Optimal. Leaf size=35 \[ -\frac {1}{a \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{a \sqrt {a+a x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {5015}
\begin {gather*} \frac {x \cot ^{-1}(x)}{a \sqrt {a x^2+a}}-\frac {1}{a \sqrt {a x^2+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 5015
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx &=-\frac {1}{a \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{a \sqrt {a+a x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 21, normalized size = 0.60 \begin {gather*} \frac {-1+x \cot ^{-1}(x)}{a \sqrt {a \left (1+x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.16, size = 68, normalized size = 1.94
method | result | size |
risch | \(\frac {i x \ln \left (i x +1\right )}{2 a \sqrt {a \left (x^{2}+1\right )}}+\frac {-i \ln \left (-i x +1\right ) x +\pi x -2}{2 a \sqrt {a \left (x^{2}+1\right )}}\) | \(55\) |
default | \(\frac {\left (\mathrm {arccot}\left (x \right )+i\right ) \left (i+x \right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{2 \left (x^{2}+1\right ) a^{2}}+\frac {\sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x -i\right ) \left (\mathrm {arccot}\left (x \right )-i\right )}{2 \left (x^{2}+1\right ) a^{2}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 31, normalized size = 0.89 \begin {gather*} \frac {x \operatorname {arccot}\left (x\right )}{\sqrt {a x^{2} + a} a} - \frac {1}{\sqrt {a x^{2} + a} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.01, size = 29, normalized size = 0.83 \begin {gather*} \frac {\sqrt {a x^{2} + a} {\left (x \operatorname {arccot}\left (x\right ) - 1\right )}}{a^{2} x^{2} + a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acot}{\left (x \right )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 33, normalized size = 0.94 \begin {gather*} \frac {x \arctan \left (\frac {1}{x}\right )}{\sqrt {a x^{2} + a} a} - \frac {1}{\sqrt {a x^{2} + a} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {acot}\left (x\right )}{{\left (a\,x^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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