Integrand size = 24, antiderivative size = 381 \[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {2 a^2 x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a^2 x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 a \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {10 a \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{c^3 x}-\frac {4 a \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.48 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5086, 5064, 5078, 5074, 5018, 197, 5020, 198} \[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {4 a \sqrt {a^2 x^2+1} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c^3 x}-\frac {5 a^2 x \arctan (a x)^2}{3 c^2 \sqrt {a^2 c x^2+c}}-\frac {10 a \arctan (a x)}{3 c^2 \sqrt {a^2 c x^2+c}}-\frac {a^2 x \arctan (a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 a \arctan (a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 i a \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i a \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{c^2 \sqrt {a^2 c x^2+c}}+\frac {94 a^2 x}{27 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 a^2 x}{27 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rule 197
Rule 198
Rule 5018
Rule 5020
Rule 5064
Rule 5074
Rule 5078
Rule 5086
Rubi steps \begin{align*} \text {integral}& = -\left (a^2 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\right )+\frac {\int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx}{c} \\ & = -\frac {2 a \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {a^2 x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {1}{9} \left (2 a^2\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac {\int \frac {\arctan (a x)^2}{x^2 \sqrt {c+a^2 c x^2}} \, dx}{c^2}-\frac {\left (2 a^2\right ) \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac {a^2 \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c} \\ & = \frac {2 a^2 x}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 a \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {10 a \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{c^3 x}+\frac {(2 a) \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx}{c^2}+\frac {\left (4 a^2\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{27 c}+\frac {\left (4 a^2\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}+\frac {\left (2 a^2\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c} \\ & = \frac {2 a^2 x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a^2 x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 a \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {10 a \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{c^3 x}+\frac {\left (2 a \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{x \sqrt {1+a^2 x^2}} \, dx}{c^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {2 a^2 x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {94 a^2 x}{27 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 a \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {10 a \arctan (a x)}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {a^2 x \arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {5 a^2 x \arctan (a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{c^3 x}-\frac {4 a \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i a \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 1.42 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.78 \[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {a \left (-378 a x+378 \arctan (a x)+189 a x \arctan (a x)^2+6 \sqrt {1+a^2 x^2} \arctan (a x) \cos (3 \arctan (a x))+27 a x \arctan (a x)^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-216 \sqrt {1+a^2 x^2} \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )+216 \sqrt {1+a^2 x^2} \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )-216 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+216 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 \sqrt {1+a^2 x^2} \sin (3 \arctan (a x))+9 \sqrt {1+a^2 x^2} \arctan (a x)^2 \sin (3 \arctan (a x))+54 \sqrt {1+a^2 x^2} \arctan (a x)^2 \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{108 c^2 \sqrt {c+a^2 c x^2}} \]
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Time = 1.22 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.52
method | result | size |
default | \(-\frac {\left (54 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) a^{5} x^{5}-54 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a^{5} x^{5}+108 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a^{3} x^{3}-54 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a x +72 \arctan \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{4} x^{4}-94 \sqrt {a^{2} x^{2}+1}\, a^{4} x^{4}+90 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+108 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) a^{3} x^{3}-108 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a^{3} x^{3}+54 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a^{5} x^{5}-108 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a^{3} x^{3}+108 \arctan \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}-96 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+96 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +54 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) a x -54 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a x -54 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a^{5} x^{5}+54 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a x +27 \arctan \left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{27 \sqrt {a^{2} x^{2}+1}\, x \,c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\) | \(579\) |
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\[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
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\[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
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\[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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