Integrand size = 22, antiderivative size = 336 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=-\frac {a^2 c^3 \arctan (a x)}{x}+a^4 c^3 x \arctan (a x)-a^3 c^3 \arctan (a x)^2-\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+a^3 c^3 \log (x)+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-a^3 c^3 \log \left (1+a^2 x^2\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \]
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Time = 0.83 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {5068, 4930, 5040, 4964, 5004, 5114, 6745, 4946, 5038, 272, 36, 29, 31, 5044, 4988, 5112, 5036, 266} \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2+3 a^4 c^3 x \arctan (a x)^3+a^4 c^3 x \arctan (a x)-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )-a^3 c^3 \arctan (a x)^2+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )+a^3 c^3 \log (x)-\frac {3 a^2 c^3 \arctan (a x)^3}{x}-\frac {a^2 c^3 \arctan (a x)}{x}-a^3 c^3 \log \left (a^2 x^2+1\right )-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {a c^3 \arctan (a x)^2}{2 x^2} \]
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 272
Rule 4930
Rule 4946
Rule 4964
Rule 4988
Rule 5004
Rule 5036
Rule 5038
Rule 5040
Rule 5044
Rule 5068
Rule 5112
Rule 5114
Rule 6745
Rubi steps \begin{align*} \text {integral}& = \int \left (3 a^4 c^3 \arctan (a x)^3+\frac {c^3 \arctan (a x)^3}{x^4}+\frac {3 a^2 c^3 \arctan (a x)^3}{x^2}+a^6 c^3 x^2 \arctan (a x)^3\right ) \, dx \\ & = c^3 \int \frac {\arctan (a x)^3}{x^4} \, dx+\left (3 a^2 c^3\right ) \int \frac {\arctan (a x)^3}{x^2} \, dx+\left (3 a^4 c^3\right ) \int \arctan (a x)^3 \, dx+\left (a^6 c^3\right ) \int x^2 \arctan (a x)^3 \, dx \\ & = -\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+\left (a c^3\right ) \int \frac {\arctan (a x)^2}{x^3 \left (1+a^2 x^2\right )} \, dx+\left (9 a^3 c^3\right ) \int \frac {\arctan (a x)^2}{x \left (1+a^2 x^2\right )} \, dx-\left (9 a^5 c^3\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx-\left (a^7 c^3\right ) \int \frac {x^3 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+\left (a c^3\right ) \int \frac {\arctan (a x)^2}{x^3} \, dx+\left (9 i a^3 c^3\right ) \int \frac {\arctan (a x)^2}{x (i+a x)} \, dx-\left (a^3 c^3\right ) \int \frac {\arctan (a x)^2}{x \left (1+a^2 x^2\right )} \, dx+\left (9 a^4 c^3\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx-\left (a^5 c^3\right ) \int x \arctan (a x)^2 \, dx+\left (a^5 c^3\right ) \int \frac {x \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+9 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+9 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )+\left (a^2 c^3\right ) \int \frac {\arctan (a x)}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (i a^3 c^3\right ) \int \frac {\arctan (a x)^2}{x (i+a x)} \, dx-\left (a^4 c^3\right ) \int \frac {\arctan (a x)^2}{i-a x} \, dx-\left (18 a^4 c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (18 a^4 c^3\right ) \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx+\left (a^6 c^3\right ) \int \frac {x^2 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-9 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+9 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\left (a^2 c^3\right ) \int \frac {\arctan (a x)}{x^2} \, dx+\left (9 i a^4 c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (9 i a^4 c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (a^4 c^3\right ) \int \arctan (a x) \, dx-2 \left (\left (a^4 c^3\right ) \int \frac {\arctan (a x)}{1+a^2 x^2} \, dx\right )+\left (2 a^4 c^3\right ) \int \frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (2 a^4 c^3\right ) \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx \\ & = -\frac {a^2 c^3 \arctan (a x)}{x}+a^4 c^3 x \arctan (a x)-a^3 c^3 \arctan (a x)^2-\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {9}{2} a^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {9}{2} a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\left (a^3 c^3\right ) \int \frac {1}{x \left (1+a^2 x^2\right )} \, dx-\left (i a^4 c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx+\left (i a^4 c^3\right ) \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (a^5 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx \\ & = -\frac {a^2 c^3 \arctan (a x)}{x}+a^4 c^3 x \arctan (a x)-a^3 c^3 \arctan (a x)^2-\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-\frac {1}{2} a^3 c^3 \log \left (1+a^2 x^2\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {a^2 c^3 \arctan (a x)}{x}+a^4 c^3 x \arctan (a x)-a^3 c^3 \arctan (a x)^2-\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-\frac {1}{2} a^3 c^3 \log \left (1+a^2 x^2\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} \left (a^3 c^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{2} \left (a^5 c^3\right ) \text {Subst}\left (\int \frac {1}{1+a^2 x} \, dx,x,x^2\right ) \\ & = -\frac {a^2 c^3 \arctan (a x)}{x}+a^4 c^3 x \arctan (a x)-a^3 c^3 \arctan (a x)^2-\frac {a c^3 \arctan (a x)^2}{2 x^2}-\frac {1}{2} a^5 c^3 x^2 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{3 x^3}-\frac {3 a^2 c^3 \arctan (a x)^3}{x}+3 a^4 c^3 x \arctan (a x)^3+\frac {1}{3} a^6 c^3 x^3 \arctan (a x)^3+a^3 c^3 \log (x)+8 a^3 c^3 \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )-a^3 c^3 \log \left (1+a^2 x^2\right )+8 a^3 c^3 \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+8 i a^3 c^3 \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+4 a^3 c^3 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\frac {c^3 \left (-2 i a^3 \pi ^3 x^3-6 a^2 x^2 \arctan (a x)+6 a^4 x^4 \arctan (a x)-3 a x \arctan (a x)^2-6 a^3 x^3 \arctan (a x)^2-3 a^5 x^5 \arctan (a x)^2-2 \arctan (a x)^3-18 a^2 x^2 \arctan (a x)^3+18 a^4 x^4 \arctan (a x)^3+2 a^6 x^6 \arctan (a x)^3+48 a^3 x^3 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+48 a^3 x^3 \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )+6 a^3 x^3 \log (a x)-6 a^3 x^3 \log \left (1+a^2 x^2\right )+48 i a^3 x^3 \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )-48 i a^3 x^3 \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+24 a^3 x^3 \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )+24 a^3 x^3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right )\right )}{6 x^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 138.35 (sec) , antiderivative size = 1948, normalized size of antiderivative = 5.80
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1948\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1949\) |
default | \(\text {Expression too large to display}\) | \(1949\) |
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\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=c^{3} \left (\int 3 a^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{4}}\, dx + \int \frac {3 a^{2} \operatorname {atan}^{3}{\left (a x \right )}}{x^{2}}\, dx + \int a^{6} x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^3}{x^4} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3}{x^4} \,d x \]
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