Integrand size = 21, antiderivative size = 72 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {2 x}{c \sqrt {c+a^2 c x^2}}+\frac {2 \arctan (a x)}{a c \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5018, 197} \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {x \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {2 x}{c \sqrt {a^2 c x^2+c}} \]
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Rule 197
Rule 5018
Rubi steps \begin{align*} \text {integral}& = \frac {2 \arctan (a x)}{a c \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}-2 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \\ & = -\frac {2 x}{c \sqrt {c+a^2 c x^2}}+\frac {2 \arctan (a x)}{a c \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.68 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (-2 a x+2 \arctan (a x)+a x \arctan (a x)^2\right )}{c^2 \left (a+a^3 x^2\right )} \]
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Result contains complex when optimal does not.
Time = 0.80 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.58
method | result | size |
default | \(\frac {\left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) a \,c^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{2 \left (a^{2} x^{2}+1\right ) a \,c^{2}}\) | \(114\) |
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Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.71 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a^{2} c x^{2} + c} {\left (a x \arctan \left (a x\right )^{2} - 2 \, a x + 2 \, \arctan \left (a x\right )\right )}}{a^{3} c^{2} x^{2} + a c^{2}} \]
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\[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.74 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {x \arctan \left (a x\right )^{2}}{\sqrt {a^{2} c x^{2} + c} c} - \frac {2 \, {\left (a x - \arctan \left (a x\right )\right )}}{\sqrt {a^{2} x^{2} + 1} a c^{\frac {3}{2}}} \]
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\[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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