Optimal. Leaf size=42 \[ \frac {1}{6 x^{3/2}}-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{2 \sqrt {x}}-\frac {1}{2} \tan ^{-1}\left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5034, 51, 63, 203} \[ \frac {1}{6 x^{3/2}}-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{2 \sqrt {x}}-\frac {1}{2} \tan ^{-1}\left (\sqrt {x}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 203
Rule 5034
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}\left (\sqrt {x}\right )}{x^3} \, dx &=-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{4} \int \frac {1}{x^{5/2} (1+x)} \, dx\\ &=\frac {1}{6 x^{3/2}}-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{2 x^2}+\frac {1}{4} \int \frac {1}{x^{3/2} (1+x)} \, dx\\ &=\frac {1}{6 x^{3/2}}-\frac {1}{2 \sqrt {x}}-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{4} \int \frac {1}{\sqrt {x} (1+x)} \, dx\\ &=\frac {1}{6 x^{3/2}}-\frac {1}{2 \sqrt {x}}-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {x}\right )\\ &=\frac {1}{6 x^{3/2}}-\frac {1}{2 \sqrt {x}}-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{2 x^2}-\frac {1}{2} \tan ^{-1}\left (\sqrt {x}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.81 \[ \frac {\, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-x\right )}{6 x^{3/2}}-\frac {\cot ^{-1}\left (\sqrt {x}\right )}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 27, normalized size = 0.64 \[ \frac {3 \, {\left (x^{2} - 1\right )} \operatorname {arccot}\left (\sqrt {x}\right ) - {\left (3 \, x - 1\right )} \sqrt {x}}{6 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 26, normalized size = 0.62 \[ -\frac {1}{2 \, \sqrt {x}} - \frac {\arctan \left (\frac {1}{\sqrt {x}}\right )}{2 \, x^{2}} + \frac {1}{6 \, x^{\frac {3}{2}}} + \frac {1}{2} \, \arctan \left (\frac {1}{\sqrt {x}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 27, normalized size = 0.64 \[ \frac {1}{6 x^{\frac {3}{2}}}-\frac {\mathrm {arccot}\left (\sqrt {x}\right )}{2 x^{2}}-\frac {\arctan \left (\sqrt {x}\right )}{2}-\frac {1}{2 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 26, normalized size = 0.62 \[ -\frac {3 \, x - 1}{6 \, x^{\frac {3}{2}}} - \frac {\operatorname {arccot}\left (\sqrt {x}\right )}{2 \, x^{2}} - \frac {1}{2} \, \arctan \left (\sqrt {x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 24, normalized size = 0.57 \[ -\frac {\mathrm {atan}\left (\sqrt {x}\right )}{2}-\frac {x-\frac {1}{3}}{2\,x^{3/2}}-\frac {\mathrm {acot}\left (\sqrt {x}\right )}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.95, size = 160, normalized size = 3.81 \[ \frac {3 x^{\frac {7}{2}} \operatorname {acot}{\left (\sqrt {x} \right )}}{6 x^{\frac {7}{2}} + 6 x^{\frac {5}{2}}} + \frac {3 x^{\frac {5}{2}} \operatorname {acot}{\left (\sqrt {x} \right )}}{6 x^{\frac {7}{2}} + 6 x^{\frac {5}{2}}} - \frac {3 x^{\frac {3}{2}} \operatorname {acot}{\left (\sqrt {x} \right )}}{6 x^{\frac {7}{2}} + 6 x^{\frac {5}{2}}} - \frac {3 \sqrt {x} \operatorname {acot}{\left (\sqrt {x} \right )}}{6 x^{\frac {7}{2}} + 6 x^{\frac {5}{2}}} - \frac {3 x^{3}}{6 x^{\frac {7}{2}} + 6 x^{\frac {5}{2}}} - \frac {2 x^{2}}{6 x^{\frac {7}{2}} + 6 x^{\frac {5}{2}}} + \frac {x}{6 x^{\frac {7}{2}} + 6 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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