\(\int x^{11} (a+b \arctan (c x^3))^2 \, dx\) [113]

Optimal result

Integrand size = 16, antiderivative size = 124 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {a b x^3}{6 c^3}+\frac {b^2 x^6}{36 c^2}+\frac {b^2 x^3 \arctan \left (c x^3\right )}{6 c^3}-\frac {b x^9 \left (a+b \arctan \left (c x^3\right )\right )}{18 c}-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{12 c^4}+\frac {1}{12} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {b^2 \log \left (1+c^2 x^6\right )}{9 c^4} \] Output:

1/6*a*b*x^3/c^3+1/36*b^2*x^6/c^2+1/6*b^2*x^3*arctan(c*x^3)/c^3-1/18*b*x^9* 
(a+b*arctan(c*x^3))/c-1/12*(a+b*arctan(c*x^3))^2/c^4+1/12*x^12*(a+b*arctan 
(c*x^3))^2-1/9*b^2*ln(c^2*x^6+1)/c^4
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {c x^3 \left (6 a b+b^2 c x^3-2 a b c^2 x^6+3 a^2 c^3 x^9\right )-2 b \left (b c x^3 \left (-3+c^2 x^6\right )+a \left (3-3 c^4 x^{12}\right )\right ) \arctan \left (c x^3\right )+3 b^2 \left (-1+c^4 x^{12}\right ) \arctan \left (c x^3\right )^2-4 b^2 \log \left (1+c^2 x^6\right )}{36 c^4} \] Input:

Integrate[x^11*(a + b*ArcTan[c*x^3])^2,x]
 

Output:

(c*x^3*(6*a*b + b^2*c*x^3 - 2*a*b*c^2*x^6 + 3*a^2*c^3*x^9) - 2*b*(b*c*x^3* 
(-3 + c^2*x^6) + a*(3 - 3*c^4*x^12))*ArcTan[c*x^3] + 3*b^2*(-1 + c^4*x^12) 
*ArcTan[c*x^3]^2 - 4*b^2*Log[1 + c^2*x^6])/(36*c^4)
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.20, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5363, 5361, 5451, 5361, 243, 49, 2009, 5451, 2009, 5419}

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.

\(\displaystyle \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx\)

\(\Big \downarrow \) 5363

\(\displaystyle \frac {1}{3} \int x^9 \left (a+b \arctan \left (c x^3\right )\right )^2dx^3\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {1}{3} \left (\frac {1}{4} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {1}{2} b c \int \frac {x^{12} \left (a+b \arctan \left (c x^3\right )\right )}{c^2 x^6+1}dx^3\right )\)

\(\Big \downarrow \) 5451

\(\displaystyle \frac {1}{3} \left (\frac {1}{4} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {1}{2} b c \left (\frac {\int x^6 \left (a+b \arctan \left (c x^3\right )\right )dx^3}{c^2}-\frac {\int \frac {x^6 \left (a+b \arctan \left (c x^3\right )\right )}{c^2 x^6+1}dx^3}{c^2}\right )\right )\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {1}{3} \left (\frac {1}{4} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {1}{2} b c \left (\frac {\frac {1}{3} x^9 \left (a+b \arctan \left (c x^3\right )\right )-\frac {1}{3} b c \int \frac {x^9}{c^2 x^6+1}dx^3}{c^2}-\frac {\int \frac {x^6 \left (a+b \arctan \left (c x^3\right )\right )}{c^2 x^6+1}dx^3}{c^2}\right )\right )\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{3} \left (\frac {1}{4} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {1}{2} b c \left (\frac {\frac {1}{3} x^9 \left (a+b \arctan \left (c x^3\right )\right )-\frac {1}{6} b c \int \frac {x^6}{c^2 x^6+1}dx^6}{c^2}-\frac {\int \frac {x^6 \left (a+b \arctan \left (c x^3\right )\right )}{c^2 x^6+1}dx^3}{c^2}\right )\right )\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {1}{3} \left (\frac {1}{4} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {1}{2} b c \left (\frac {\frac {1}{3} x^9 \left (a+b \arctan \left (c x^3\right )\right )-\frac {1}{6} b c \int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (c^2 x^6+1\right )}\right )dx^6}{c^2}-\frac {\int \frac {x^6 \left (a+b \arctan \left (c x^3\right )\right )}{c^2 x^6+1}dx^3}{c^2}\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (\frac {1}{4} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {1}{2} b c \left (\frac {\frac {1}{3} x^9 \left (a+b \arctan \left (c x^3\right )\right )-\frac {1}{6} b c \left (\frac {x^6}{c^2}-\frac {\log \left (c^2 x^6+1\right )}{c^4}\right )}{c^2}-\frac {\int \frac {x^6 \left (a+b \arctan \left (c x^3\right )\right )}{c^2 x^6+1}dx^3}{c^2}\right )\right )\)

\(\Big \downarrow \) 5451

\(\displaystyle \frac {1}{3} \left (\frac {1}{4} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {1}{2} b c \left (\frac {\frac {1}{3} x^9 \left (a+b \arctan \left (c x^3\right )\right )-\frac {1}{6} b c \left (\frac {x^6}{c^2}-\frac {\log \left (c^2 x^6+1\right )}{c^4}\right )}{c^2}-\frac {\frac {\int \left (a+b \arctan \left (c x^3\right )\right )dx^3}{c^2}-\frac {\int \frac {a+b \arctan \left (c x^3\right )}{c^2 x^6+1}dx^3}{c^2}}{c^2}\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (\frac {1}{4} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {1}{2} b c \left (\frac {\frac {1}{3} x^9 \left (a+b \arctan \left (c x^3\right )\right )-\frac {1}{6} b c \left (\frac {x^6}{c^2}-\frac {\log \left (c^2 x^6+1\right )}{c^4}\right )}{c^2}-\frac {\frac {a x^3+b x^3 \arctan \left (c x^3\right )-\frac {b \log \left (c^2 x^6+1\right )}{2 c}}{c^2}-\frac {\int \frac {a+b \arctan \left (c x^3\right )}{c^2 x^6+1}dx^3}{c^2}}{c^2}\right )\right )\)

\(\Big \downarrow \) 5419

\(\displaystyle \frac {1}{3} \left (\frac {1}{4} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {1}{2} b c \left (\frac {\frac {1}{3} x^9 \left (a+b \arctan \left (c x^3\right )\right )-\frac {1}{6} b c \left (\frac {x^6}{c^2}-\frac {\log \left (c^2 x^6+1\right )}{c^4}\right )}{c^2}-\frac {\frac {a x^3+b x^3 \arctan \left (c x^3\right )-\frac {b \log \left (c^2 x^6+1\right )}{2 c}}{c^2}-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{2 b c^3}}{c^2}\right )\right )\)

Input:

Int[x^11*(a + b*ArcTan[c*x^3])^2,x]
 

Output:

((x^12*(a + b*ArcTan[c*x^3])^2)/4 - (b*c*(((x^9*(a + b*ArcTan[c*x^3]))/3 - 
 (b*c*(x^6/c^2 - Log[1 + c^2*x^6]/c^4))/6)/c^2 - (-1/2*(a + b*ArcTan[c*x^3 
])^2/(b*c^3) + (a*x^3 + b*x^3*ArcTan[c*x^3] - (b*Log[1 + c^2*x^6])/(2*c))/ 
c^2)/c^2))/2)/3
 

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5361

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> 
 Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 
1))   Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], 
x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & 
& IntegerQ[m])) && NeQ[m, -1]
 

rule 5363

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> 
Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTan[c*x])^p, x], 
 x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simplif 
y[(m + 1)/n]]
 

rule 5419

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

rule 5451

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[f^2/e   Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] 
)^p, x], x] - Simp[d*(f^2/e)   Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/(d 
+ e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
 

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.22

method	result	size
default	\(\frac {a^{2} x^{12}}{12}+\frac {b^{2} x^{12} \arctan \left (c \,x^{3}\right )^{2}}{12}-\frac {b^{2} \arctan \left (c \,x^{3}\right ) x^{9}}{18 c}+\frac {b^{2} x^{3} \arctan \left (c \,x^{3}\right )}{6 c^{3}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2}}{12 c^{4}}+\frac {b^{2} x^{6}}{36 c^{2}}-\frac {b^{2} \ln \left (c^{2} x^{6}+1\right )}{9 c^{4}}+\frac {a b \,x^{12} \arctan \left (c \,x^{3}\right )}{6}-\frac {a b \,x^{9}}{18 c}+\frac {a b \,x^{3}}{6 c^{3}}-\frac {a b \arctan \left (c \,x^{3}\right )}{6 c^{4}}\)	\(151\)
parts	\(\frac {a^{2} x^{12}}{12}+\frac {b^{2} x^{12} \arctan \left (c \,x^{3}\right )^{2}}{12}-\frac {b^{2} \arctan \left (c \,x^{3}\right ) x^{9}}{18 c}+\frac {b^{2} x^{3} \arctan \left (c \,x^{3}\right )}{6 c^{3}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2}}{12 c^{4}}+\frac {b^{2} x^{6}}{36 c^{2}}-\frac {b^{2} \ln \left (c^{2} x^{6}+1\right )}{9 c^{4}}+\frac {a b \,x^{12} \arctan \left (c \,x^{3}\right )}{6}-\frac {a b \,x^{9}}{18 c}+\frac {a b \,x^{3}}{6 c^{3}}-\frac {a b \arctan \left (c \,x^{3}\right )}{6 c^{4}}\)	\(151\)
parallelrisch	\(-\frac {-3 b^{2} \arctan \left (c \,x^{3}\right )^{2} x^{12} c^{4}-6 a b \arctan \left (c \,x^{3}\right ) x^{12} c^{4}-3 c^{4} a^{2} x^{12}+2 b^{2} \arctan \left (c \,x^{3}\right ) x^{9} c^{3}+2 a b \,c^{3} x^{9}-x^{6} b^{2} c^{2}-6 b^{2} \arctan \left (c \,x^{3}\right ) x^{3} c -6 a b c \,x^{3}+3 b^{2} \arctan \left (c \,x^{3}\right )^{2}+4 b^{2} \ln \left (c^{2} x^{6}+1\right )+6 a b \arctan \left (c \,x^{3}\right )+b^{2}}{36 c^{4}}\)	\(155\)
risch	\(-\frac {b^{2} \left (c^{4} x^{12}-1\right ) \ln \left (i c \,x^{3}+1\right )^{2}}{48 c^{4}}-\frac {i b \left (6 a \,c^{4} x^{12}+3 i b \,c^{4} x^{12} \ln \left (-i c \,x^{3}+1\right )-2 b \,c^{3} x^{9}+6 b c \,x^{3}-3 i b \ln \left (-i c \,x^{3}+1\right )\right ) \ln \left (i c \,x^{3}+1\right )}{72 c^{4}}-\frac {b^{2} x^{12} \ln \left (-i c \,x^{3}+1\right )^{2}}{48}+\frac {i a b \,x^{12} \ln \left (-i c \,x^{3}+1\right )}{12}+\frac {a^{2} x^{12}}{12}-\frac {i b^{2} x^{9} \ln \left (-i c \,x^{3}+1\right )}{36 c}-\frac {a b \,x^{9}}{18 c}+\frac {b^{2} x^{6}}{36 c^{2}}+\frac {i b^{2} x^{3} \ln \left (-i c \,x^{3}+1\right )}{12 c^{3}}+\frac {a b \,x^{3}}{6 c^{3}}+\frac {b^{2} \ln \left (-i c \,x^{3}+1\right )^{2}}{48 c^{4}}-\frac {a b \arctan \left (c \,x^{3}\right )}{6 c^{4}}-\frac {b^{2} \ln \left (c^{2} x^{6}+1\right )}{9 c^{4}}\)	\(280\)

Input:

int(x^11*(a+b*arctan(c*x^3))^2,x,method=_RETURNVERBOSE)
 

Output:

1/12*a^2*x^12+1/12*b^2*x^12*arctan(c*x^3)^2-1/18*b^2*arctan(c*x^3)/c*x^9+1 
/6*b^2*x^3*arctan(c*x^3)/c^3-1/12*b^2/c^4*arctan(c*x^3)^2+1/36*b^2*x^6/c^2 
-1/9*b^2*ln(c^2*x^6+1)/c^4+1/6*a*b*x^12*arctan(c*x^3)-1/18*a*b/c*x^9+1/6*a 
*b*x^3/c^3-1/6*a*b/c^4*arctan(c*x^3)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {3 \, a^{2} c^{4} x^{12} - 2 \, a b c^{3} x^{9} + b^{2} c^{2} x^{6} + 6 \, a b c x^{3} + 3 \, {\left (b^{2} c^{4} x^{12} - b^{2}\right )} \arctan \left (c x^{3}\right )^{2} - 4 \, b^{2} \log \left (c^{2} x^{6} + 1\right ) + 2 \, {\left (3 \, a b c^{4} x^{12} - b^{2} c^{3} x^{9} + 3 \, b^{2} c x^{3} - 3 \, a b\right )} \arctan \left (c x^{3}\right )}{36 \, c^{4}} \] Input:

integrate(x^11*(a+b*arctan(c*x^3))^2,x, algorithm="fricas")
 

Output:

1/36*(3*a^2*c^4*x^12 - 2*a*b*c^3*x^9 + b^2*c^2*x^6 + 6*a*b*c*x^3 + 3*(b^2* 
c^4*x^12 - b^2)*arctan(c*x^3)^2 - 4*b^2*log(c^2*x^6 + 1) + 2*(3*a*b*c^4*x^ 
12 - b^2*c^3*x^9 + 3*b^2*c*x^3 - 3*a*b)*arctan(c*x^3))/c^4
 

Sympy [F(-1)]

Timed out. \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\text {Timed out} \] Input:

integrate(x**11*(a+b*atan(c*x**3))**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.36 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {1}{12} \, b^{2} x^{12} \arctan \left (c x^{3}\right )^{2} + \frac {1}{12} \, a^{2} x^{12} + \frac {1}{18} \, {\left (3 \, x^{12} \arctan \left (c x^{3}\right ) - c {\left (\frac {c^{2} x^{9} - 3 \, x^{3}}{c^{4}} + \frac {3 \, \arctan \left (c x^{3}\right )}{c^{5}}\right )}\right )} a b - \frac {1}{36} \, {\left (2 \, c {\left (\frac {c^{2} x^{9} - 3 \, x^{3}}{c^{4}} + \frac {3 \, \arctan \left (c x^{3}\right )}{c^{5}}\right )} \arctan \left (c x^{3}\right ) - \frac {c^{2} x^{6} + 3 \, \arctan \left (c x^{3}\right )^{2} - 3 \, \log \left (18 \, c^{7} x^{6} + 18 \, c^{5}\right ) - \log \left (c^{2} x^{6} + 1\right )}{c^{4}}\right )} b^{2} \] Input:

integrate(x^11*(a+b*arctan(c*x^3))^2,x, algorithm="maxima")
 

Output:

1/12*b^2*x^12*arctan(c*x^3)^2 + 1/12*a^2*x^12 + 1/18*(3*x^12*arctan(c*x^3) 
 - c*((c^2*x^9 - 3*x^3)/c^4 + 3*arctan(c*x^3)/c^5))*a*b - 1/36*(2*c*((c^2* 
x^9 - 3*x^3)/c^4 + 3*arctan(c*x^3)/c^5)*arctan(c*x^3) - (c^2*x^6 + 3*arcta 
n(c*x^3)^2 - 3*log(18*c^7*x^6 + 18*c^5) - log(c^2*x^6 + 1))/c^4)*b^2
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.17 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {3 \, a^{2} c x^{12} + 2 \, {\left (3 \, c x^{12} \arctan \left (c x^{3}\right ) - \frac {3 \, \arctan \left (c x^{3}\right )}{c^{3}} - \frac {c^{9} x^{9} - 3 \, c^{7} x^{3}}{c^{9}}\right )} a b + {\left (3 \, c x^{12} \arctan \left (c x^{3}\right )^{2} - \frac {2 \, c^{3} x^{9} \arctan \left (c x^{3}\right ) - c^{2} x^{6} - 6 \, c x^{3} \arctan \left (c x^{3}\right ) + 3 \, \arctan \left (c x^{3}\right )^{2} + 4 \, \log \left (c^{2} x^{6} + 1\right )}{c^{3}}\right )} b^{2}}{36 \, c} \] Input:

integrate(x^11*(a+b*arctan(c*x^3))^2,x, algorithm="giac")
 

Output:

1/36*(3*a^2*c*x^12 + 2*(3*c*x^12*arctan(c*x^3) - 3*arctan(c*x^3)/c^3 - (c^ 
9*x^9 - 3*c^7*x^3)/c^9)*a*b + (3*c*x^12*arctan(c*x^3)^2 - (2*c^3*x^9*arcta 
n(c*x^3) - c^2*x^6 - 6*c*x^3*arctan(c*x^3) + 3*arctan(c*x^3)^2 + 4*log(c^2 
*x^6 + 1))/c^3)*b^2)/c
 

Mupad [B] (verification not implemented)

Time = 1.47 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.21 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {a^2\,x^{12}}{12}-\frac {b^2\,{\mathrm {atan}\left (c\,x^3\right )}^2}{12\,c^4}+\frac {b^2\,x^{12}\,{\mathrm {atan}\left (c\,x^3\right )}^2}{12}-\frac {b^2\,\ln \left (c^2\,x^6+1\right )}{9\,c^4}+\frac {b^2\,x^6}{36\,c^2}+\frac {b^2\,x^3\,\mathrm {atan}\left (c\,x^3\right )}{6\,c^3}-\frac {b^2\,x^9\,\mathrm {atan}\left (c\,x^3\right )}{18\,c}+\frac {a\,b\,x^3}{6\,c^3}-\frac {a\,b\,x^9}{18\,c}-\frac {a\,b\,\mathrm {atan}\left (c\,x^3\right )}{6\,c^4}+\frac {a\,b\,x^{12}\,\mathrm {atan}\left (c\,x^3\right )}{6} \] Input:

int(x^11*(a + b*atan(c*x^3))^2,x)
                                                                                    
                                                                                    
 

Output:

(a^2*x^12)/12 - (b^2*atan(c*x^3)^2)/(12*c^4) + (b^2*x^12*atan(c*x^3)^2)/12 
 - (b^2*log(c^2*x^6 + 1))/(9*c^4) + (b^2*x^6)/(36*c^2) + (b^2*x^3*atan(c*x 
^3))/(6*c^3) - (b^2*x^9*atan(c*x^3))/(18*c) + (a*b*x^3)/(6*c^3) - (a*b*x^9 
)/(18*c) - (a*b*atan(c*x^3))/(6*c^4) + (a*b*x^12*atan(c*x^3))/6
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.57 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {3 \mathit {atan} \left (c \,x^{3}\right )^{2} b^{2} c^{4} x^{12}-3 \mathit {atan} \left (c \,x^{3}\right )^{2} b^{2}+6 \mathit {atan} \left (c \,x^{3}\right ) a b \,c^{4} x^{12}-6 \mathit {atan} \left (c \,x^{3}\right ) a b -2 \mathit {atan} \left (c \,x^{3}\right ) b^{2} c^{3} x^{9}+6 \mathit {atan} \left (c \,x^{3}\right ) b^{2} c \,x^{3}-4 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b^{2}-4 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}+c^{\frac {1}{3}} \sqrt {3}\, x +1\right ) b^{2}-4 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}+1\right ) b^{2}+3 a^{2} c^{4} x^{12}-2 a b \,c^{3} x^{9}+6 a b c \,x^{3}+b^{2} c^{2} x^{6}}{36 c^{4}} \] Input:

int(x^11*(a+b*atan(c*x^3))^2,x)
 

Output:

(3*atan(c*x**3)**2*b**2*c**4*x**12 - 3*atan(c*x**3)**2*b**2 + 6*atan(c*x** 
3)*a*b*c**4*x**12 - 6*atan(c*x**3)*a*b - 2*atan(c*x**3)*b**2*c**3*x**9 + 6 
*atan(c*x**3)*b**2*c*x**3 - 4*log(c**(2/3)*x**2 - c**(1/3)*sqrt(3)*x + 1)* 
b**2 - 4*log(c**(2/3)*x**2 + c**(1/3)*sqrt(3)*x + 1)*b**2 - 4*log(c**(2/3) 
*x**2 + 1)*b**2 + 3*a**2*c**4*x**12 - 2*a*b*c**3*x**9 + 6*a*b*c*x**3 + b** 
2*c**2*x**6)/(36*c**4)