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 = 0.06, antiderivative size = 55, normalized size of antiderivative = 1.00, 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 5114
Rule 5115
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*}
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Mathematica [A] time = 0.17, 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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fricas [A] time = 0.75, size = 50, normalized size = 0.91 \[ \frac {\sqrt {a x^{2} + a} {\left (3 \, x^{3} - 3 \, x^{2} + 6 \, x - 4\right )} e^{\operatorname {arccot}\relax (x)}}{10 \, {\left (a^{3} x^{4} + 2 \, a^{3} x^{2} + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {arccot}\relax (x)}}{{\left (a x^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 35, normalized size = 0.64 \[ \frac {\left (x^{2}+1\right ) \left (3 x^{3}-3 x^{2}+6 x -4\right ) {\mathrm e}^{\mathrm {arccot}\relax (x )}}{10 \left (a \,x^{2}+a \right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {arccot}\relax (x)}}{{\left (a x^{2} + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 42, normalized size = 0.76 \[ -\frac {4\,{\mathrm {e}}^{\mathrm {acot}\relax (x)}-6\,x\,{\mathrm {e}}^{\mathrm {acot}\relax (x)}+3\,x^2\,{\mathrm {e}}^{\mathrm {acot}\relax (x)}-3\,x^3\,{\mathrm {e}}^{\mathrm {acot}\relax (x)}}{10\,a\,{\left (a\,x^2+a\right )}^{3/2}} \]
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 \[ \int \frac {e^{\operatorname {acot}{\relax (x )}}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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