Optimal. Leaf size=118 \[ -\frac {8}{15 a^3 \sqrt {a x^2+a}}+\frac {8 x \cot ^{-1}(x)}{15 a^3 \sqrt {a x^2+a}}-\frac {4}{45 a^2 \left (a x^2+a\right )^{3/2}}+\frac {4 x \cot ^{-1}(x)}{15 a^2 \left (a x^2+a\right )^{3/2}}-\frac {1}{25 a \left (a x^2+a\right )^{5/2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a x^2+a\right )^{5/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4897, 4895} \[ -\frac {8}{15 a^3 \sqrt {a x^2+a}}-\frac {4}{45 a^2 \left (a x^2+a\right )^{3/2}}+\frac {8 x \cot ^{-1}(x)}{15 a^3 \sqrt {a x^2+a}}+\frac {4 x \cot ^{-1}(x)}{15 a^2 \left (a x^2+a\right )^{3/2}}-\frac {1}{25 a \left (a x^2+a\right )^{5/2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a x^2+a\right )^{5/2}} \]
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 )^{7/2}} \, dx &=-\frac {1}{25 a \left (a+a x^2\right )^{5/2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a+a x^2\right )^{5/2}}+\frac {4 \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{5/2}} \, dx}{5 a}\\ &=-\frac {1}{25 a \left (a+a x^2\right )^{5/2}}-\frac {4}{45 a^2 \left (a+a x^2\right )^{3/2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a+a x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(x)}{15 a^2 \left (a+a x^2\right )^{3/2}}+\frac {8 \int \frac {\cot ^{-1}(x)}{\left (a+a x^2\right )^{3/2}} \, dx}{15 a^2}\\ &=-\frac {1}{25 a \left (a+a x^2\right )^{5/2}}-\frac {4}{45 a^2 \left (a+a x^2\right )^{3/2}}-\frac {8}{15 a^3 \sqrt {a+a x^2}}+\frac {x \cot ^{-1}(x)}{5 a \left (a+a x^2\right )^{5/2}}+\frac {4 x \cot ^{-1}(x)}{15 a^2 \left (a+a x^2\right )^{3/2}}+\frac {8 x \cot ^{-1}(x)}{15 a^3 \sqrt {a+a x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 0.40 \[ \frac {-120 x^4-260 x^2+15 \left (8 x^4+20 x^2+15\right ) x \cot ^{-1}(x)-149}{225 a \left (a \left (x^2+1\right )\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 70, normalized size = 0.59 \[ -\frac {{\left (120 \, x^{4} + 260 \, x^{2} - 15 \, {\left (8 \, x^{5} + 20 \, x^{3} + 15 \, x\right )} \operatorname {arccot}\relax (x) + 149\right )} \sqrt {a x^{2} + a}}{225 \, {\left (a^{4} x^{6} + 3 \, a^{4} x^{4} + 3 \, a^{4} x^{2} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 83, normalized size = 0.70 \[ \frac {{\left (4 \, x^{2} {\left (\frac {2 \, x^{2}}{a} + \frac {5}{a}\right )} + \frac {15}{a}\right )} x \arctan \left (\frac {1}{x}\right )}{15 \, {\left (a x^{2} + a\right )}^{\frac {5}{2}}} - \frac {120 \, {\left (a x^{2} + a\right )}^{2} + 20 \, {\left (a x^{2} + a\right )} a + 9 \, a^{2}}{225 \, {\left (a x^{2} + a\right )}^{\frac {5}{2}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.85, size = 258, normalized size = 2.19 \[ \frac {\left (i+5 \,\mathrm {arccot}\relax (x )\right ) \left (x^{5}+5 i x^{4}-10 x^{3}-10 i x^{2}+5 x +i\right ) \sqrt {a \left (x +i\right ) \left (x -i\right )}}{800 \left (x^{2}+1\right )^{3} a^{4}}+\frac {5 \left (i+\mathrm {arccot}\relax (x )\right ) \left (x +i\right ) \sqrt {a \left (x +i\right ) \left (x -i\right )}}{16 \left (x^{2}+1\right ) a^{4}}+\frac {5 \sqrt {a \left (x +i\right ) \left (x -i\right )}\, \left (x -i\right ) \left (\mathrm {arccot}\relax (x )-i\right )}{16 \left (x^{2}+1\right ) a^{4}}-\frac {5 \left (-i+3 \,\mathrm {arccot}\relax (x )\right ) \sqrt {a \left (x +i\right ) \left (x -i\right )}\, \left (x^{3}-3 i x^{2}-3 x +i\right )}{288 \left (x^{4}+2 x^{2}+1\right ) a^{4}}-\frac {\sqrt {a \left (x +i\right ) \left (x -i\right )}\, \left (x -i\right ) \left (67 i+165 \,\mathrm {arccot}\relax (x )\right ) \cos \left (4 \,\mathrm {arccot}\relax (x )\right )}{3600 \left (x^{2}+1\right ) a^{4}}-\frac {\sqrt {a \left (x +i\right ) \left (x -i\right )}\, \left (i x +1\right ) \left (29 i+105 \,\mathrm {arccot}\relax (x )\right ) \sin \left (4 \,\mathrm {arccot}\relax (x )\right )}{1800 \left (x^{2}+1\right ) a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 93, normalized size = 0.79 \[ \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {a x^{2} + a} a^{3}} + \frac {4 \, x}{{\left (a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} + \frac {3 \, x}{{\left (a x^{2} + a\right )}^{\frac {5}{2}} a}\right )} \operatorname {arccot}\relax (x) - \frac {8}{15 \, \sqrt {a x^{2} + a} a^{3}} - \frac {4}{45 \, {\left (a x^{2} + a\right )}^{\frac {3}{2}} a^{2}} - \frac {1}{25 \, {\left (a x^{2} + a\right )}^{\frac {5}{2}} a} \]
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 \[ \int \frac {\mathrm {acot}\relax (x)}{{\left (a\,x^2+a\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\operatorname {acot}{\relax (x )}}{\left (a \left (x^{2} + 1\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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