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 = 0.0440098, antiderivative size = 79, normalized size of antiderivative = 1., 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 4897
Rule 4895
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*}
Mathematica [A] time = 0.0305638, size = 37, normalized size = 0.47 \[ \frac{-6 x^2+\left (6 x^3+9 x\right ) \cot ^{-1}(x)-7}{9 a \left (a \left (x^2+1\right )\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.386, size = 165, normalized size = 2.1 \begin{align*} -{\frac{ \left ( i+3\,{\rm arccot} \left (x\right ) \right ) \left ( 3\,i{x}^{2}+{x}^{3}-i-3\,x \right ) }{72\, \left ({x}^{2}+1 \right ) ^{2}{a}^{3}}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}+{\frac{ \left ( 3\,{\rm arccot} \left (x\right )+3\,i \right ) \left ( x+i \right ) }{8\,{a}^{3} \left ({x}^{2}+1 \right ) }\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}+{\frac{ \left ( 3\,x-3\,i \right ) \left ({\rm arccot} \left (x\right )-i \right ) }{8\,{a}^{3} \left ({x}^{2}+1 \right ) }\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }}-{\frac{ \left ( -i+3\,{\rm arccot} \left (x\right ) \right ) \left ( -3\,x-3\,i{x}^{2}+{x}^{3}+i \right ) }{ \left ( 72\,{x}^{4}+144\,{x}^{2}+72 \right ){a}^{3}}\sqrt{a \left ( x+i \right ) \left ( x-i \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45611, size = 85, normalized size = 1.08 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20266, size = 122, normalized size = 1.54 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}{\left (x \right )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14799, size = 74, normalized size = 0.94 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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