Optimal. Leaf size=55 \[ -\frac{3 (1-x) e^{\cot ^{-1}(x)}}{10 a^2 \sqrt{a x^2+a}}-\frac{(1-3 x) e^{\cot ^{-1}(x)}}{10 a \left (a x^2+a\right )^{3/2}} \]
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Rubi [A] time = 0.0588136, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5115, 5114} \[ -\frac{3 (1-x) e^{\cot ^{-1}(x)}}{10 a^2 \sqrt{a x^2+a}}-\frac{(1-3 x) e^{\cot ^{-1}(x)}}{10 a \left (a x^2+a\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5115
Rule 5114
Rubi steps
\begin{align*} \int \frac{e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{5/2}} \, dx &=-\frac{e^{\cot ^{-1}(x)} (1-3 x)}{10 a \left (a+a x^2\right )^{3/2}}+\frac{3 \int \frac{e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{3/2}} \, dx}{5 a}\\ &=-\frac{e^{\cot ^{-1}(x)} (1-3 x)}{10 a \left (a+a x^2\right )^{3/2}}-\frac{3 e^{\cot ^{-1}(x)} (1-x)}{10 a^2 \sqrt{a+a x^2}}\\ \end{align*}
Mathematica [A] time = 0.161773, size = 51, normalized size = 0.93 \[ \frac{e^{\cot ^{-1}(x)} \left (-3 \sqrt{\frac{1}{x^2}+1} x \cos \left (3 \cot ^{-1}(x)\right )+15 x+2 \cos \left (2 \cot ^{-1}(x)\right )-14\right )}{40 a^2 \sqrt{a \left (x^2+1\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 35, normalized size = 0.6 \begin{align*}{\frac{ \left ({x}^{2}+1 \right ) \left ( 3\,{x}^{3}-3\,{x}^{2}+6\,x-4 \right ){{\rm e}^{{\rm arccot} \left (x\right )}}}{10} \left ( a{x}^{2}+a \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{arccot}\left (x\right )}}{{\left (a x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51251, size = 119, normalized size = 2.16 \begin{align*} \frac{\sqrt{a x^{2} + a}{\left (3 \, x^{3} - 3 \, x^{2} + 6 \, x - 4\right )} e^{\operatorname{arccot}\left (x\right )}}{10 \,{\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{arccot}\left (x\right )}}{{\left (a x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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