Integrand size = 22, antiderivative size = 158 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {a}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{c^3 x}-\frac {a \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{c^{5/2}} \]
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Time = 0.25 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {5086, 5064, 272, 65, 214, 5014, 5016} \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c^3 x}-\frac {5 a^2 x \arctan (a x)}{3 c^2 \sqrt {a^2 c x^2+c}}-\frac {a^2 x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{c^{5/2}}-\frac {5 a}{3 c^2 \sqrt {a^2 c x^2+c}}-\frac {a}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rule 65
Rule 214
Rule 272
Rule 5014
Rule 5016
Rule 5064
Rule 5086
Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\right )+\frac {\int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx}{c} \\ & = -\frac {a}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {a^2 x \arctan (a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\int \frac {\arctan (a x)}{x^2 \sqrt {c+a^2 c x^2}} \, dx}{c^2}-\frac {\left (2 a^2\right ) \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac {a^2 \int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c} \\ & = -\frac {a}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{c^3 x}+\frac {a \int \frac {1}{x \sqrt {c+a^2 c x^2}} \, dx}{c^2} \\ & = -\frac {a}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{c^3 x}+\frac {a \text {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )}{2 c^2} \\ & = -\frac {a}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{c^3 x}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c+a^2 c x^2}\right )}{a c^3} \\ & = -\frac {a}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{c^3 x}-\frac {a \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{c^{5/2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.96 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {-3 \sqrt {c+a^2 c x^2} \left (3+12 a^2 x^2+8 a^4 x^4\right ) \arctan (a x)+a x \left (-\left (\left (16+15 a^2 x^2\right ) \sqrt {c+a^2 c x^2}\right )+9 \sqrt {c} \left (1+a^2 x^2\right )^2 \log (x)-9 \sqrt {c} \left (1+a^2 x^2\right )^2 \log \left (c+\sqrt {c} \sqrt {c+a^2 c x^2}\right )\right )}{9 c^3 x \left (1+a^2 x^2\right )^2} \]
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Result contains complex when optimal does not.
Time = 0.47 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.06
method | result | size |
default | \(-\frac {\left (9 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) a^{5} x^{5}-9 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right ) a^{5} x^{5}+24 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{4} x^{4}+15 \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+18 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) a^{3} x^{3}-18 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right ) a^{3} x^{3}+36 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+16 \sqrt {a^{2} x^{2}+1}\, a x +9 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) a x -9 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right ) a x +9 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{9 \sqrt {a^{2} x^{2}+1}\, x \,c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\) | \(325\) |
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Time = 0.29 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.90 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {9 \, {\left (a^{5} x^{5} + 2 \, a^{3} x^{3} + a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {a^{2} c x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 2 \, {\left (15 \, a^{3} x^{3} + 16 \, a x + 3 \, {\left (8 \, a^{4} x^{4} + 12 \, a^{2} x^{2} + 3\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{18 \, {\left (a^{4} c^{3} x^{5} + 2 \, a^{2} c^{3} x^{3} + c^{3} x\right )}} \]
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\[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {atan}{\left (a x \right )}}{x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
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\[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )}{x^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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