Optimal. Leaf size=79 \[ -\frac {2}{3 a^2 \sqrt {a x^2+a}}+\frac {2 x \cot ^{-1}(x)}{3 a^2 \sqrt {a x^2+a}}-\frac {1}{9 a \left (a x^2+a\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a x^2+a\right )^{3/2}} \]
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Rubi [A] time = 0.04, 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 = {4897, 4895} \[ -\frac {2}{3 a^2 \sqrt {a x^2+a}}+\frac {2 x \cot ^{-1}(x)}{3 a^2 \sqrt {a x^2+a}}-\frac {1}{9 a \left (a x^2+a\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{3 a \left (a x^2+a\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4895
Rule 4897
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 = 0.03, size = 37, normalized size = 0.47 \[ \frac {\left (6 x^3+9 x\right ) \cot ^{-1}(x)-6 x^2-7}{9 a \left (a \left (x^2+1\right )\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 52, normalized size = 0.66 \[ -\frac {\sqrt {a x^{2} + a} {\left (6 \, x^{2} - 3 \, {\left (2 \, x^{3} + 3 \, x\right )} \operatorname {arccot}\relax (x) + 7\right )}}{9 \, {\left (a^{3} x^{4} + 2 \, a^{3} x^{2} + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 55, normalized size = 0.70 \[ \frac {x {\left (\frac {2 \, x^{2}}{a} + \frac {3}{a}\right )} \arctan \left (\frac {1}{x}\right )}{3 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}}} - \frac {6 \, a x^{2} + 7 \, a}{9 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.75, size = 165, normalized size = 2.09 \[ -\frac {\left (i+3 \,\mathrm {arccot}\relax (x )\right ) \left (x^{3}+3 i x^{2}-3 x -i\right ) \sqrt {a \left (x +i\right ) \left (x -i\right )}}{72 \left (x^{2}+1\right )^{2} a^{3}}+\frac {3 \left (i+\mathrm {arccot}\relax (x )\right ) \left (x +i\right ) \sqrt {a \left (x +i\right ) \left (x -i\right )}}{8 a^{3} \left (x^{2}+1\right )}+\frac {3 \sqrt {a \left (x +i\right ) \left (x -i\right )}\, \left (x -i\right ) \left (\mathrm {arccot}\relax (x )-i\right )}{8 a^{3} \left (x^{2}+1\right )}-\frac {\left (-i+3 \,\mathrm {arccot}\relax (x )\right ) \sqrt {a \left (x +i\right ) \left (x -i\right )}\, \left (x^{3}-3 i x^{2}-3 x +i\right )}{72 \left (x^{4}+2 x^{2}+1\right ) a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 63, normalized size = 0.80 \[ \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {a x^{2} + a} a^{2}} + \frac {x}{{\left (a x^{2} + a\right )}^{\frac {3}{2}} a}\right )} \operatorname {arccot}\relax (x) - \frac {2}{3 \, \sqrt {a x^{2} + a} a^{2}} - \frac {1}{9 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a} \]
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 \[ \int \frac {\mathrm {acot}\relax (x)}{{\left (a\,x^2+a\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\operatorname {acot}{\relax (x )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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