )}{\sqrt {x}} + \log \left (\frac {1}{x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.67, size = 20, normalized size = 0.91 \begin {gather*} \ln \left (x+1\right )-2\,\ln \left (\sqrt {x}\right )-\frac {2\,\mathrm {acot}\left (\sqrt {x}\right )}{\sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________