\(\int x (c+a^2 c x^2)^2 \arctan (a x)^2 \, dx\) [268]

Optimal result

Integrand size = 20, antiderivative size = 153 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {2 c^2 \left (1+a^2 x^2\right )}{45 a^2}+\frac {c^2 \left (1+a^2 x^2\right )^2}{60 a^2}-\frac {8 c^2 x \arctan (a x)}{45 a}-\frac {4 c^2 x \left (1+a^2 x^2\right ) \arctan (a x)}{45 a}-\frac {c^2 x \left (1+a^2 x^2\right )^2 \arctan (a x)}{15 a}+\frac {c^2 \left (1+a^2 x^2\right )^3 \arctan (a x)^2}{6 a^2}+\frac {4 c^2 \log \left (1+a^2 x^2\right )}{45 a^2} \] Output:

2/45*c^2*(a^2*x^2+1)/a^2+1/60*c^2*(a^2*x^2+1)^2/a^2-8/45*c^2*x*arctan(a*x) 
/a-4/45*c^2*x*(a^2*x^2+1)*arctan(a*x)/a-1/15*c^2*x*(a^2*x^2+1)^2*arctan(a* 
x)/a+1/6*c^2*(a^2*x^2+1)^3*arctan(a*x)^2/a^2+4/45*c^2*ln(a^2*x^2+1)/a^2
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.55 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {c^2 \left (14 a^2 x^2+3 a^4 x^4-4 a x \left (15+10 a^2 x^2+3 a^4 x^4\right ) \arctan (a x)+30 \left (1+a^2 x^2\right )^3 \arctan (a x)^2+16 \log \left (1+a^2 x^2\right )\right )}{180 a^2} \] Input:

Integrate[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]
 

Output:

(c^2*(14*a^2*x^2 + 3*a^4*x^4 - 4*a*x*(15 + 10*a^2*x^2 + 3*a^4*x^4)*ArcTan[ 
a*x] + 30*(1 + a^2*x^2)^3*ArcTan[a*x]^2 + 16*Log[1 + a^2*x^2]))/(180*a^2)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5465, 27, 5413, 5413, 5345, 240}

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.

\(\displaystyle \int x \arctan (a x)^2 \left (a^2 c x^2+c\right )^2 \, dx\)

\(\Big \downarrow \) 5465

\(\displaystyle \frac {c^2 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{6 a^2}-\frac {\int c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)dx}{3 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {c^2 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{6 a^2}-\frac {c^2 \int \left (a^2 x^2+1\right )^2 \arctan (a x)dx}{3 a}\)

\(\Big \downarrow \) 5413

\(\displaystyle \frac {c^2 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{6 a^2}-\frac {c^2 \left (\frac {4}{5} \int \left (a^2 x^2+1\right ) \arctan (a x)dx+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )}{3 a}\)

\(\Big \downarrow \) 5413

\(\displaystyle \frac {c^2 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{6 a^2}-\frac {c^2 \left (\frac {4}{5} \left (\frac {2}{3} \int \arctan (a x)dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )}{3 a}\)

\(\Big \downarrow \) 5345

\(\displaystyle \frac {c^2 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{6 a^2}-\frac {c^2 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)-a \int \frac {x}{a^2 x^2+1}dx\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )}{3 a}\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {c^2 \left (a^2 x^2+1\right )^3 \arctan (a x)^2}{6 a^2}-\frac {c^2 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )}{3 a}\)

Input:

Int[x*(c + a^2*c*x^2)^2*ArcTan[a*x]^2,x]
 

Output:

(c^2*(1 + a^2*x^2)^3*ArcTan[a*x]^2)/(6*a^2) - (c^2*(-1/20*(1 + a^2*x^2)^2/ 
a + (x*(1 + a^2*x^2)^2*ArcTan[a*x])/5 + (4*(-1/6*(1 + a^2*x^2)/a + (x*(1 + 
 a^2*x^2)*ArcTan[a*x])/3 + (2*(x*ArcTan[a*x] - Log[1 + a^2*x^2]/(2*a)))/3) 
)/5))/(3*a)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 5345

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])
 

rule 5413

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbo 
l] :> Simp[(-b)*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2) 
^q*((a + b*ArcTan[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1))   Int[(d + 
e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && 
 EqQ[e, c^2*d] && GtQ[q, 0]
 

rule 5465

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]
 

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.86

method	result	size
parts	\(\frac {a^{4} c^{2} x^{6} \arctan \left (a x \right )^{2}}{6}+\frac {a^{2} c^{2} x^{4} \arctan \left (a x \right )^{2}}{2}+\frac {c^{2} x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c^{2} \arctan \left (a x \right )^{2}}{6 a^{2}}-\frac {c^{2} \left (\frac {\arctan \left (a x \right ) a^{5} x^{5}}{5}+\frac {2 \arctan \left (a x \right ) x^{3} a^{3}}{3}+\arctan \left (a x \right ) a x -\frac {a^{4} x^{4}}{20}-\frac {7 a^{2} x^{2}}{30}-\frac {4 \ln \left (a^{2} x^{2}+1\right )}{15}\right )}{3 a^{2}}\)	\(132\)
derivativedivides	\(\frac {\frac {c^{2} \arctan \left (a x \right )^{2} a^{6} x^{6}}{6}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )^{2}}{2}+\frac {a^{2} c^{2} x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c^{2} \arctan \left (a x \right )^{2}}{6}-\frac {c^{2} \left (\frac {\arctan \left (a x \right ) a^{5} x^{5}}{5}+\frac {2 \arctan \left (a x \right ) x^{3} a^{3}}{3}+\arctan \left (a x \right ) a x -\frac {a^{4} x^{4}}{20}-\frac {7 a^{2} x^{2}}{30}-\frac {4 \ln \left (a^{2} x^{2}+1\right )}{15}\right )}{3}}{a^{2}}\)	\(133\)
default	\(\frac {\frac {c^{2} \arctan \left (a x \right )^{2} a^{6} x^{6}}{6}+\frac {a^{4} c^{2} x^{4} \arctan \left (a x \right )^{2}}{2}+\frac {a^{2} c^{2} x^{2} \arctan \left (a x \right )^{2}}{2}+\frac {c^{2} \arctan \left (a x \right )^{2}}{6}-\frac {c^{2} \left (\frac {\arctan \left (a x \right ) a^{5} x^{5}}{5}+\frac {2 \arctan \left (a x \right ) x^{3} a^{3}}{3}+\arctan \left (a x \right ) a x -\frac {a^{4} x^{4}}{20}-\frac {7 a^{2} x^{2}}{30}-\frac {4 \ln \left (a^{2} x^{2}+1\right )}{15}\right )}{3}}{a^{2}}\)	\(133\)
parallelrisch	\(\frac {30 c^{2} \arctan \left (a x \right )^{2} a^{6} x^{6}-12 a^{5} c^{2} \arctan \left (a x \right ) x^{5}+90 a^{4} c^{2} x^{4} \arctan \left (a x \right )^{2}+3 a^{4} c^{2} x^{4}-40 a^{3} c^{2} x^{3} \arctan \left (a x \right )+90 a^{2} c^{2} x^{2} \arctan \left (a x \right )^{2}+14 x^{2} c^{2} a^{2}-60 a \,c^{2} x \arctan \left (a x \right )+30 c^{2} \arctan \left (a x \right )^{2}+16 c^{2} \ln \left (a^{2} x^{2}+1\right )}{180 a^{2}}\)	\(147\)
risch	\(-\frac {c^{2} \left (a^{2} x^{2}+1\right )^{3} \ln \left (i a x +1\right )^{2}}{24 a^{2}}+\frac {c^{2} \left (15 a^{6} x^{6} \ln \left (-i a x +1\right )+6 i a^{5} x^{5}+45 x^{4} \ln \left (-i a x +1\right ) a^{4}+20 i a^{3} x^{3}+45 a^{2} x^{2} \ln \left (-i a x +1\right )+30 i a x +15 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{180 a^{2}}-\frac {c^{2} a^{4} x^{6} \ln \left (-i a x +1\right )^{2}}{24}-\frac {i c^{2} a^{3} x^{5} \ln \left (-i a x +1\right )}{30}-\frac {c^{2} a^{2} x^{4} \ln \left (-i a x +1\right )^{2}}{8}-\frac {i c^{2} a \,x^{3} \ln \left (-i a x +1\right )}{9}+\frac {a^{2} c^{2} x^{4}}{60}-\frac {c^{2} x^{2} \ln \left (-i a x +1\right )^{2}}{8}-\frac {i c^{2} x \ln \left (-i a x +1\right )}{6 a}+\frac {7 c^{2} x^{2}}{90}-\frac {c^{2} \ln \left (-i a x +1\right )^{2}}{24 a^{2}}+\frac {4 c^{2} \ln \left (-a^{2} x^{2}-1\right )}{45 a^{2}}+\frac {49 c^{2}}{540 a^{2}}\)	\(309\)

Input:

int(x*(a^2*c*x^2+c)^2*arctan(a*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/6*a^4*c^2*x^6*arctan(a*x)^2+1/2*a^2*c^2*x^4*arctan(a*x)^2+1/2*c^2*x^2*ar 
ctan(a*x)^2+1/6*c^2*arctan(a*x)^2/a^2-1/3*c^2/a^2*(1/5*arctan(a*x)*a^5*x^5 
+2/3*arctan(a*x)*x^3*a^3+arctan(a*x)*a*x-1/20*a^4*x^4-7/30*a^2*x^2-4/15*ln 
(a^2*x^2+1))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.80 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {3 \, a^{4} c^{2} x^{4} + 14 \, a^{2} c^{2} x^{2} + 30 \, {\left (a^{6} c^{2} x^{6} + 3 \, a^{4} c^{2} x^{4} + 3 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{2} + 16 \, c^{2} \log \left (a^{2} x^{2} + 1\right ) - 4 \, {\left (3 \, a^{5} c^{2} x^{5} + 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \arctan \left (a x\right )}{180 \, a^{2}} \] Input:

integrate(x*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="fricas")
 

Output:

1/180*(3*a^4*c^2*x^4 + 14*a^2*c^2*x^2 + 30*(a^6*c^2*x^6 + 3*a^4*c^2*x^4 + 
3*a^2*c^2*x^2 + c^2)*arctan(a*x)^2 + 16*c^2*log(a^2*x^2 + 1) - 4*(3*a^5*c^ 
2*x^5 + 10*a^3*c^2*x^3 + 15*a*c^2*x)*arctan(a*x))/a^2
 

Sympy [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.03 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\begin {cases} \frac {a^{4} c^{2} x^{6} \operatorname {atan}^{2}{\left (a x \right )}}{6} - \frac {a^{3} c^{2} x^{5} \operatorname {atan}{\left (a x \right )}}{15} + \frac {a^{2} c^{2} x^{4} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {a^{2} c^{2} x^{4}}{60} - \frac {2 a c^{2} x^{3} \operatorname {atan}{\left (a x \right )}}{9} + \frac {c^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{2} + \frac {7 c^{2} x^{2}}{90} - \frac {c^{2} x \operatorname {atan}{\left (a x \right )}}{3 a} + \frac {4 c^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{45 a^{2}} + \frac {c^{2} \operatorname {atan}^{2}{\left (a x \right )}}{6 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

integrate(x*(a**2*c*x**2+c)**2*atan(a*x)**2,x)
 

Output:

Piecewise((a**4*c**2*x**6*atan(a*x)**2/6 - a**3*c**2*x**5*atan(a*x)/15 + a 
**2*c**2*x**4*atan(a*x)**2/2 + a**2*c**2*x**4/60 - 2*a*c**2*x**3*atan(a*x) 
/9 + c**2*x**2*atan(a*x)**2/2 + 7*c**2*x**2/90 - c**2*x*atan(a*x)/(3*a) + 
4*c**2*log(x**2 + a**(-2))/(45*a**2) + c**2*atan(a*x)**2/(6*a**2), Ne(a, 0 
)), (0, True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.73 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{6 \, a^{2} c} + \frac {{\left (3 \, a^{2} c^{3} x^{4} + 14 \, c^{3} x^{2} + \frac {16 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a - 4 \, {\left (3 \, a^{4} c^{3} x^{5} + 10 \, a^{2} c^{3} x^{3} + 15 \, c^{3} x\right )} \arctan \left (a x\right )}{180 \, a c} \] Input:

integrate(x*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="maxima")
 

Output:

1/6*(a^2*c*x^2 + c)^3*arctan(a*x)^2/(a^2*c) + 1/180*((3*a^2*c^3*x^4 + 14*c 
^3*x^2 + 16*c^3*log(a^2*x^2 + 1)/a^2)*a - 4*(3*a^4*c^3*x^5 + 10*a^2*c^3*x^ 
3 + 15*c^3*x)*arctan(a*x))/(a*c)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.05 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{6 \, a^{2} c} - \frac {3 \, {\left (4 \, x^{5} \arctan \left (a x\right ) - a {\left (\frac {a^{2} x^{4} - 2 \, x^{2}}{a^{4}} + \frac {2 \, \log \left (a^{2} x^{2} + 1\right )}{a^{6}}\right )}\right )} a^{4} c^{2} + 20 \, {\left (2 \, x^{3} \arctan \left (a x\right ) - a {\left (\frac {x^{2}}{a^{2}} - \frac {\log \left (a^{2} x^{2} + 1\right )}{a^{4}}\right )}\right )} a^{2} c^{2} + \frac {30 \, {\left (2 \, a x \arctan \left (a x\right ) - \log \left (a^{2} x^{2} + 1\right )\right )} c^{2}}{a}}{180 \, a} \] Input:

integrate(x*(a^2*c*x^2+c)^2*arctan(a*x)^2,x, algorithm="giac")
 

Output:

1/6*(a^2*c*x^2 + c)^3*arctan(a*x)^2/(a^2*c) - 1/180*(3*(4*x^5*arctan(a*x) 
- a*((a^2*x^4 - 2*x^2)/a^4 + 2*log(a^2*x^2 + 1)/a^6))*a^4*c^2 + 20*(2*x^3* 
arctan(a*x) - a*(x^2/a^2 - log(a^2*x^2 + 1)/a^4))*a^2*c^2 + 30*(2*a*x*arct 
an(a*x) - log(a^2*x^2 + 1))*c^2/a)/a
 

Mupad [B] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.88 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {\frac {c^2\,\left (30\,{\mathrm {atan}\left (a\,x\right )}^2+16\,\ln \left (a^2\,x^2+1\right )\right )}{180}-\frac {a\,c^2\,x\,\mathrm {atan}\left (a\,x\right )}{3}}{a^2}+\frac {c^2\,\left (90\,x^2\,{\mathrm {atan}\left (a\,x\right )}^2+14\,x^2\right )}{180}+\frac {a^2\,c^2\,\left (90\,x^4\,{\mathrm {atan}\left (a\,x\right )}^2+3\,x^4\right )}{180}-\frac {a^3\,c^2\,x^5\,\mathrm {atan}\left (a\,x\right )}{15}+\frac {a^4\,c^2\,x^6\,{\mathrm {atan}\left (a\,x\right )}^2}{6}-\frac {2\,a\,c^2\,x^3\,\mathrm {atan}\left (a\,x\right )}{9} \] Input:

int(x*atan(a*x)^2*(c + a^2*c*x^2)^2,x)
 

Output:

((c^2*(16*log(a^2*x^2 + 1) + 30*atan(a*x)^2))/180 - (a*c^2*x*atan(a*x))/3) 
/a^2 + (c^2*(90*x^2*atan(a*x)^2 + 14*x^2))/180 + (a^2*c^2*(90*x^4*atan(a*x 
)^2 + 3*x^4))/180 - (a^3*c^2*x^5*atan(a*x))/15 + (a^4*c^2*x^6*atan(a*x)^2) 
/6 - (2*a*c^2*x^3*atan(a*x))/9
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.78 \[ \int x \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {c^{2} \left (30 \mathit {atan} \left (a x \right )^{2} a^{6} x^{6}+90 \mathit {atan} \left (a x \right )^{2} a^{4} x^{4}+90 \mathit {atan} \left (a x \right )^{2} a^{2} x^{2}+30 \mathit {atan} \left (a x \right )^{2}-12 \mathit {atan} \left (a x \right ) a^{5} x^{5}-40 \mathit {atan} \left (a x \right ) a^{3} x^{3}-60 \mathit {atan} \left (a x \right ) a x +16 \,\mathrm {log}\left (a^{2} x^{2}+1\right )+3 a^{4} x^{4}+14 a^{2} x^{2}\right )}{180 a^{2}} \] Input:

int(x*(a^2*c*x^2+c)^2*atan(a*x)^2,x)
 

Output:

(c**2*(30*atan(a*x)**2*a**6*x**6 + 90*atan(a*x)**2*a**4*x**4 + 90*atan(a*x 
)**2*a**2*x**2 + 30*atan(a*x)**2 - 12*atan(a*x)*a**5*x**5 - 40*atan(a*x)*a 
**3*x**3 - 60*atan(a*x)*a*x + 16*log(a**2*x**2 + 1) + 3*a**4*x**4 + 14*a** 
2*x**2))/(180*a**2)