\(\int \frac {(c+a^2 c x^2)^3 \arctan (a x)^2}{x^2} \, dx\) [279]

Optimal result

Integrand size = 22, antiderivative size = 251 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^2} \, dx=\frac {7}{10} a^2 c^3 x+\frac {1}{30} a^4 c^3 x^3-\frac {7}{10} a c^3 \arctan (a x)-\frac {4}{5} a^3 c^3 x^2 \arctan (a x)-\frac {1}{10} a^5 c^3 x^4 \arctan (a x)+\frac {6}{5} i a c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{x}+3 a^2 c^3 x \arctan (a x)^2+a^4 c^3 x^3 \arctan (a x)^2+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^2+\frac {22}{5} a c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {11}{5} i a c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right ) \]

[Out]

7/10*a^2*c^3*x+1/30*a^4*c^3*x^3-7/10*a*c^3*arctan(a*x)-4/5*a^3*c^3*x^2*arctan(a*x)-1/10*a^5*c^3*x^4*arctan(a*x
)+6/5*I*a*c^3*arctan(a*x)^2-c^3*arctan(a*x)^2/x+3*a^2*c^3*x*arctan(a*x)^2+a^4*c^3*x^3*arctan(a*x)^2+1/5*a^6*c^
3*x^5*arctan(a*x)^2+22/5*a*c^3*arctan(a*x)*ln(2/(1+I*a*x))+2*a*c^3*arctan(a*x)*ln(2-2/(1-I*a*x))-I*a*c^3*polyl
og(2,-1+2/(1-I*a*x))+11/5*I*a*c^3*polylog(2,1-2/(1+I*a*x))

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {5068, 4930, 5040, 4964, 2449, 2352, 4946, 5044, 4988, 2497, 5036, 327, 209, 308} \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^2} \, dx=\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^2-\frac {1}{10} a^5 c^3 x^4 \arctan (a x)+a^4 c^3 x^3 \arctan (a x)^2+\frac {1}{30} a^4 c^3 x^3-\frac {4}{5} a^3 c^3 x^2 \arctan (a x)+3 a^2 c^3 x \arctan (a x)^2+\frac {7}{10} a^2 c^3 x+\frac {6}{5} i a c^3 \arctan (a x)^2-\frac {7}{10} a c^3 \arctan (a x)-\frac {c^3 \arctan (a x)^2}{x}+\frac {22}{5} a c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )+\frac {11}{5} i a c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right ) \]

[In]

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

[Out]

(7*a^2*c^3*x)/10 + (a^4*c^3*x^3)/30 - (7*a*c^3*ArcTan[a*x])/10 - (4*a^3*c^3*x^2*ArcTan[a*x])/5 - (a^5*c^3*x^4*
ArcTan[a*x])/10 + ((6*I)/5)*a*c^3*ArcTan[a*x]^2 - (c^3*ArcTan[a*x]^2)/x + 3*a^2*c^3*x*ArcTan[a*x]^2 + a^4*c^3*
x^3*ArcTan[a*x]^2 + (a^6*c^3*x^5*ArcTan[a*x]^2)/5 + (22*a*c^3*ArcTan[a*x]*Log[2/(1 + I*a*x)])/5 + 2*a*c^3*ArcT
an[a*x]*Log[2 - 2/(1 - I*a*x)] - I*a*c^3*PolyLog[2, -1 + 2/(1 - I*a*x)] + ((11*I)/5)*a*c^3*PolyLog[2, 1 - 2/(1
+ I*a*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 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 327

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-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]

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 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[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 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 4964

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] + Dist[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]

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 5036

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

Rule 5040

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*e*(p + 1))), x] - Dist[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]

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]

Rule 5068

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e,
c^2*d] && IGtQ[p, 0] && IGtQ[q, 1] && (EqQ[p, 1] || IntegerQ[m])

Rubi steps \begin{align*} \text {integral}& = \int \left (3 a^2 c^3 \arctan (a x)^2+\frac {c^3 \arctan (a x)^2}{x^2}+3 a^4 c^3 x^2 \arctan (a x)^2+a^6 c^3 x^4 \arctan (a x)^2\right ) \, dx \\ & = c^3 \int \frac {\arctan (a x)^2}{x^2} \, dx+\left (3 a^2 c^3\right ) \int \arctan (a x)^2 \, dx+\left (3 a^4 c^3\right ) \int x^2 \arctan (a x)^2 \, dx+\left (a^6 c^3\right ) \int x^4 \arctan (a x)^2 \, dx \\ & = -\frac {c^3 \arctan (a x)^2}{x}+3 a^2 c^3 x \arctan (a x)^2+a^4 c^3 x^3 \arctan (a x)^2+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^2+\left (2 a c^3\right ) \int \frac {\arctan (a x)}{x \left (1+a^2 x^2\right )} \, dx-\left (6 a^3 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx-\left (2 a^5 c^3\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (2 a^7 c^3\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = 2 i a c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{x}+3 a^2 c^3 x \arctan (a x)^2+a^4 c^3 x^3 \arctan (a x)^2+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^2+\left (2 i a c^3\right ) \int \frac {\arctan (a x)}{x (i+a x)} \, dx+\left (6 a^2 c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx-\left (2 a^3 c^3\right ) \int x \arctan (a x) \, dx+\left (2 a^3 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (2 a^5 c^3\right ) \int x^3 \arctan (a x) \, dx+\frac {1}{5} \left (2 a^5 c^3\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -a^3 c^3 x^2 \arctan (a x)-\frac {1}{10} a^5 c^3 x^4 \arctan (a x)+i a c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{x}+3 a^2 c^3 x \arctan (a x)^2+a^4 c^3 x^3 \arctan (a x)^2+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^2+6 a c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\left (2 a^2 c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx-\left (2 a^2 c^3\right ) \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (6 a^2 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{5} \left (2 a^3 c^3\right ) \int x \arctan (a x) \, dx-\frac {1}{5} \left (2 a^3 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx+\left (a^4 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx+\frac {1}{10} \left (a^6 c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx \\ & = a^2 c^3 x-\frac {4}{5} a^3 c^3 x^2 \arctan (a x)-\frac {1}{10} a^5 c^3 x^4 \arctan (a x)+\frac {6}{5} i a c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{x}+3 a^2 c^3 x \arctan (a x)^2+a^4 c^3 x^3 \arctan (a x)^2+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^2+4 a c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\left (6 i a c^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )+\frac {1}{5} \left (2 a^2 c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx-\left (a^2 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx+\left (2 a^2 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac {1}{5} \left (a^4 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx+\frac {1}{10} \left (a^6 c^3\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = \frac {7}{10} a^2 c^3 x+\frac {1}{30} a^4 c^3 x^3-a c^3 \arctan (a x)-\frac {4}{5} a^3 c^3 x^2 \arctan (a x)-\frac {1}{10} a^5 c^3 x^4 \arctan (a x)+\frac {6}{5} i a c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{x}+3 a^2 c^3 x \arctan (a x)^2+a^4 c^3 x^3 \arctan (a x)^2+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^2+\frac {22}{5} a c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+3 i a c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\left (2 i a c^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )+\frac {1}{10} \left (a^2 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^2 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx-\frac {1}{5} \left (2 a^2 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx \\ & = \frac {7}{10} a^2 c^3 x+\frac {1}{30} a^4 c^3 x^3-\frac {7}{10} a c^3 \arctan (a x)-\frac {4}{5} a^3 c^3 x^2 \arctan (a x)-\frac {1}{10} a^5 c^3 x^4 \arctan (a x)+\frac {6}{5} i a c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{x}+3 a^2 c^3 x \arctan (a x)^2+a^4 c^3 x^3 \arctan (a x)^2+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^2+\frac {22}{5} a c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+2 i a c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {1}{5} \left (2 i a c^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right ) \\ & = \frac {7}{10} a^2 c^3 x+\frac {1}{30} a^4 c^3 x^3-\frac {7}{10} a c^3 \arctan (a x)-\frac {4}{5} a^3 c^3 x^2 \arctan (a x)-\frac {1}{10} a^5 c^3 x^4 \arctan (a x)+\frac {6}{5} i a c^3 \arctan (a x)^2-\frac {c^3 \arctan (a x)^2}{x}+3 a^2 c^3 x \arctan (a x)^2+a^4 c^3 x^3 \arctan (a x)^2+\frac {1}{5} a^6 c^3 x^5 \arctan (a x)^2+\frac {22}{5} a c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )+2 a c^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-i a c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+\frac {11}{5} i a c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.80 \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^2} \, dx=\frac {c^3 \left (21 a^2 x^2+a^4 x^4-21 a x \arctan (a x)-24 a^3 x^3 \arctan (a x)-3 a^5 x^5 \arctan (a x)-30 \arctan (a x)^2-96 i a x \arctan (a x)^2+90 a^2 x^2 \arctan (a x)^2+30 a^4 x^4 \arctan (a x)^2+6 a^6 x^6 \arctan (a x)^2+60 a x \arctan (a x) \log \left (1-e^{2 i \arctan (a x)}\right )+132 a x \arctan (a x) \log \left (1+e^{2 i \arctan (a x)}\right )-66 i a x \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )-30 i a x \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )\right )}{30 x} \]

[In]

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

[Out]

(c^3*(21*a^2*x^2 + a^4*x^4 - 21*a*x*ArcTan[a*x] - 24*a^3*x^3*ArcTan[a*x] - 3*a^5*x^5*ArcTan[a*x] - 30*ArcTan[a
*x]^2 - (96*I)*a*x*ArcTan[a*x]^2 + 90*a^2*x^2*ArcTan[a*x]^2 + 30*a^4*x^4*ArcTan[a*x]^2 + 6*a^6*x^6*ArcTan[a*x]
^2 + 60*a*x*ArcTan[a*x]*Log[1 - E^((2*I)*ArcTan[a*x])] + 132*a*x*ArcTan[a*x]*Log[1 + E^((2*I)*ArcTan[a*x])] -
(66*I)*a*x*PolyLog[2, -E^((2*I)*ArcTan[a*x])] - (30*I)*a*x*PolyLog[2, E^((2*I)*ArcTan[a*x])]))/(30*x)

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.28

method	result	size
derivativedivides	\(a \left (\frac {a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}+3 a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {c^{3} \arctan \left (a x \right )^{2}}{a x}-\frac {2 c^{3} \left (\frac {\arctan \left (a x \right ) a^{4} x^{4}}{4}+2 a^{2} \arctan \left (a x \right ) x^{2}+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-5 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {a^{3} x^{3}}{12}-\frac {7 a x}{4}+\frac {7 \arctan \left (a x \right )}{4}-\frac {5 i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {5 i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {5 i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {5 i \operatorname {dilog}\left (-i a x +1\right )}{2}+4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{5}\right )\)	\(321\)
default	\(a \left (\frac {a^{5} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )^{2}+3 a \,c^{3} x \arctan \left (a x \right )^{2}-\frac {c^{3} \arctan \left (a x \right )^{2}}{a x}-\frac {2 c^{3} \left (\frac {\arctan \left (a x \right ) a^{4} x^{4}}{4}+2 a^{2} \arctan \left (a x \right ) x^{2}+8 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-5 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {a^{3} x^{3}}{12}-\frac {7 a x}{4}+\frac {7 \arctan \left (a x \right )}{4}-\frac {5 i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {5 i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {5 i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {5 i \operatorname {dilog}\left (-i a x +1\right )}{2}+4 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-4 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{5}\right )\)	\(321\)
parts	\(\frac {a^{6} c^{3} x^{5} \arctan \left (a x \right )^{2}}{5}+a^{4} c^{3} x^{3} \arctan \left (a x \right )^{2}+3 a^{2} c^{3} x \arctan \left (a x \right )^{2}-\frac {c^{3} \arctan \left (a x \right )^{2}}{x}-\frac {2 c^{3} \left (\frac {\arctan \left (a x \right ) a^{5} x^{4}}{4}+2 \arctan \left (a x \right ) x^{2} a^{3}+8 a \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-5 a \arctan \left (a x \right ) \ln \left (a x \right )-\frac {a \left (\frac {a^{3} x^{3}}{3}+7 a x -7 \arctan \left (a x \right )+10 i \ln \left (a x \right ) \ln \left (i a x +1\right )-10 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+10 i \operatorname {dilog}\left (i a x +1\right )-10 i \operatorname {dilog}\left (-i a x +1\right )-16 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+16 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{4}\right )}{5}\)	\(324\)

[In]

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

[Out]

a*(1/5*a^5*c^3*x^5*arctan(a*x)^2+a^3*c^3*x^3*arctan(a*x)^2+3*a*c^3*x*arctan(a*x)^2-c^3*arctan(a*x)^2/a/x-2/5*c
^3*(1/4*arctan(a*x)*a^4*x^4+2*a^2*arctan(a*x)*x^2+8*arctan(a*x)*ln(a^2*x^2+1)-5*arctan(a*x)*ln(a*x)-1/12*a^3*x
^3-7/4*a*x+7/4*arctan(a*x)-5/2*I*ln(a*x)*ln(1+I*a*x)+5/2*I*ln(a*x)*ln(1-I*a*x)-5/2*I*dilog(1+I*a*x)+5/2*I*dilo
g(1-I*a*x)+4*I*(ln(a*x-I)*ln(a^2*x^2+1)-dilog(-1/2*I*(I+a*x))-ln(a*x-I)*ln(-1/2*I*(I+a*x))-1/2*ln(a*x-I)^2)-4*
I*(ln(I+a*x)*ln(a^2*x^2+1)-dilog(1/2*I*(a*x-I))-ln(I+a*x)*ln(1/2*I*(a*x-I))-1/2*ln(I+a*x)^2)))

Fricas [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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^2, x)

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^2} \, dx=c^{3} \left (\int 3 a^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{2}}\, dx + \int 3 a^{4} x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{4} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]

[In]

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

[Out]

c**3*(Integral(3*a**2*atan(a*x)**2, x) + Integral(atan(a*x)**2/x**2, x) + Integral(3*a**4*x**2*atan(a*x)**2, x
) + Integral(a**6*x**4*atan(a*x)**2, x))

Maxima [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

1/80*(4*(a^6*c^3*x^6 + 5*a^4*c^3*x^4 + 15*a^2*c^3*x^2 - 5*c^3)*arctan(a*x)^2 - (a^6*c^3*x^6 + 5*a^4*c^3*x^4 +
15*a^2*c^3*x^2 - 5*c^3)*log(a^2*x^2 + 1)^2 + 80*(60*a^8*c^3*integrate(1/80*x^8*arctan(a*x)^2/(a^2*x^4 + x^2),
x) + 5*a^8*c^3*integrate(1/80*x^8*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 4*a^8*c^3*integrate(1/80*x^8*log(a^
2*x^2 + 1)/(a^2*x^4 + x^2), x) - 8*a^7*c^3*integrate(1/80*x^7*arctan(a*x)/(a^2*x^4 + x^2), x) + 240*a^6*c^3*in
tegrate(1/80*x^6*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 20*a^6*c^3*integrate(1/80*x^6*log(a^2*x^2 + 1)^2/(a^2*x^4
+ x^2), x) + 20*a^6*c^3*integrate(1/80*x^6*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 40*a^5*c^3*integrate(1/80*x
^5*arctan(a*x)/(a^2*x^4 + x^2), x) + 360*a^4*c^3*integrate(1/80*x^4*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 30*a^4
*c^3*integrate(1/80*x^4*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 60*a^4*c^3*integrate(1/80*x^4*log(a^2*x^2 + 1
)/(a^2*x^4 + x^2), x) + a*c^3*arctan(a*x)^3 - 120*a^3*c^3*integrate(1/80*x^3*arctan(a*x)/(a^2*x^4 + x^2), x) +
20*a^2*c^3*integrate(1/80*x^2*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 20*a^2*c^3*integrate(1/80*x^2*log(a^2*
x^2 + 1)/(a^2*x^4 + x^2), x) + 40*a*c^3*integrate(1/80*x*arctan(a*x)/(a^2*x^4 + x^2), x) + 60*c^3*integrate(1/
80*arctan(a*x)^2/(a^2*x^4 + x^2), x) + 5*c^3*integrate(1/80*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x))*x)/x

Giac [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^3 \arctan (a x)^2}{x^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3}{x^2} \,d x \]

[In]

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

[Out]

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