Optimal. Leaf size=82 \[ -\frac{3 (1-x) e^{\cot ^{-1}(x)}}{13 a^3 \sqrt{a x^2+a}}-\frac{(1-3 x) e^{\cot ^{-1}(x)}}{13 a^2 \left (a x^2+a\right )^{3/2}}-\frac{(1-5 x) e^{\cot ^{-1}(x)}}{26 a \left (a x^2+a\right )^{5/2}} \]
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Rubi [A] time = 0.090235, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5115, 5114} \[ -\frac{3 (1-x) e^{\cot ^{-1}(x)}}{13 a^3 \sqrt{a x^2+a}}-\frac{(1-3 x) e^{\cot ^{-1}(x)}}{13 a^2 \left (a x^2+a\right )^{3/2}}-\frac{(1-5 x) e^{\cot ^{-1}(x)}}{26 a \left (a x^2+a\right )^{5/2}} \]
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 )^{7/2}} \, dx &=-\frac{e^{\cot ^{-1}(x)} (1-5 x)}{26 a \left (a+a x^2\right )^{5/2}}+\frac{10 \int \frac{e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{5/2}} \, dx}{13 a}\\ &=-\frac{e^{\cot ^{-1}(x)} (1-5 x)}{26 a \left (a+a x^2\right )^{5/2}}-\frac{e^{\cot ^{-1}(x)} (1-3 x)}{13 a^2 \left (a+a x^2\right )^{3/2}}+\frac{6 \int \frac{e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^{3/2}} \, dx}{13 a^2}\\ &=-\frac{e^{\cot ^{-1}(x)} (1-5 x)}{26 a \left (a+a x^2\right )^{5/2}}-\frac{e^{\cot ^{-1}(x)} (1-3 x)}{13 a^2 \left (a+a x^2\right )^{3/2}}-\frac{3 e^{\cot ^{-1}(x)} (1-x)}{13 a^3 \sqrt{a+a x^2}}\\ \end{align*}
Mathematica [A] time = 0.208714, size = 95, normalized size = 1.16 \[ \frac{e^{\cot ^{-1}(x)} \left (-39 \sqrt{\frac{1}{x^2}+1} x \cos \left (3 \cot ^{-1}(x)\right )+5 \sqrt{\frac{1}{x^2}+1} x \cos \left (5 \cot ^{-1}(x)\right )+13 \sqrt{\frac{1}{x^2}+1} x \sin \left (3 \cot ^{-1}(x)\right )-\sqrt{\frac{1}{x^2}+1} x \sin \left (5 \cot ^{-1}(x)\right )+130 x-130\right )}{416 a^3 \sqrt{a \left (x^2+1\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 45, normalized size = 0.6 \begin{align*}{\frac{ \left ({x}^{2}+1 \right ) \left ( 6\,{x}^{5}-6\,{x}^{4}+18\,{x}^{3}-14\,{x}^{2}+17\,x-9 \right ){{\rm e}^{{\rm arccot} \left (x\right )}}}{26} \left ( a{x}^{2}+a \right ) ^{-{\frac{7}{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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.4986, size = 161, normalized size = 1.96 \begin{align*} \frac{{\left (6 \, x^{5} - 6 \, x^{4} + 18 \, x^{3} - 14 \, x^{2} + 17 \, x - 9\right )} \sqrt{a x^{2} + a} e^{\operatorname{arccot}\left (x\right )}}{26 \,{\left (a^{4} x^{6} + 3 \, a^{4} x^{4} + 3 \, a^{4} x^{2} + a^{4}\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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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