Optimal. Leaf size=287 \[ \frac {29}{180} a^2 c^3 x^2+\frac {1}{60} a^4 c^3 x^4-\frac {11}{6} a c^3 x \text {ArcTan}(a x)-\frac {7}{18} a^3 c^3 x^3 \text {ArcTan}(a x)-\frac {1}{15} a^5 c^3 x^5 \text {ArcTan}(a x)+\frac {11}{12} c^3 \text {ArcTan}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \text {ArcTan}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \text {ArcTan}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \text {ArcTan}(a x)^2+2 c^3 \text {ArcTan}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {34}{45} c^3 \log \left (1+a^2 x^2\right )-i c^3 \text {ArcTan}(a x) \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i c^3 \text {ArcTan}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \text {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.54, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps
used = 38, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {5068, 4942,
5108, 5004, 5114, 6745, 4946, 5036, 4930, 266, 272, 45} \begin {gather*} \frac {1}{6} a^6 c^3 x^6 \text {ArcTan}(a x)^2-\frac {1}{15} a^5 c^3 x^5 \text {ArcTan}(a x)+\frac {3}{4} a^4 c^3 x^4 \text {ArcTan}(a x)^2+\frac {1}{60} a^4 c^3 x^4-\frac {7}{18} a^3 c^3 x^3 \text {ArcTan}(a x)+\frac {3}{2} a^2 c^3 x^2 \text {ArcTan}(a x)^2+\frac {29}{180} a^2 c^3 x^2+\frac {34}{45} c^3 \log \left (a^2 x^2+1\right )-i c^3 \text {ArcTan}(a x) \text {Li}_2\left (1-\frac {2}{i a x+1}\right )+i c^3 \text {ArcTan}(a x) \text {Li}_2\left (\frac {2}{i a x+1}-1\right )-\frac {11}{6} a c^3 x \text {ArcTan}(a x)+\frac {11}{12} c^3 \text {ArcTan}(a x)^2+2 c^3 \text {ArcTan}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \text {Li}_3\left (1-\frac {2}{i a x+1}\right )+\frac {1}{2} c^3 \text {Li}_3\left (\frac {2}{i a x+1}-1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 266
Rule 272
Rule 4930
Rule 4942
Rule 4946
Rule 5004
Rule 5036
Rule 5068
Rule 5108
Rule 5114
Rule 6745
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2}{x} \, dx &=\int \left (\frac {c^3 \tan ^{-1}(a x)^2}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^2+3 a^4 c^3 x^3 \tan ^{-1}(a x)^2+a^6 c^3 x^5 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^3 \int \frac {\tan ^{-1}(a x)^2}{x} \, dx+\left (3 a^2 c^3\right ) \int x \tan ^{-1}(a x)^2 \, dx+\left (3 a^4 c^3\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx+\left (a^6 c^3\right ) \int x^5 \tan ^{-1}(a x)^2 \, dx\\ &=\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\left (4 a c^3\right ) \int \frac {\tan ^{-1}(a x) \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^5 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^7 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\left (2 a c^3\right ) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (2 a c^3\right ) \int \frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a c^3\right ) \int \tan ^{-1}(a x) \, dx+\left (3 a c^3\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^3 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx+\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^5 c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx+\frac {1}{3} \left (a^5 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-3 a c^3 x \tan ^{-1}(a x)-\frac {1}{2} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac {1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac {3}{2} c^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-i c^3 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )+\left (i a c^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (i a c^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{2} \left (3 a c^3\right ) \int \tan ^{-1}(a x) \, dx-\frac {1}{2} \left (3 a c^3\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (3 a^2 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx+\frac {1}{3} \left (a^3 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx-\frac {1}{3} \left (a^3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{2} \left (a^4 c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{15} \left (a^6 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx\\ &=-\frac {3}{2} a c^3 x \tan ^{-1}(a x)-\frac {7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac {1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac {3}{4} c^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \log \left (1+a^2 x^2\right )-i c^3 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{3} \left (a c^3\right ) \int \tan ^{-1}(a x) \, dx+\frac {1}{3} \left (a c^3\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^2 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{9} \left (a^4 c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{4} \left (a^4 c^3\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{30} \left (a^6 c^3\right ) \text {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {11}{6} a c^3 x \tan ^{-1}(a x)-\frac {7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac {1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac {11}{12} c^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {3}{4} c^3 \log \left (1+a^2 x^2\right )-i c^3 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{3} \left (a^2 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{18} \left (a^4 c^3\right ) \text {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{4} \left (a^4 c^3\right ) \text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {1}{30} \left (a^6 c^3\right ) \text {Subst}\left (\int \left (-\frac {1}{a^4}+\frac {x}{a^2}+\frac {1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {13}{60} a^2 c^3 x^2+\frac {1}{60} a^4 c^3 x^4-\frac {11}{6} a c^3 x \tan ^{-1}(a x)-\frac {7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac {1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac {11}{12} c^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {7}{10} c^3 \log \left (1+a^2 x^2\right )-i c^3 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{18} \left (a^4 c^3\right ) \text {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {29}{180} a^2 c^3 x^2+\frac {1}{60} a^4 c^3 x^4-\frac {11}{6} a c^3 x \tan ^{-1}(a x)-\frac {7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac {1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac {11}{12} c^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {34}{45} c^3 \log \left (1+a^2 x^2\right )-i c^3 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.38, size = 252, normalized size = 0.88 \begin {gather*} \frac {1}{360} c^3 \left (52-15 i \pi ^3+58 a^2 x^2+6 a^4 x^4-660 a x \text {ArcTan}(a x)-140 a^3 x^3 \text {ArcTan}(a x)-24 a^5 x^5 \text {ArcTan}(a x)+330 \text {ArcTan}(a x)^2+540 a^2 x^2 \text {ArcTan}(a x)^2+270 a^4 x^4 \text {ArcTan}(a x)^2+60 a^6 x^6 \text {ArcTan}(a x)^2+240 i \text {ArcTan}(a x)^3+360 \text {ArcTan}(a x)^2 \log \left (1-e^{-2 i \text {ArcTan}(a x)}\right )-360 \text {ArcTan}(a x)^2 \log \left (1+e^{2 i \text {ArcTan}(a x)}\right )+272 \log \left (1+a^2 x^2\right )+360 i \text {ArcTan}(a x) \text {PolyLog}\left (2,e^{-2 i \text {ArcTan}(a x)}\right )+360 i \text {ArcTan}(a x) \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(a x)}\right )+180 \text {PolyLog}\left (3,e^{-2 i \text {ArcTan}(a x)}\right )-180 \text {PolyLog}\left (3,-e^{2 i \text {ArcTan}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 12.04, size = 1405, normalized size = 4.90
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1405\) |
default | \(\text {Expression too large to display}\) | \(1405\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c^{3} \left (\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x}\, dx + \int 3 a^{2} x \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{4} x^{3} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{5} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________