Integrand size = 19, antiderivative size = 268 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {38 c^3 x}{105}+\frac {19}{315} a^2 c^3 x^3+\frac {1}{105} a^4 c^3 x^5-\frac {8 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}{35 a}-\frac {c^3 \left (1+a^2 x^2\right )^3 \arctan (a x)}{21 a}+\frac {16 i c^3 \arctan (a x)^2}{35 a}+\frac {16}{35} c^3 x \arctan (a x)^2+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)^2+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^2+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)^2+\frac {32 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{35 a}+\frac {16 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{35 a} \] Output:
38/105*c^3*x+19/315*a^2*c^3*x^3+1/105*a^4*c^3*x^5-8/35*c^3*(a^2*x^2+1)*arc tan(a*x)/a-3/35*c^3*(a^2*x^2+1)^2*arctan(a*x)/a-1/21*c^3*(a^2*x^2+1)^3*arc tan(a*x)/a+16/35*I*c^3*arctan(a*x)^2/a+16/35*c^3*x*arctan(a*x)^2+8/35*c^3* x*(a^2*x^2+1)*arctan(a*x)^2+6/35*c^3*x*(a^2*x^2+1)^2*arctan(a*x)^2+1/7*c^3 *x*(a^2*x^2+1)^3*arctan(a*x)^2+32/35*c^3*arctan(a*x)*ln(2/(1+I*a*x))/a+16/ 35*I*c^3*polylog(2,1-2/(1+I*a*x))/a
Time = 0.83 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.51 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {c^3 \left (a x \left (114+19 a^2 x^2+3 a^4 x^4\right )+9 \left (-16 i+35 a x+35 a^3 x^3+21 a^5 x^5+5 a^7 x^7\right ) \arctan (a x)^2-3 \arctan (a x) \left (38+57 a^2 x^2+24 a^4 x^4+5 a^6 x^6-96 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-144 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{315 a} \] Input:
Integrate[(c + a^2*c*x^2)^3*ArcTan[a*x]^2,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 - 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)*Pol yLog[2, -E^((2*I)*ArcTan[a*x])]))/(315*a)
Time = 1.23 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {5415, 27, 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.
| \(\displaystyle \int \arctan (a x)^2 \left (a^2 c x^2+c\right )^3 \, dx\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {6}{7} c \int c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^2dx+\frac {1}{21} c \int \left (a^2 c x^2+c\right )^2dx+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{7} c^3 \int \left (a^2 x^2+1\right )^2 \arctan (a x)^2dx+\frac {1}{21} c \int \left (a^2 c x^2+c\right )^2dx+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle \frac {6}{7} c^3 \int \left (a^2 x^2+1\right )^2 \arctan (a x)^2dx+\frac {1}{21} c \int \left (a^4 c^2 x^4+2 a^2 c^2 x^2+c^2\right )dx+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6}{7} c^3 \int \left (a^2 x^2+1\right )^2 \arctan (a x)^2dx+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} c \left (\frac {1}{5} a^4 c^2 x^5+\frac {2}{3} a^2 c^2 x^3+c^2 x\right )\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \int \left (a^2 x^2+1\right ) \arctan (a x)^2dx+\frac {1}{10} \int \left (a^2 x^2+1\right )dx+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} c \left (\frac {1}{5} a^4 c^2 x^5+\frac {2}{3} a^2 c^2 x^3+c^2 x\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \int \left (a^2 x^2+1\right ) \arctan (a x)^2dx+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} c \left (\frac {1}{5} a^4 c^2 x^5+\frac {2}{3} a^2 c^2 x^3+c^2 x\right )\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \int \arctan (a x)^2dx+\frac {\int 1dx}{3}+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} c \left (\frac {1}{5} a^4 c^2 x^5+\frac {2}{3} a^2 c^2 x^3+c^2 x\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \int \arctan (a x)^2dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {x}{3}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} c \left (\frac {1}{5} a^4 c^2 x^5+\frac {2}{3} a^2 c^2 x^3+c^2 x\right )\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^2-2 a \int \frac {x \arctan (a x)}{a^2 x^2+1}dx\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {x}{3}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} c \left (\frac {1}{5} a^4 c^2 x^5+\frac {2}{3} a^2 c^2 x^3+c^2 x\right )\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^2-2 a \left (-\frac {\int \frac {\arctan (a x)}{i-a x}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {x}{3}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} c \left (\frac {1}{5} a^4 c^2 x^5+\frac {2}{3} a^2 c^2 x^3+c^2 x\right )\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^2-2 a \left (-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}-\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {x}{3}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} c \left (\frac {1}{5} a^4 c^2 x^5+\frac {2}{3} a^2 c^2 x^3+c^2 x\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^2-2 a \left (-\frac {\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}+\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}}{a}-\frac {i \arctan (a x)^2}{2 a^2}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {x}{3}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} c \left (\frac {1}{5} a^4 c^2 x^5+\frac {2}{3} a^2 c^2 x^3+c^2 x\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)^2-2 a \left (-\frac {i \arctan (a x)^2}{2 a^2}-\frac {\frac {\arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}\right )\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)^2-\frac {\left (a^2 x^2+1\right ) \arctan (a x)}{3 a}+\frac {x}{3}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)^2-\frac {\left (a^2 x^2+1\right )^2 \arctan (a x)}{10 a}+\frac {1}{10} \left (\frac {a^2 x^3}{3}+x\right )\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)^2-\frac {c^3 \left (a^2 x^2+1\right )^3 \arctan (a x)}{21 a}+\frac {1}{21} c \left (\frac {1}{5} a^4 c^2 x^5+\frac {2}{3} a^2 c^2 x^3+c^2 x\right )\) |
Input:
Int[(c + a^2*c*x^2)^3*ArcTan[a*x]^2,x]
Output:
(c*(c^2*x + (2*a^2*c^2*x^3)/3 + (a^4*c^2*x^5)/5))/21 - (c^3*(1 + a^2*x^2)^ 3*ArcTan[a*x])/(21*a) + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/7 + (6*c^3*( (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)*Arc Tan[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
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]
Time = 2.62 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.04
| method | result | size |
| derivativedivides | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}+a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {2 c^{3} \left (\frac {5 a^{6} \arctan \left (a x \right ) x^{6}}{6}+4 x^{4} \arctan \left (a x \right ) a^{4}+\frac {19 x^{2} a^{2} \arctan \left (a x \right )}{2}+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{5} x^{5}}{6}-\frac {19 a^{3} x^{3}}{18}-\frac {19 a x}{3}+\frac {19 \arctan \left (a x \right )}{3}+4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )-4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )\right )}{35}}{a}\) | \(280\) |
| default | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}+a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {2 c^{3} \left (\frac {5 a^{6} \arctan \left (a x \right ) x^{6}}{6}+4 x^{4} \arctan \left (a x \right ) a^{4}+\frac {19 x^{2} a^{2} \arctan \left (a x \right )}{2}+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{5} x^{5}}{6}-\frac {19 a^{3} x^{3}}{18}-\frac {19 a x}{3}+\frac {19 \arctan \left (a x \right )}{3}+4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )-4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )\right )}{35}}{a}\) | \(280\) |
| parts | \(\frac {c^{3} \arctan \left (a x \right )^{2} a^{6} x^{7}}{7}+\frac {3 c^{3} \arctan \left (a x \right )^{2} a^{4} x^{5}}{5}+c^{3} \arctan \left (a x \right )^{2} a^{2} x^{3}+c^{3} x \arctan \left (a x \right )^{2}-\frac {2 c^{3} \left (\frac {5 a^{5} \arctan \left (a x \right ) x^{6}}{6}+4 a^{3} \arctan \left (a x \right ) x^{4}+\frac {19 x^{2} \arctan \left (a x \right ) a}{2}+\frac {8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{a}-\frac {a^{5} x^{5}+\frac {19 a^{3} x^{3}}{3}+38 a x -38 \arctan \left (a x \right )-24 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )+24 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{6 a}\right )}{35}\) | \(281\) |
| risch | \(\frac {19 c^{3} x^{3} a^{2}}{315}+\frac {a^{4} c^{3} x^{5}}{105}+\frac {38 c^{3} x}{105}-\frac {38 c^{3} \arctan \left (a x \right )}{105 a}-\frac {3 c^{3} a^{4} \ln \left (-i a x +1\right )^{2} x^{5}}{20}-\frac {c^{3} a^{6} \ln \left (-i a x +1\right )^{2} x^{7}}{28}-\frac {c^{3} a^{2} \ln \left (-i a x +1\right )^{2} x^{3}}{4}+\frac {c^{3} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x}{2}-\frac {c^{3} a^{6} \ln \left (i a x +1\right )^{2} x^{7}}{28}-\frac {c^{3} a^{2} \ln \left (i a x +1\right )^{2} x^{3}}{4}-\frac {3 c^{3} a^{4} \ln \left (i a x +1\right )^{2} x^{5}}{20}+\frac {16 i c^{3} \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{35 a}-\frac {4 i c^{3} \ln \left (-i a x +1\right )^{2}}{35 a}+\frac {4 i c^{3} \ln \left (i a x +1\right )^{2}}{35 a}-\frac {19 i c^{3} a \ln \left (-i a x +1\right ) x^{2}}{70}+\frac {4 i c^{3} a^{3} \ln \left (i a x +1\right ) x^{4}}{35}+\frac {19 i c^{3} a \ln \left (i a x +1\right ) x^{2}}{70}+\frac {8 i c^{3} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{35 a}-\frac {c^{3} \ln \left (i a x +1\right )^{2} x}{4}+\frac {20469 i c^{3}}{42875 a}-\frac {c^{3} \ln \left (-i a x +1\right )^{2} x}{4}+\frac {c^{3} a^{6} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{7}}{14}+\frac {3 c^{3} a^{4} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{5}}{10}+\frac {c^{3} a^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{2}-\frac {16 i c^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{35 a}+\frac {16 i c^{3} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{35 a}-\frac {i c^{3} a^{5} \ln \left (-i a x +1\right ) x^{6}}{42}+\frac {i c^{3} a^{5} \ln \left (i a x +1\right ) x^{6}}{42}-\frac {4 i c^{3} a^{3} \ln \left (-i a x +1\right ) x^{4}}{35}\) | \(558\) |
Input:
int((a^2*c*x^2+c)^3*arctan(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/a*(1/7*c^3*arctan(a*x)^2*a^7*x^7+3/5*a^5*c^3*x^5*arctan(a*x)^2+a^3*c^3*x ^3*arctan(a*x)^2+a*c^3*x*arctan(a*x)^2-2/35*c^3*(5/6*a^6*arctan(a*x)*x^6+4 *x^4*arctan(a*x)*a^4+19/2*x^2*a^2*arctan(a*x)+8*arctan(a*x)*ln(a^2*x^2+1)- 1/6*a^5*x^5-19/18*a^3*x^3-19/3*a*x+19/3*arctan(a*x)+4*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))) -4*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)))))
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2} \,d x } \] Input:
integrate((a^2*c*x^2+c)^3*arctan(a*x)^2,x, algorithm="fricas")
Output:
integral((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 + c^3)*arctan(a*x)^2 , x)
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=c^{3} \left (\int 3 a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{4} x^{4} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{6} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \] Input:
integrate((a**2*c*x**2+c)**3*atan(a*x)**2,x)
Output:
c**3*(Integral(3*a**2*x**2*atan(a*x)**2, x) + Integral(3*a**4*x**4*atan(a* x)**2, x) + Integral(a**6*x**6*atan(a*x)**2, x) + Integral(atan(a*x)**2, x ))
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2} \,d x } \] Input:
integrate((a^2*c*x^2+c)^3*arctan(a*x)^2,x, algorithm="maxima")
Output:
420*a^8*c^3*integrate(1/560*x^8*arctan(a*x)^2/(a^2*x^2 + 1), x) + 35*a^8*c ^3*integrate(1/560*x^8*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 20*a^8*c^3*i ntegrate(1/560*x^8*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) - 40*a^7*c^3*integra te(1/560*x^7*arctan(a*x)/(a^2*x^2 + 1), x) + 1680*a^6*c^3*integrate(1/560* x^6*arctan(a*x)^2/(a^2*x^2 + 1), x) + 140*a^6*c^3*integrate(1/560*x^6*log( a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 84*a^6*c^3*integrate(1/560*x^6*log(a^2* x^2 + 1)/(a^2*x^2 + 1), x) - 168*a^5*c^3*integrate(1/560*x^5*arctan(a*x)/( a^2*x^2 + 1), x) + 2520*a^4*c^3*integrate(1/560*x^4*arctan(a*x)^2/(a^2*x^2 + 1), x) + 210*a^4*c^3*integrate(1/560*x^4*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 140*a^4*c^3*integrate(1/560*x^4*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x ) - 280*a^3*c^3*integrate(1/560*x^3*arctan(a*x)/(a^2*x^2 + 1), x) + 1680*a ^2*c^3*integrate(1/560*x^2*arctan(a*x)^2/(a^2*x^2 + 1), x) + 140*a^2*c^3*i ntegrate(1/560*x^2*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 140*a^2*c^3*inte grate(1/560*x^2*log(a^2*x^2 + 1)/(a^2*x^2 + 1), x) + 1/4*c^3*arctan(a*x)^3 /a - 280*a*c^3*integrate(1/560*x*arctan(a*x)/(a^2*x^2 + 1), x) + 35*c^3*in tegrate(1/560*log(a^2*x^2 + 1)^2/(a^2*x^2 + 1), x) + 1/140*(5*a^6*c^3*x^7 + 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 + 35*c^3*x)*arctan(a*x)^2 - 1/560*(5*a^6 *c^3*x^7 + 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 + 35*c^3*x)*log(a^2*x^2 + 1)^2
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2} \,d x } \] Input:
integrate((a^2*c*x^2+c)^3*arctan(a*x)^2,x, algorithm="giac")
Output:
integrate((a^2*c*x^2 + c)^3*arctan(a*x)^2, x)
Timed out. \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\int {\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \] Input:
int(atan(a*x)^2*(c + a^2*c*x^2)^3,x)
Output:
int(atan(a*x)^2*(c + a^2*c*x^2)^3, x)
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x)^2 \, dx=\frac {c^{3} \left (45 \mathit {atan} \left (a x \right )^{2} a^{7} x^{7}+189 \mathit {atan} \left (a x \right )^{2} a^{5} x^{5}+315 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}+315 \mathit {atan} \left (a x \right )^{2} a x -15 \mathit {atan} \left (a x \right ) a^{6} x^{6}-72 \mathit {atan} \left (a x \right ) a^{4} x^{4}-171 \mathit {atan} \left (a x \right ) a^{2} x^{2}-114 \mathit {atan} \left (a x \right )-288 \left (\int \frac {\mathit {atan} \left (a x \right ) x}{a^{2} x^{2}+1}d x \right ) a^{2}+3 a^{5} x^{5}+19 a^{3} x^{3}+114 a x \right )}{315 a} \] Input:
int((a^2*c*x^2+c)^3*atan(a*x)^2,x)
Output:
(c**3*(45*atan(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))/( 315*a)