Integrand size = 20, antiderivative size = 308 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=-\frac {19 c^3 x}{140 a}-\frac {19}{840} a c^3 x^3-\frac {1}{280} a^3 c^3 x^5+\frac {3 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a^2}+\frac {9 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{280 a^2}+\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{56 a^2}-\frac {6 i c^3 \arctan (a x)^2}{35 a^2}-\frac {6 c^3 x \arctan (a x)^2}{35 a}-\frac {3 c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2}{35 a}-\frac {9 c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2}{140 a}-\frac {3 c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{56 a}+\frac {c^3 \left (1+a^2 x^2\right )^4 \arctan (a x)^3}{8 a^2}-\frac {12 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{35 a^2}-\frac {6 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a^2} \] Output:
-19/140*c^3*x/a-19/840*a*c^3*x^3-1/280*a^3*c^3*x^5+3/35*c^3*(a^2*x^2+1)*ar ctan(a*x)/a^2+9/280*c^3*(a^2*x^2+1)^2*arctan(a*x)/a^2+1/56*c^3*(a^2*x^2+1) ^3*arctan(a*x)/a^2-6/35*I*c^3*arctan(a*x)^2/a^2-6/35*c^3*x*arctan(a*x)^2/a -3/35*c^3*x*(a^2*x^2+1)*arctan(a*x)^2/a-9/140*c^3*x*(a^2*x^2+1)^2*arctan(a *x)^2/a-3/56*c^3*x*(a^2*x^2+1)^3*arctan(a*x)^2/a+1/8*c^3*(a^2*x^2+1)^4*arc tan(a*x)^3/a^2-12/35*c^3*arctan(a*x)*ln(2/(1+I*a*x))/a^2-6/35*I*c^3*polylo g(2,1-2/(1+I*a*x))/a^2
Time = 0.99 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.51 \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {c^3 \left (-a x \left (114+19 a^2 x^2+3 a^4 x^4\right )-9 \left (-16 i+35 a x+35 a^3 x^3+21 a^5 x^5+5 a^7 x^7\right ) \arctan (a x)^2+105 \left (1+a^2 x^2\right )^4 \arctan (a x)^3+3 \arctan (a x) \left (38+57 a^2 x^2+24 a^4 x^4+5 a^6 x^6-96 \log \left (1+e^{2 i \arctan (a x)}\right )\right )+144 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{840 a^2} \] Input:
Integrate[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^3,x]
Output:
(c^3*(-(a*x*(114 + 19*a^2*x^2 + 3*a^4*x^4)) - 9*(-16*I + 35*a*x + 35*a^3*x ^3 + 21*a^5*x^5 + 5*a^7*x^7)*ArcTan[a*x]^2 + 105*(1 + a^2*x^2)^4*ArcTan[a* x]^3 + 3*ArcTan[a*x]*(38 + 57*a^2*x^2 + 24*a^4*x^4 + 5*a^6*x^6 - 96*Log[1 + E^((2*I)*ArcTan[a*x])]) + (144*I)*PolyLog[2, -E^((2*I)*ArcTan[a*x])]))/( 840*a^2)
Time = 1.31 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5465, 27, 5415, 210, 2009, 5415, 2009, 5415, 24, 5345, 5455, 5379, 2849, 2752}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x \arctan (a x)^3 \left (a^2 c x^2+c\right )^3 \, dx\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 \int c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)^2dx}{8 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \int \left (a^2 x^2+1\right )^3 \arctan (a x)^2dx}{8 a}\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \left (\frac {6}{7} \int \left (a^2 x^2+1\right )^2 \arctan (a x)^2dx+\frac {1}{21} \int \left (a^2 x^2+1\right )^2dx+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}\right )}{8 a}\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \left (\frac {6}{7} \int \left (a^2 x^2+1\right )^2 \arctan (a x)^2dx+\frac {1}{21} \int \left (a^4 x^4+2 a^2 x^2+1\right )dx+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}\right )}{8 a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \left (\frac {6}{7} \int \left (a^2 x^2+1\right )^2 \arctan (a x)^2dx+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} \left (\frac {a^4 x^5}{5}+\frac {2 a^2 x^3}{3}+x\right )\right )}{8 a}\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \left (\frac {6}{7} \left (\frac {4}{5} \int \left (a^2 x^2+1\right ) \arctan (a x)^2dx+\frac {1}{10} \int \left (a^2 x^2+1\right )dx+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}\right )+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} \left (\frac {a^4 x^5}{5}+\frac {2 a^2 x^3}{3}+x\right )\right )}{8 a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \left (\frac {6}{7} \left (\frac {4}{5} \int \left (a^2 x^2+1\right ) \arctan (a x)^2dx+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} \left (\frac {a^4 x^5}{5}+\frac {2 a^2 x^3}{3}+x\right )\right )}{8 a}\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \arctan (a x)^2dx+\frac {\int 1dx}{3}+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} \left (\frac {a^4 x^5}{5}+\frac {2 a^2 x^3}{3}+x\right )\right )}{8 a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \arctan (a x)^2dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {x}{3}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} \left (\frac {a^4 x^5}{5}+\frac {2 a^2 x^3}{3}+x\right )\right )}{8 a}\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {x}{3}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} \left (\frac {a^4 x^5}{5}+\frac {2 a^2 x^3}{3}+x\right )\right )}{8 a}\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^2-2 a \left (-\frac {\int \frac {\arctan (a x)}{i-a x}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {x}{3}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} \left (\frac {a^4 x^5}{5}+\frac {2 a^2 x^3}{3}+x\right )\right )}{8 a}\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^2-2 a \left (-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}-\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {x}{3}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} \left (\frac {a^4 x^5}{5}+\frac {2 a^2 x^3}{3}+x\right )\right )}{8 a}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^2-2 a \left (-\frac {\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {x}{3}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} \left (\frac {a^4 x^5}{5}+\frac {2 a^2 x^3}{3}+x\right )\right )}{8 a}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {c^3 \left (a^2 x^2+1\right )^4 \arctan (a x)^3}{8 a^2}-\frac {3 c^3 \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^2-2 a \left (-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {x}{3}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} \left (\frac {a^4 x^5}{5}+\frac {2 a^2 x^3}{3}+x\right )\right )}{8 a}\) |
Input:
Int[x*(c + a^2*c*x^2)^3*ArcTan[a*x]^3,x]
Output:
(c^3*(1 + a^2*x^2)^4*ArcTan[a*x]^3)/(8*a^2) - (3*c^3*((x + (2*a^2*x^3)/3 + (a^4*x^5)/5)/21 - ((1 + a^2*x^2)^3*ArcTan[a*x])/(21*a) + (x*(1 + a^2*x^2) ^3*ArcTan[a*x]^2)/7 + (6*((x + (a^2*x^3)/3)/10 - ((1 + a^2*x^2)^2*ArcTan[a *x])/(10*a) + (x*(1 + a^2*x^2)^2*ArcTan[a*x]^2)/5 + (4*(x/3 - ((1 + a^2*x^ 2)*ArcTan[a*x])/(3*a) + (x*(1 + a^2*x^2)*ArcTan[a*x]^2)/3 + (2*(x*ArcTan[a *x]^2 - 2*a*(((-1/2*I)*ArcTan[a*x]^2)/a^2 - ((ArcTan[a*x]*Log[2/(1 + I*a*x )])/a + ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/a)/a)))/3))/5))/7))/(8*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( p/e) Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x^2)) , x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0 ]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^q*((a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2 *q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTan[c*x])^p/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1))) Int[(d + e*x^2)^(q - 1)*( a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si mp[1/(c*d) Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Time = 7.21 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.12
| method | result | size |
| parts | \(\frac {c^{3} \arctan \left (a x \right )^{3} a^{6} x^{8}}{8}+\frac {c^{3} \arctan \left (a x \right )^{3} a^{4} x^{6}}{2}+\frac {3 c^{3} \arctan \left (a x \right )^{3} a^{2} x^{4}}{4}+\frac {c^{3} \arctan \left (a x \right )^{3} x^{2}}{2}+\frac {c^{3} \arctan \left (a x \right )^{3}}{8 a^{2}}-\frac {3 c^{3} \left (\frac {\arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+a^{3} \arctan \left (a x \right )^{2} x^{3}+a \arctan \left (a x \right )^{2} x -\frac {a^{6} \arctan \left (a x \right ) x^{6}}{21}-\frac {8 x^{4} \arctan \left (a x \right ) a^{4}}{35}-\frac {19 x^{2} a^{2} \arctan \left (a x \right )}{35}-\frac {16 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{35}+\frac {a^{5} x^{5}}{105}+\frac {19 a^{3} x^{3}}{315}+\frac {38 a x}{105}-\frac {38 \arctan \left (a x \right )}{105}-\frac {8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{35}+\frac {8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{35}\right )}{8 a^{2}}\) | \(346\) |
| derivativedivides | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{3} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{3}}{2}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )^{3}}{4}+\frac {a^{2} c^{3} x^{2} \arctan \left (a x \right )^{3}}{2}+\frac {c^{3} \arctan \left (a x \right )^{3}}{8}-\frac {3 c^{3} \left (\frac {\arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+a^{3} \arctan \left (a x \right )^{2} x^{3}+a \arctan \left (a x \right )^{2} x -\frac {a^{6} \arctan \left (a x \right ) x^{6}}{21}-\frac {8 x^{4} \arctan \left (a x \right ) a^{4}}{35}-\frac {19 x^{2} a^{2} \arctan \left (a x \right )}{35}-\frac {16 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{35}+\frac {a^{5} x^{5}}{105}+\frac {19 a^{3} x^{3}}{315}+\frac {38 a x}{105}-\frac {38 \arctan \left (a x \right )}{105}-\frac {8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{35}+\frac {8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{35}\right )}{8}}{a^{2}}\) | \(347\) |
| default | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{3} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{3}}{2}+\frac {3 a^{4} c^{3} x^{4} \arctan \left (a x \right )^{3}}{4}+\frac {a^{2} c^{3} x^{2} \arctan \left (a x \right )^{3}}{2}+\frac {c^{3} \arctan \left (a x \right )^{3}}{8}-\frac {3 c^{3} \left (\frac {\arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+a^{3} \arctan \left (a x \right )^{2} x^{3}+a \arctan \left (a x \right )^{2} x -\frac {a^{6} \arctan \left (a x \right ) x^{6}}{21}-\frac {8 x^{4} \arctan \left (a x \right ) a^{4}}{35}-\frac {19 x^{2} a^{2} \arctan \left (a x \right )}{35}-\frac {16 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{35}+\frac {a^{5} x^{5}}{105}+\frac {19 a^{3} x^{3}}{315}+\frac {38 a x}{105}-\frac {38 \arctan \left (a x \right )}{105}-\frac {8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{35}+\frac {8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{35}\right )}{8}}{a^{2}}\) | \(347\) |
Input:
int(x*(a^2*c*x^2+c)^3*arctan(a*x)^3,x,method=_RETURNVERBOSE)
Output:
1/8*c^3*arctan(a*x)^3*a^6*x^8+1/2*c^3*arctan(a*x)^3*a^4*x^6+3/4*c^3*arctan (a*x)^3*a^2*x^4+1/2*c^3*arctan(a*x)^3*x^2+1/8*c^3*arctan(a*x)^3/a^2-3/8/a^ 2*c^3*(1/7*arctan(a*x)^2*a^7*x^7+3/5*a^5*arctan(a*x)^2*x^5+a^3*arctan(a*x) ^2*x^3+a*arctan(a*x)^2*x-1/21*a^6*arctan(a*x)*x^6-8/35*x^4*arctan(a*x)*a^4 -19/35*x^2*a^2*arctan(a*x)-16/35*arctan(a*x)*ln(a^2*x^2+1)+1/105*a^5*x^5+1 9/315*a^3*x^3+38/105*a*x-38/105*arctan(a*x)-8/35*I*(ln(a*x-I)*ln(a^2*x^2+1 )-1/2*ln(a*x-I)^2-dilog(-1/2*I*(a*x+I))-ln(a*x-I)*ln(-1/2*I*(a*x+I)))+8/35 *I*(ln(a*x+I)*ln(a^2*x^2+1)-1/2*ln(a*x+I)^2-dilog(1/2*I*(a*x-I))-ln(a*x+I) *ln(1/2*I*(a*x-I))))
\[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right )^{3} \,d x } \] Input:
integrate(x*(a^2*c*x^2+c)^3*arctan(a*x)^3,x, algorithm="fricas")
Output:
integral((a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x)*arctan(a*x) ^3, x)
\[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=c^{3} \left (\int x \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{2} x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{7} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \] Input:
integrate(x*(a**2*c*x**2+c)**3*atan(a*x)**3,x)
Output:
c**3*(Integral(x*atan(a*x)**3, x) + Integral(3*a**2*x**3*atan(a*x)**3, x) + Integral(3*a**4*x**5*atan(a*x)**3, x) + Integral(a**6*x**7*atan(a*x)**3, x))
\[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right )^{3} \,d x } \] Input:
integrate(x*(a^2*c*x^2+c)^3*arctan(a*x)^3,x, algorithm="maxima")
Output:
1/4480*(280*(a^8*c^3*x^8 + 4*a^6*c^3*x^6 + 6*a^4*c^3*x^4 + 4*a^2*c^3*x^2 + c^3)*arctan(a*x)^3 + 140*(71680*a^9*c^3*integrate(1/4480*x^9*arctan(a*x)^ 3/(a^3*x^2 + a), x) - 13440*a^8*c^3*integrate(1/4480*x^8*arctan(a*x)^2/(a^ 3*x^2 + a), x) - 3360*a^8*c^3*integrate(1/4480*x^8*log(a^2*x^2 + 1)^2/(a^3 *x^2 + a), x) - 1920*a^8*c^3*integrate(1/4480*x^8*log(a^2*x^2 + 1)/(a^3*x^ 2 + a), x) + 286720*a^7*c^3*integrate(1/4480*x^7*arctan(a*x)^3/(a^3*x^2 + a), x) + 3840*a^7*c^3*integrate(1/4480*x^7*arctan(a*x)/(a^3*x^2 + a), x) - 53760*a^6*c^3*integrate(1/4480*x^6*arctan(a*x)^2/(a^3*x^2 + a), x) - 1344 0*a^6*c^3*integrate(1/4480*x^6*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x) - 8064 *a^6*c^3*integrate(1/4480*x^6*log(a^2*x^2 + 1)/(a^3*x^2 + a), x) + 430080* a^5*c^3*integrate(1/4480*x^5*arctan(a*x)^3/(a^3*x^2 + a), x) + 16128*a^5*c ^3*integrate(1/4480*x^5*arctan(a*x)/(a^3*x^2 + a), x) - 80640*a^4*c^3*inte grate(1/4480*x^4*arctan(a*x)^2/(a^3*x^2 + a), x) - 20160*a^4*c^3*integrate (1/4480*x^4*log(a^2*x^2 + 1)^2/(a^3*x^2 + a), x) - 13440*a^4*c^3*integrate (1/4480*x^4*log(a^2*x^2 + 1)/(a^3*x^2 + a), x) + 286720*a^3*c^3*integrate( 1/4480*x^3*arctan(a*x)^3/(a^3*x^2 + a), x) + 26880*a^3*c^3*integrate(1/448 0*x^3*arctan(a*x)/(a^3*x^2 + a), x) - 53760*a^2*c^3*integrate(1/4480*x^2*a rctan(a*x)^2/(a^3*x^2 + a), x) - 13440*a^2*c^3*integrate(1/4480*x^2*log(a^ 2*x^2 + 1)^2/(a^3*x^2 + a), x) - 13440*a^2*c^3*integrate(1/4480*x^2*log(a^ 2*x^2 + 1)/(a^3*x^2 + a), x) + 71680*a*c^3*integrate(1/4480*x*arctan(a*...
\[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x \arctan \left (a x\right )^{3} \,d x } \] Input:
integrate(x*(a^2*c*x^2+c)^3*arctan(a*x)^3,x, algorithm="giac")
Output:
integrate((a^2*c*x^2 + c)^3*x*arctan(a*x)^3, x)
Timed out. \[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \] Input:
int(x*atan(a*x)^3*(c + a^2*c*x^2)^3,x)
Output:
int(x*atan(a*x)^3*(c + a^2*c*x^2)^3, x)
\[ \int x \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {c^{3} \left (105 \mathit {atan} \left (a x \right )^{3} a^{8} x^{8}+420 \mathit {atan} \left (a x \right )^{3} a^{6} x^{6}+630 \mathit {atan} \left (a x \right )^{3} a^{4} x^{4}+420 \mathit {atan} \left (a x \right )^{3} a^{2} x^{2}+105 \mathit {atan} \left (a x \right )^{3}-45 \mathit {atan} \left (a x \right )^{2} a^{7} x^{7}-189 \mathit {atan} \left (a x \right )^{2} a^{5} x^{5}-315 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}-315 \mathit {atan} \left (a x \right )^{2} a x +15 \mathit {atan} \left (a x \right ) a^{6} x^{6}+72 \mathit {atan} \left (a x \right ) a^{4} x^{4}+171 \mathit {atan} \left (a x \right ) a^{2} x^{2}+114 \mathit {atan} \left (a x \right )+288 \left (\int \frac {\mathit {atan} \left (a x \right ) x}{a^{2} x^{2}+1}d x \right ) a^{2}-3 a^{5} x^{5}-19 a^{3} x^{3}-114 a x \right )}{840 a^{2}} \] Input:
int(x*(a^2*c*x^2+c)^3*atan(a*x)^3,x)
Output:
(c**3*(105*atan(a*x)**3*a**8*x**8 + 420*atan(a*x)**3*a**6*x**6 + 630*atan( a*x)**3*a**4*x**4 + 420*atan(a*x)**3*a**2*x**2 + 105*atan(a*x)**3 - 45*ata n(a*x)**2*a**7*x**7 - 189*atan(a*x)**2*a**5*x**5 - 315*atan(a*x)**2*a**3*x **3 - 315*atan(a*x)**2*a*x + 15*atan(a*x)*a**6*x**6 + 72*atan(a*x)*a**4*x* *4 + 171*atan(a*x)*a**2*x**2 + 114*atan(a*x) + 288*int((atan(a*x)*x)/(a**2 *x**2 + 1),x)*a**2 - 3*a**5*x**5 - 19*a**3*x**3 - 114*a*x))/(840*a**2)