3.2.41 \(\int x^2 (a+b \text {ArcTan}(\frac {c}{x}))^2 \, dx\) [141]

Optimal. Leaf size=152 \[ \frac {1}{3} b^2 c^2 x+\frac {1}{3} b^2 c^3 \cot ^{-1}\left (\frac {x}{c}\right )+\frac {1}{3} b c x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{3} i c^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{3} x^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {2}{3} b c^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1-\frac {i c}{x}}\right )-\frac {1}{3} i b^2 c^3 \text {PolyLog}\left (2,-1+\frac {2}{1-\frac {i c}{x}}\right ) \]

[Out]

1/3*b^2*c^2*x+1/3*b^2*c^3*arccot(x/c)+1/3*b*c*x^2*(a+b*arccot(x/c))-1/3*I*c^3*(a+b*arccot(x/c))^2+1/3*x^3*(a+b
*arccot(x/c))^2+2/3*b*c^3*(a+b*arccot(x/c))*ln(2-2/(1-I*c/x))-1/3*I*b^2*c^3*polylog(2,-1+2/(1-I*c/x))

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4948, 4946, 5038, 331, 209, 5044, 4988, 2497} \begin {gather*} -\frac {1}{3} i c^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {2}{3} b c^3 \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{3} x^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{3} b c x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )-\frac {1}{3} i b^2 c^3 \text {Li}_2\left (\frac {2}{1-\frac {i c}{x}}-1\right )+\frac {1}{3} b^2 c^3 \cot ^{-1}\left (\frac {x}{c}\right )+\frac {1}{3} b^2 c^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcTan[c/x])^2,x]

[Out]

(b^2*c^2*x)/3 + (b^2*c^3*ArcCot[x/c])/3 + (b*c*x^2*(a + b*ArcCot[x/c]))/3 - (I/3)*c^3*(a + b*ArcCot[x/c])^2 +
(x^3*(a + b*ArcCot[x/c])^2)/3 + (2*b*c^3*(a + b*ArcCot[x/c])*Log[2 - 2/(1 - (I*c)/x)])/3 - (I/3)*b^2*c^3*PolyL
og[2, -1 + 2/(1 - (I*c)/x)]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 4946

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] - Dist[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 4948

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[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[Sim
plify[(m + 1)/n]]

Rule 4988

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTan[c*x])
^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 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]

Rule 5038

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/(d*f^2), 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] && LtQ[m, -1]

Rule 5044

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*d*(p + 1))), x] + Dist[I/d, Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]

Rubi steps

\begin {align*} \int x^2 \left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{2} b x^2 \left (-2 i a+b \log \left (1-\frac {i c}{x}\right )\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} b^2 x^2 \log ^2\left (1+\frac {i c}{x}\right )\right ) \, dx\\ &=\frac {1}{4} \int x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2 \, dx+\frac {1}{2} b \int x^2 \left (-2 i a+b \log \left (1-\frac {i c}{x}\right )\right ) \log \left (1+\frac {i c}{x}\right ) \, dx-\frac {1}{4} b^2 \int x^2 \log ^2\left (1+\frac {i c}{x}\right ) \, dx\\ &=-\left (\frac {1}{4} \text {Subst}\left (\int \frac {(2 a+i b \log (1-i c x))^2}{x^4} \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{2} b \int \left (-2 i a x^2 \log \left (1+\frac {i c}{x}\right )+b x^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )\right ) \, dx+\frac {1}{4} b^2 \text {Subst}\left (\int \frac {\log ^2(1+i c x)}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{12} x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {i c}{x}\right )-(i a b) \int x^2 \log \left (1+\frac {i c}{x}\right ) \, dx+\frac {1}{2} b^2 \int x^2 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right ) \, dx-\frac {1}{6} (b c) \text {Subst}\left (\int \frac {2 a+i b \log (1-i c x)}{x^3 (1-i c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{6} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+i c x)}{x^3 (1+i c x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{12} x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{3} i a b x^3 \log \left (1+\frac {i c}{x}\right )+\frac {1}{6} b^2 x^3 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{6} (i b) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )^3} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{2} b^2 \int \frac {c x^2 \log \left (1-\frac {i c}{x}\right )}{3 (-c+i x)} \, dx-\frac {1}{2} b^2 \int \frac {c x^2 \log \left (1+\frac {i c}{x}\right )}{3 (-c-i x)} \, dx+\frac {1}{3} (a b c) \int \frac {x}{1+\frac {i c}{x}} \, dx+\frac {1}{6} \left (i b^2 c\right ) \text {Subst}\left (\int \left (\frac {\log (1+i c x)}{x^3}-\frac {i c \log (1+i c x)}{x^2}-\frac {c^2 \log (1+i c x)}{x}+\frac {c^3 \log (1+i c x)}{-i+c x}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{12} x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{3} i a b x^3 \log \left (1+\frac {i c}{x}\right )+\frac {1}{6} b^2 x^3 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{6} (i b) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{\left (-\frac {i}{c}+\frac {i x}{c}\right )^3} \, dx,x,1-\frac {i c}{x}\right )+\frac {1}{6} (b c) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )^2} \, dx,x,1-\frac {i c}{x}\right )+\frac {1}{3} (a b c) \int \frac {x^2}{i c+x} \, dx+\frac {1}{6} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+i c x)}{x^3} \, dx,x,\frac {1}{x}\right )-\frac {1}{6} \left (b^2 c\right ) \int \frac {x^2 \log \left (1-\frac {i c}{x}\right )}{-c+i x} \, dx-\frac {1}{6} \left (b^2 c\right ) \int \frac {x^2 \log \left (1+\frac {i c}{x}\right )}{-c-i x} \, dx+\frac {1}{6} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (1+i c x)}{x^2} \, dx,x,\frac {1}{x}\right )-\frac {1}{6} \left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+i c x)}{x} \, dx,x,\frac {1}{x}\right )+\frac {1}{6} \left (i b^2 c^4\right ) \text {Subst}\left (\int \frac {\log (1+i c x)}{-i+c x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{6} b^2 c^2 x \log \left (1+\frac {i c}{x}\right )-\frac {1}{12} i b^2 c x^2 \log \left (1+\frac {i c}{x}\right )-\frac {1}{3} i a b x^3 \log \left (1+\frac {i c}{x}\right )+\frac {1}{6} b^2 x^3 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {i c}{x}\right )+\frac {1}{6} i b^2 c^3 \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{6} (b c) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{\left (-\frac {i}{c}+\frac {i x}{c}\right )^2} \, dx,x,1-\frac {i c}{x}\right )+\frac {1}{3} (a b c) \int \left (-i c+\frac {i c^2}{c-i x}+x\right ) \, dx-\frac {1}{12} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )^2} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{6} \left (b^2 c\right ) \int \left (-c \log \left (1-\frac {i c}{x}\right )+\frac {c^2 \log \left (1-\frac {i c}{x}\right )}{c-i x}-i x \log \left (1-\frac {i c}{x}\right )\right ) \, dx-\frac {1}{6} \left (b^2 c\right ) \int \left (-c \log \left (1+\frac {i c}{x}\right )+\frac {c^2 \log \left (1+\frac {i c}{x}\right )}{c+i x}+i x \log \left (1+\frac {i c}{x}\right )\right ) \, dx+\frac {1}{6} \left (i b c^2\right ) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x^2 (1+i c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{6} \left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x (1+i c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{6} \left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\frac {i c}{x}\right )\\ &=-\frac {1}{3} i a b c^2 x+\frac {1}{6} a b c x^2+\frac {1}{6} i b c^2 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{6} b^2 c^2 x \log \left (1+\frac {i c}{x}\right )-\frac {1}{12} i b^2 c x^2 \log \left (1+\frac {i c}{x}\right )-\frac {1}{3} i a b x^3 \log \left (1+\frac {i c}{x}\right )+\frac {1}{6} b^2 x^3 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{12} i b^2 c^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{3} a b c^3 \log (c-i x)+\frac {1}{6} i b^2 c^3 \text {Li}_2\left (-\frac {i c}{x}\right )-\frac {1}{12} \left (i b^2 c\right ) \text {Subst}\left (\int \left (-\frac {c^2}{(-1+x)^2}+\frac {c^2}{-1+x}-\frac {c^2}{x}\right ) \, dx,x,1-\frac {i c}{x}\right )+\frac {1}{6} \left (i b^2 c\right ) \int x \log \left (1-\frac {i c}{x}\right ) \, dx-\frac {1}{6} \left (i b^2 c\right ) \int x \log \left (1+\frac {i c}{x}\right ) \, dx+\frac {1}{6} \left (i b c^2\right ) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{-\frac {i}{c}+\frac {i x}{c}} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {i c}{x}+\frac {i c^2}{-i+c x}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{6} \left (b^2 c^2\right ) \int \log \left (1-\frac {i c}{x}\right ) \, dx+\frac {1}{6} \left (b^2 c^2\right ) \int \log \left (1+\frac {i c}{x}\right ) \, dx+\frac {1}{6} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {i}{c}+\frac {i x}{c}} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{6} \left (b c^3\right ) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{x} \, dx,x,1-\frac {i c}{x}\right )+\frac {1}{6} \left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\frac {1}{x}\right )-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log \left (1-\frac {i c}{x}\right )}{c-i x} \, dx-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{c+i x} \, dx+\frac {1}{6} \left (b^2 c^4\right ) \text {Subst}\left (\int \frac {1}{1+i c x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{3} i a b c^2 x+\frac {1}{6} b^2 c^2 x+\frac {1}{6} a b c x^2-\frac {1}{4} i b^2 c^3 \log \left (i-\frac {c}{x}\right )+\frac {1}{12} i b^2 c^3 \log \left (1-\frac {i c}{x}\right )+\frac {1}{6} b^2 c^2 x \log \left (1-\frac {i c}{x}\right )+\frac {1}{12} i b^2 c x^2 \log \left (1-\frac {i c}{x}\right )+\frac {1}{6} i b c^2 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} i c^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{12} x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{6} i b^2 c x^2 \log \left (1+\frac {i c}{x}\right )-\frac {1}{3} i a b x^3 \log \left (1+\frac {i c}{x}\right )+\frac {1}{6} b^2 x^3 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{12} i b^2 c^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{3} a b c^3 \log (c-i x)-\frac {1}{6} i b^2 c^3 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)+\frac {1}{6} i b^2 c^3 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)-\frac {1}{3} a b c^3 \log (x)+\frac {1}{6} i b^2 c^3 \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{12} \left (b^2 c^2\right ) \int \frac {1}{1-\frac {i c}{x}} \, dx+\frac {1}{12} \left (b^2 c^2\right ) \int \frac {1}{1+\frac {i c}{x}} \, dx-\frac {1}{6} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (x)}{-\frac {i}{c}+\frac {i x}{c}} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{6} \left (i b^2 c^3\right ) \int \frac {1}{\left (1-\frac {i c}{x}\right ) x} \, dx+\frac {1}{6} \left (i b^2 c^3\right ) \int \frac {1}{\left (1+\frac {i c}{x}\right ) x} \, dx-\frac {1}{6} \left (b^2 c^4\right ) \int \frac {\log (c-i x)}{\left (1-\frac {i c}{x}\right ) x^2} \, dx-\frac {1}{6} \left (b^2 c^4\right ) \int \frac {\log (c+i x)}{\left (1+\frac {i c}{x}\right ) x^2} \, dx\\ &=-\frac {1}{3} i a b c^2 x+\frac {1}{6} b^2 c^2 x+\frac {1}{6} a b c x^2-\frac {1}{4} i b^2 c^3 \log \left (i-\frac {c}{x}\right )+\frac {1}{12} i b^2 c^3 \log \left (1-\frac {i c}{x}\right )+\frac {1}{6} b^2 c^2 x \log \left (1-\frac {i c}{x}\right )+\frac {1}{12} i b^2 c x^2 \log \left (1-\frac {i c}{x}\right )+\frac {1}{6} i b c^2 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} i c^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{12} x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{6} i b^2 c x^2 \log \left (1+\frac {i c}{x}\right )-\frac {1}{3} i a b x^3 \log \left (1+\frac {i c}{x}\right )+\frac {1}{6} b^2 x^3 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{12} i b^2 c^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{3} a b c^3 \log (c-i x)-\frac {1}{6} i b^2 c^3 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)+\frac {1}{6} i b^2 c^3 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)-\frac {1}{3} a b c^3 \log (x)+\frac {1}{6} i b^2 c^3 \text {Li}_2\left (-\frac {i c}{x}\right )-\frac {1}{6} i b^2 c^3 \text {Li}_2\left (\frac {i c}{x}\right )+\frac {1}{12} \left (b^2 c^2\right ) \int \frac {x}{-i c+x} \, dx+\frac {1}{12} \left (b^2 c^2\right ) \int \frac {x}{i c+x} \, dx-\frac {1}{6} \left (i b^2 c^3\right ) \int \frac {1}{-i c+x} \, dx+\frac {1}{6} \left (i b^2 c^3\right ) \int \frac {1}{i c+x} \, dx-\frac {1}{6} \left (b^2 c^4\right ) \int \left (\frac {\log (c-i x)}{c (c+i x)}+\frac {i \log (c-i x)}{c x}\right ) \, dx-\frac {1}{6} \left (b^2 c^4\right ) \int \left (\frac {\log (c+i x)}{c (c-i x)}-\frac {i \log (c+i x)}{c x}\right ) \, dx\\ &=-\frac {1}{3} i a b c^2 x+\frac {1}{6} b^2 c^2 x+\frac {1}{6} a b c x^2-\frac {1}{4} i b^2 c^3 \log \left (i-\frac {c}{x}\right )+\frac {1}{12} i b^2 c^3 \log \left (1-\frac {i c}{x}\right )+\frac {1}{6} b^2 c^2 x \log \left (1-\frac {i c}{x}\right )+\frac {1}{12} i b^2 c x^2 \log \left (1-\frac {i c}{x}\right )+\frac {1}{6} i b c^2 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} i c^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{12} x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{6} i b^2 c x^2 \log \left (1+\frac {i c}{x}\right )-\frac {1}{3} i a b x^3 \log \left (1+\frac {i c}{x}\right )+\frac {1}{6} b^2 x^3 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{12} i b^2 c^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{3} a b c^3 \log (c-i x)+\frac {1}{6} i b^2 c^3 \log (c-i x)-\frac {1}{6} i b^2 c^3 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)-\frac {1}{6} i b^2 c^3 \log (c+i x)+\frac {1}{6} i b^2 c^3 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)-\frac {1}{3} a b c^3 \log (x)+\frac {1}{6} i b^2 c^3 \text {Li}_2\left (-\frac {i c}{x}\right )-\frac {1}{6} i b^2 c^3 \text {Li}_2\left (\frac {i c}{x}\right )+\frac {1}{12} \left (b^2 c^2\right ) \int \left (1-\frac {c}{c-i x}\right ) \, dx+\frac {1}{12} \left (b^2 c^2\right ) \int \left (1-\frac {c}{c+i x}\right ) \, dx-\frac {1}{6} \left (i b^2 c^3\right ) \int \frac {\log (c-i x)}{x} \, dx+\frac {1}{6} \left (i b^2 c^3\right ) \int \frac {\log (c+i x)}{x} \, dx-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log (c-i x)}{c+i x} \, dx-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log (c+i x)}{c-i x} \, dx\\ &=-\frac {1}{3} i a b c^2 x+\frac {1}{3} b^2 c^2 x+\frac {1}{6} a b c x^2-\frac {1}{4} i b^2 c^3 \log \left (i-\frac {c}{x}\right )+\frac {1}{12} i b^2 c^3 \log \left (1-\frac {i c}{x}\right )+\frac {1}{6} b^2 c^2 x \log \left (1-\frac {i c}{x}\right )+\frac {1}{12} i b^2 c x^2 \log \left (1-\frac {i c}{x}\right )+\frac {1}{6} i b c^2 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} i c^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{12} x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{6} i b^2 c x^2 \log \left (1+\frac {i c}{x}\right )-\frac {1}{3} i a b x^3 \log \left (1+\frac {i c}{x}\right )+\frac {1}{6} b^2 x^3 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{12} i b^2 c^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{3} a b c^3 \log (c-i x)+\frac {1}{12} i b^2 c^3 \log (c-i x)-\frac {1}{6} i b^2 c^3 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)-\frac {1}{12} i b^2 c^3 \log (c+i x)+\frac {1}{6} i b^2 c^3 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)-\frac {1}{6} i b^2 c^3 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x)+\frac {1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right )-\frac {1}{3} a b c^3 \log (x)+\frac {1}{6} i b^2 c^3 \log (c+i x) \log \left (-\frac {i x}{c}\right )-\frac {1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac {i x}{c}\right )+\frac {1}{6} i b^2 c^3 \text {Li}_2\left (-\frac {i c}{x}\right )-\frac {1}{6} i b^2 c^3 \text {Li}_2\left (\frac {i c}{x}\right )-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log \left (\frac {c-i x}{2 c}\right )}{c+i x} \, dx-\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log \left (\frac {c+i x}{2 c}\right )}{c-i x} \, dx+\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log \left (-\frac {i x}{c}\right )}{c+i x} \, dx+\frac {1}{6} \left (b^2 c^3\right ) \int \frac {\log \left (\frac {i x}{c}\right )}{c-i x} \, dx\\ &=-\frac {1}{3} i a b c^2 x+\frac {1}{3} b^2 c^2 x+\frac {1}{6} a b c x^2-\frac {1}{4} i b^2 c^3 \log \left (i-\frac {c}{x}\right )+\frac {1}{12} i b^2 c^3 \log \left (1-\frac {i c}{x}\right )+\frac {1}{6} b^2 c^2 x \log \left (1-\frac {i c}{x}\right )+\frac {1}{12} i b^2 c x^2 \log \left (1-\frac {i c}{x}\right )+\frac {1}{6} i b c^2 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} i c^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{12} x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{6} i b^2 c x^2 \log \left (1+\frac {i c}{x}\right )-\frac {1}{3} i a b x^3 \log \left (1+\frac {i c}{x}\right )+\frac {1}{6} b^2 x^3 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{12} i b^2 c^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{3} a b c^3 \log (c-i x)+\frac {1}{12} i b^2 c^3 \log (c-i x)-\frac {1}{6} i b^2 c^3 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)-\frac {1}{12} i b^2 c^3 \log (c+i x)+\frac {1}{6} i b^2 c^3 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)-\frac {1}{6} i b^2 c^3 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x)+\frac {1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right )-\frac {1}{3} a b c^3 \log (x)+\frac {1}{6} i b^2 c^3 \log (c+i x) \log \left (-\frac {i x}{c}\right )-\frac {1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac {i x}{c}\right )+\frac {1}{6} i b^2 c^3 \text {Li}_2\left (-\frac {i c}{x}\right )-\frac {1}{6} i b^2 c^3 \text {Li}_2\left (\frac {i c}{x}\right )-\frac {1}{6} i b^2 c^3 \text {Li}_2\left (1-\frac {i x}{c}\right )+\frac {1}{6} i b^2 c^3 \text {Li}_2\left (1+\frac {i x}{c}\right )-\frac {1}{6} \left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c-i x\right )+\frac {1}{6} \left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c+i x\right )\\ &=-\frac {1}{3} i a b c^2 x+\frac {1}{3} b^2 c^2 x+\frac {1}{6} a b c x^2-\frac {1}{4} i b^2 c^3 \log \left (i-\frac {c}{x}\right )+\frac {1}{12} i b^2 c^3 \log \left (1-\frac {i c}{x}\right )+\frac {1}{6} b^2 c^2 x \log \left (1-\frac {i c}{x}\right )+\frac {1}{12} i b^2 c x^2 \log \left (1-\frac {i c}{x}\right )+\frac {1}{6} i b c^2 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} b c x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{12} i c^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{12} x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{6} i b^2 c x^2 \log \left (1+\frac {i c}{x}\right )-\frac {1}{3} i a b x^3 \log \left (1+\frac {i c}{x}\right )+\frac {1}{6} b^2 x^3 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{12} i b^2 c^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{3} a b c^3 \log (c-i x)+\frac {1}{12} i b^2 c^3 \log (c-i x)-\frac {1}{6} i b^2 c^3 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)-\frac {1}{12} i b^2 c^3 \log (c+i x)+\frac {1}{6} i b^2 c^3 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)-\frac {1}{6} i b^2 c^3 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x)+\frac {1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right )-\frac {1}{3} a b c^3 \log (x)+\frac {1}{6} i b^2 c^3 \log (c+i x) \log \left (-\frac {i x}{c}\right )-\frac {1}{6} i b^2 c^3 \log (c-i x) \log \left (\frac {i x}{c}\right )+\frac {1}{6} i b^2 c^3 \text {Li}_2\left (\frac {c-i x}{2 c}\right )-\frac {1}{6} i b^2 c^3 \text {Li}_2\left (\frac {c+i x}{2 c}\right )+\frac {1}{6} i b^2 c^3 \text {Li}_2\left (-\frac {i c}{x}\right )-\frac {1}{6} i b^2 c^3 \text {Li}_2\left (\frac {i c}{x}\right )-\frac {1}{6} i b^2 c^3 \text {Li}_2\left (1-\frac {i x}{c}\right )+\frac {1}{6} i b^2 c^3 \text {Li}_2\left (1+\frac {i x}{c}\right )\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 152, normalized size = 1.00 \begin {gather*} \frac {1}{3} \left (b^2 c^2 x+a b c x^2+a^2 x^3+b^2 \left (-i c^3+x^3\right ) \text {ArcTan}\left (\frac {c}{x}\right )^2+b \text {ArcTan}\left (\frac {c}{x}\right ) \left (2 a x^3+b c \left (c^2+x^2\right )+2 b c^3 \log \left (1-e^{2 i \text {ArcTan}\left (\frac {c}{x}\right )}\right )\right )-a b c^3 \log \left (1+\frac {c^2}{x^2}\right )+2 a b c^3 \log \left (\frac {c}{x}\right )-i b^2 c^3 \text {PolyLog}\left (2,e^{2 i \text {ArcTan}\left (\frac {c}{x}\right )}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*c^2*x + a*b*c*x^2 + a^2*x^3 + b^2*((-I)*c^3 + x^3)*ArcTan[c/x]^2 + b*ArcTan[c/x]*(2*a*x^3 + b*c*(c^2 + x^
2) + 2*b*c^3*Log[1 - E^((2*I)*ArcTan[c/x])]) - a*b*c^3*Log[1 + c^2/x^2] + 2*a*b*c^3*Log[c/x] - I*b^2*c^3*PolyL
og[2, E^((2*I)*ArcTan[c/x])])/3

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (134 ) = 268\).
time = 0.38, size = 412, normalized size = 2.71

method	result	size
derivativedivides	\(-c^{3} \left (-\frac {a^{2} x^{3}}{3 c^{3}}-\frac {b^{2} x^{3} \arctan \left (\frac {c}{x}\right )^{2}}{3 c^{3}}+\frac {b^{2} \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{3}-\frac {b^{2} \arctan \left (\frac {c}{x}\right ) x^{2}}{3 c^{2}}-\frac {2 b^{2} \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )}{3}+\frac {i b^{2} \dilog \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{6}-\frac {i b^{2} \ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )}{6}-\frac {i b^{2} \ln \left (\frac {c}{x}-i\right )^{2}}{12}-\frac {i b^{2} \dilog \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )}{6}-\frac {i b^{2} \ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{6}+\frac {i b^{2} \dilog \left (1-\frac {i c}{x}\right )}{3}+\frac {i b^{2} \ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{6}+\frac {i b^{2} \ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{6}-\frac {b^{2} \arctan \left (\frac {c}{x}\right )}{3}-\frac {b^{2} x}{3 c}+\frac {i b^{2} \ln \left (\frac {c}{x}+i\right )^{2}}{12}+\frac {i b^{2} \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{3}-\frac {i b^{2} \dilog \left (1+\frac {i c}{x}\right )}{3}-\frac {i b^{2} \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{3}-\frac {2 a b \,x^{3} \arctan \left (\frac {c}{x}\right )}{3 c^{3}}+\frac {a b \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{3}-\frac {a b \,x^{2}}{3 c^{2}}-\frac {2 a b \ln \left (\frac {c}{x}\right )}{3}\right )\)	\(412\)
default	\(-c^{3} \left (-\frac {a^{2} x^{3}}{3 c^{3}}-\frac {b^{2} x^{3} \arctan \left (\frac {c}{x}\right )^{2}}{3 c^{3}}+\frac {b^{2} \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{3}-\frac {b^{2} \arctan \left (\frac {c}{x}\right ) x^{2}}{3 c^{2}}-\frac {2 b^{2} \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )}{3}+\frac {i b^{2} \dilog \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{6}-\frac {i b^{2} \ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )}{6}-\frac {i b^{2} \ln \left (\frac {c}{x}-i\right )^{2}}{12}-\frac {i b^{2} \dilog \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )}{6}-\frac {i b^{2} \ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{6}+\frac {i b^{2} \dilog \left (1-\frac {i c}{x}\right )}{3}+\frac {i b^{2} \ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )}{6}+\frac {i b^{2} \ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{6}-\frac {b^{2} \arctan \left (\frac {c}{x}\right )}{3}-\frac {b^{2} x}{3 c}+\frac {i b^{2} \ln \left (\frac {c}{x}+i\right )^{2}}{12}+\frac {i b^{2} \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{3}-\frac {i b^{2} \dilog \left (1+\frac {i c}{x}\right )}{3}-\frac {i b^{2} \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{3}-\frac {2 a b \,x^{3} \arctan \left (\frac {c}{x}\right )}{3 c^{3}}+\frac {a b \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{3}-\frac {a b \,x^{2}}{3 c^{2}}-\frac {2 a b \ln \left (\frac {c}{x}\right )}{3}\right )\)	\(412\)
risch	\(\text {Expression too large to display}\)	\(25023\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^3*(-1/3*a^2/c^3*x^3-1/3*b^2/c^3*x^3*arctan(c/x)^2+1/3*b^2*arctan(c/x)*ln(1+c^2/x^2)-1/3*b^2*arctan(c/x)/c^2
*x^2-2/3*b^2*ln(c/x)*arctan(c/x)-1/6*I*b^2*dilog(-1/2*I*(c/x+I))-1/6*I*b^2*ln(c/x-I)*ln(-1/2*I*(c/x+I))-1/12*I
*b^2*ln(c/x-I)^2+1/3*I*b^2*dilog(1-I*c/x)+1/6*I*b^2*dilog(1/2*I*(c/x-I))-1/6*I*b^2*ln(c/x+I)*ln(1+c^2/x^2)+1/6
*I*b^2*ln(c/x+I)*ln(1/2*I*(c/x-I))+1/6*I*b^2*ln(c/x-I)*ln(1+c^2/x^2)-1/3*b^2*arctan(c/x)-1/3*b^2/c*x+1/12*I*b^
2*ln(c/x+I)^2+1/3*I*b^2*ln(c/x)*ln(1-I*c/x)-1/3*I*b^2*ln(c/x)*ln(1+I*c/x)-1/3*I*b^2*dilog(1+I*c/x)-2/3*a*b/c^3
*x^3*arctan(c/x)+1/3*a*b*ln(1+c^2/x^2)-1/3*a*b/c^2*x^2-2/3*a*b*ln(c/x))

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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]

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

[Out]

1/3*a^2*x^3 + 1/3*(2*x^3*arctan(c/x) - (c^2*log(c^2 + x^2) - x^2)*c)*a*b + 1/48*(4*x^3*arctan2(c, x)^2 - x^3*l
og(c^2 + x^2)^2 + 48*integrate(1/48*(36*c^2*x^2*arctan2(c, x)^2 + 36*x^4*arctan2(c, x)^2 + 8*c*x^3*arctan2(c,
x) + 4*x^4*log(c^2 + x^2) + 3*(c^2*x^2 + x^4)*log(c^2 + x^2)^2)/(c^2 + x^2), x))*b^2

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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]

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

[Out]

integral(b^2*x^2*arctan(c/x)^2 + 2*a*b*x^2*arctan(c/x) + a^2*x^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \operatorname {atan}{\left (\frac {c}{x} \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a + b*atan(c/x))**2, x)

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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]

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

[Out]

integrate((b*arctan(c/x) + a)^2*x^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {atan}\left (\frac {c}{x}\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*atan(c/x))^2,x)

[Out]

int(x^2*(a + b*atan(c/x))^2, x)

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