Optimal. Leaf size=79 \[ -\frac {1}{9 a \left (a+a x^2\right )^{3/2}}-\frac {2}{3 a^2 \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a+a x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(x)}{3 a^2 \sqrt {a+a x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5017, 5015}
\begin {gather*} -\frac {2}{3 a^2 \sqrt {a x^2+a}}+\frac {2 x \cot ^{-1}(x)}{3 a^2 \sqrt {a x^2+a}}-\frac {1}{9 a \left (a x^2+a\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a x^2+a\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 5015
Rule 5017
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx &=-\frac {1}{9 a \left (a+a x^2\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a+a x^2\right )^{3/2}}+\frac {2 \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx}{3 a}\\ &=-\frac {1}{9 a \left (a+a x^2\right )^{3/2}}-\frac {2}{3 a^2 \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a+a x^2\right )^{3/2}}+\frac {2 x \cot ^{-1}(x)}{3 a^2 \sqrt {a+a x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 37, normalized size = 0.47 \begin {gather*} \frac {-7-6 x^2+\left (9 x+6 x^3\right ) \cot ^{-1}(x)}{9 a \left (a \left (1+x^2\right )\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.20, size = 165, normalized size = 2.09
method | result | size |
risch | \(\frac {i x \left (2 x^{2}+3\right ) \ln \left (i x +1\right )}{6 a^{2} \left (x^{2}+1\right ) \sqrt {a \left (x^{2}+1\right )}}+\frac {-6 i x^{3} \ln \left (-i x +1\right )+6 \pi \,x^{3}-9 i \ln \left (-i x +1\right ) x +9 \pi x -12 x^{2}-14}{18 a^{2} \left (x^{2}+1\right ) \sqrt {a \left (x^{2}+1\right )}}\) | \(101\) |
default | \(-\frac {\left (3 \,\mathrm {arccot}\left (x \right )+i\right ) \left (x^{3}+3 i x^{2}-3 x -i\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{72 \left (x^{2}+1\right )^{2} a^{3}}+\frac {3 \left (\mathrm {arccot}\left (x \right )+i\right ) \left (i+x \right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}}{8 a^{3} \left (x^{2}+1\right )}+\frac {3 \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x -i\right ) \left (\mathrm {arccot}\left (x \right )-i\right )}{8 a^{3} \left (x^{2}+1\right )}-\frac {\left (-i+3 \,\mathrm {arccot}\left (x \right )\right ) \sqrt {a \left (i+x \right ) \left (x -i\right )}\, \left (x^{3}-3 i x^{2}-3 x +i\right )}{72 \left (x^{4}+2 x^{2}+1\right ) a^{3}}\) | \(165\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 63, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {a x^{2} + a} a^{2}} + \frac {x}{{\left (a x^{2} + a\right )}^{\frac {3}{2}} a}\right )} \operatorname {arccot}\left (x\right ) - \frac {2}{3 \, \sqrt {a x^{2} + a} a^{2}} - \frac {1}{9 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.32, size = 52, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {a x^{2} + a} {\left (6 \, x^{2} - 3 \, {\left (2 \, x^{3} + 3 \, x\right )} \operatorname {arccot}\left (x\right ) + 7\right )}}{9 \, {\left (a^{3} x^{4} + 2 \, a^{3} x^{2} + a^{3}\right )}} \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 {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 55, normalized size = 0.70 \begin {gather*} \frac {x {\left (\frac {2 \, x^{2}}{a} + \frac {3}{a}\right )} \arctan \left (\frac {1}{x}\right )}{3 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}}} - \frac {6 \, a x^{2} + 7 \, a}{9 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} \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 = -0.01 \begin {gather*} \int \frac {\mathrm {acot}\left (x\right )}{{\left (a\,x^2+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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