Integrand size = 20, antiderivative size = 138 \[ \int \frac {x \arctan (a x)^3}{c+a^2 c x^2} \, dx=-\frac {i \arctan (a x)^4}{4 a^2 c}-\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^2 c}-\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^2 c}+\frac {3 i \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )}{4 a^2 c} \]
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Time = 0.15 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5040, 4964, 5004, 5114, 5118, 6745} \[ \int \frac {x \arctan (a x)^3}{c+a^2 c x^2} \, dx=-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a^2 c}-\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{2 a^2 c}-\frac {i \arctan (a x)^4}{4 a^2 c}-\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}+\frac {3 i \operatorname {PolyLog}\left (4,1-\frac {2}{i a x+1}\right )}{4 a^2 c} \]
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Rule 4964
Rule 5004
Rule 5040
Rule 5114
Rule 5118
Rule 6745
Rubi steps \begin{align*} \text {integral}& = -\frac {i \arctan (a x)^4}{4 a^2 c}-\frac {\int \frac {\arctan (a x)^3}{i-a x} \, dx}{a c} \\ & = -\frac {i \arctan (a x)^4}{4 a^2 c}-\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}+\frac {3 \int \frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a c} \\ & = -\frac {i \arctan (a x)^4}{4 a^2 c}-\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^2 c}+\frac {(3 i) \int \frac {\arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a c} \\ & = -\frac {i \arctan (a x)^4}{4 a^2 c}-\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^2 c}-\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^2 c}+\frac {3 \int \frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a c} \\ & = -\frac {i \arctan (a x)^4}{4 a^2 c}-\frac {\arctan (a x)^3 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {3 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{2 a^2 c}-\frac {3 \arctan (a x) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^2 c}+\frac {3 i \operatorname {PolyLog}\left (4,1-\frac {2}{1+i a x}\right )}{4 a^2 c} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.86 \[ \int \frac {x \arctan (a x)^3}{c+a^2 c x^2} \, dx=-\frac {i \left (\arctan (a x)^4-4 i \arctan (a x)^3 \log \left (\frac {2 i}{i-a x}\right )+6 \arctan (a x)^2 \operatorname {PolyLog}\left (2,\frac {i+a x}{-i+a x}\right )-6 i \arctan (a x) \operatorname {PolyLog}\left (3,\frac {i+a x}{-i+a x}\right )-3 \operatorname {PolyLog}\left (4,\frac {i+a x}{-i+a x}\right )\right )}{4 a^2 c} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 15.79 (sec) , antiderivative size = 789, normalized size of antiderivative = 5.72
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{3}}{2 c}-\frac {3 \left (\frac {2 \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{3}-\frac {i \arctan \left (a x \right )^{4}}{6}+\frac {\left (-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 (\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}+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 )}{\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 (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}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{3}}{6}-i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\frac {i \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}\right )}{2 c}}{a^{2}}\) | \(789\) |
default | \(\frac {\frac {\ln \left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{3}}{2 c}-\frac {3 \left (\frac {2 \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{3}-\frac {i \arctan \left (a x \right )^{4}}{6}+\frac {\left (-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 (\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}+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 )}{\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 (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}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{3}}{6}-i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )+\frac {i \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2}\right )}{2 c}}{a^{2}}\) | \(789\) |
parts | \(\frac {\ln \left (a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{3}}{2 a^{2} c}-\frac {3 \left (\frac {2 \arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{3 a}-\frac {i \arctan \left (a x \right )^{4}}{6 a}+\frac {\left (-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 (\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}+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 )}{\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 (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}+4 \ln \left (2\right )\right ) \arctan \left (a x \right )^{3}}{6 a}-\frac {i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{a}+\frac {\arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{a}+\frac {i \operatorname {polylog}\left (4, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 a}\right )}{2 a c}\) | \(809\) |
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\[ \int \frac {x \arctan (a x)^3}{c+a^2 c x^2} \, dx=\int { \frac {x \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c} \,d x } \]
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\[ \int \frac {x \arctan (a x)^3}{c+a^2 c x^2} \, dx=\frac {\int \frac {x \operatorname {atan}^{3}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
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\[ \int \frac {x \arctan (a x)^3}{c+a^2 c x^2} \, dx=\int { \frac {x \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c} \,d x } \]
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\[ \int \frac {x \arctan (a x)^3}{c+a^2 c x^2} \, dx=\int { \frac {x \arctan \left (a x\right )^{3}}{a^{2} c x^{2} + c} \,d x } \]
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Timed out. \[ \int \frac {x \arctan (a x)^3}{c+a^2 c x^2} \, dx=\int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^3}{c\,a^2\,x^2+c} \,d x \]
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