Integrand size = 22, antiderivative size = 192 \[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {1}{4 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)}{2 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^2}{4 a^4 c^2}+\frac {\arctan (a x)^2}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^3}{3 a^4 c^2}-\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^4 c^2}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4 c^2}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^4 c^2} \]
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Time = 0.22 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5084, 5040, 4964, 5004, 5114, 6745, 5050, 5012, 267} \[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^4 c^2}-\frac {i \arctan (a x)^3}{3 a^4 c^2}-\frac {\arctan (a x)^2}{4 a^4 c^2}-\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^4 c^2}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{2 a^4 c^2}+\frac {\arctan (a x)^2}{2 a^4 c^2 \left (a^2 x^2+1\right )}-\frac {1}{4 a^4 c^2 \left (a^2 x^2+1\right )}-\frac {x \arctan (a x)}{2 a^3 c^2 \left (a^2 x^2+1\right )} \]
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Rule 267
Rule 4964
Rule 5004
Rule 5012
Rule 5040
Rule 5050
Rule 5084
Rule 5114
Rule 6745
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {x \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx}{a^2}+\frac {\int \frac {x \arctan (a x)^2}{c+a^2 c x^2} \, dx}{a^2 c} \\ & = \frac {\arctan (a x)^2}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^3}{3 a^4 c^2}-\frac {\int \frac {\arctan (a x)}{\left (c+a^2 c x^2\right )^2} \, dx}{a^3}-\frac {\int \frac {\arctan (a x)^2}{i-a x} \, dx}{a^3 c^2} \\ & = -\frac {x \arctan (a x)}{2 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^2}{4 a^4 c^2}+\frac {\arctan (a x)^2}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^3}{3 a^4 c^2}-\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^4 c^2}+\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^2} \, dx}{2 a^2}+\frac {2 \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3 c^2} \\ & = -\frac {1}{4 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)}{2 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^2}{4 a^4 c^2}+\frac {\arctan (a x)^2}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^3}{3 a^4 c^2}-\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^4 c^2}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4 c^2}+\frac {i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3 c^2} \\ & = -\frac {1}{4 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)}{2 a^3 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)^2}{4 a^4 c^2}+\frac {\arctan (a x)^2}{2 a^4 c^2 \left (1+a^2 x^2\right )}-\frac {i \arctan (a x)^3}{3 a^4 c^2}-\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^4 c^2}-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^4 c^2}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^4 c^2} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.61 \[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\frac {1}{3} i \arctan (a x)^3+\frac {1}{8} \left (-1+2 \arctan (a x)^2\right ) \cos (2 \arctan (a x))-\arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )-\frac {1}{4} \arctan (a x) \sin (2 \arctan (a x))}{a^4 c^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 9.50 (sec) , antiderivative size = 855, normalized size of antiderivative = 4.45
method | result | size |
derivativedivides | \(\frac {\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c^{2}}+\frac {\arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {i \arctan \left (a x \right )^{3}}{3}-\frac {i \arctan \left (a x \right ) \left (a x +i\right )}{8 a x -8 i}-\frac {a x +i}{16 \left (a x -i\right )}+\frac {i \arctan \left (a x \right ) \left (a x -i\right )}{8 a x +8 i}-\frac {a x -i}{16 \left (a x +i\right )}-i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {\left (i \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-2 i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )-i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}-i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}+i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}-4 \ln \left (2\right )-1\right ) \arctan \left (a x \right )^{2}}{4}}{c^{2}}}{a^{4}}\) | \(855\) |
default | \(\frac {\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c^{2}}+\frac {\arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\frac {i \arctan \left (a x \right )^{3}}{3}-\frac {i \arctan \left (a x \right ) \left (a x +i\right )}{8 a x -8 i}-\frac {a x +i}{16 \left (a x -i\right )}+\frac {i \arctan \left (a x \right ) \left (a x -i\right )}{8 a x +8 i}-\frac {a x -i}{16 \left (a x +i\right )}-i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}-\frac {\left (i \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-2 i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )-i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}-i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}+i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}-4 \ln \left (2\right )-1\right ) \arctan \left (a x \right )^{2}}{4}}{c^{2}}}{a^{4}}\) | \(855\) |
parts | \(\frac {\arctan \left (a x \right )^{2}}{2 a^{4} c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2 c^{2} a^{4}}-\frac {a \left (\frac {\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{5}}-\frac {i \arctan \left (a x \right )^{3}}{3 a^{5}}-\frac {i \arctan \left (a x \right ) \left (a x +i\right )}{8 a^{5} \left (a x -i\right )}-\frac {a x +i}{16 a^{5} \left (a x -i\right )}+\frac {i \arctan \left (a x \right ) \left (a x -i\right )}{8 a^{5} \left (a x +i\right )}-\frac {a x -i}{16 a^{5} \left (a x +i\right )}-\frac {i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{a^{5}}+\frac {\operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 a^{5}}-\frac {\left (i \pi {\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )}^{2} \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )-2 i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )-i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )}^{2} \operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{2}-i \pi {\operatorname {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )}^{3}+i \pi \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3}-i \pi \,\operatorname {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2}-4 \ln \left (2\right )-1\right ) \arctan \left (a x \right )^{2}}{4 a^{5}}\right )}{c^{2}}\) | \(883\) |
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\[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{a^{4} x^{4} + 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
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\[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
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\[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]
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