Optimal. Leaf size=22 \[ -\log (x)+\log (x+1)-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \]
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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 = {5034, 36, 29, 31} \[ -\log (x)+\log (x+1)-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 5034
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*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ -\log (x)+\log (x+1)-\frac {2 \cot ^{-1}\left (\sqrt {x}\right )}{\sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.09, size = 25, normalized size = 1.14 \[ \frac {x \log \left (x + 1\right ) - x \log \relax (x) - 2 \, \sqrt {x} \operatorname {arccot}\left (\sqrt {x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 16, normalized size = 0.73 \[ -\frac {2 \, \arctan \left (\frac {1}{\sqrt {x}}\right )}{\sqrt {x}} + \log \left (\frac {1}{x} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 19, normalized size = 0.86 \[ -\ln \relax (x )+\ln \left (x +1\right )-\frac {2 \,\mathrm {arccot}\left (\sqrt {x}\right )}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 18, normalized size = 0.82 \[ -\frac {2 \, \operatorname {arccot}\left (\sqrt {x}\right )}{\sqrt {x}} + \log \left (x + 1\right ) - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 20, normalized size = 0.91 \[ \ln \left (x+1\right )-2\,\ln \left (\sqrt {x}\right )-\frac {2\,\mathrm {acot}\left (\sqrt {x}\right )}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.18, size = 20, normalized size = 0.91 \[ - \log {\relax (x )} + \log {\left (x + 1 \right )} - \frac {2 \operatorname {acot}{\left (\sqrt {x} \right )}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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