\(\int x^2 (c+a^2 c x^2)^2 \arctan (a x)^2 \, dx\) [267]

Optimal result

Integrand size = 22, antiderivative size = 225 \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=-\frac {c^2 x}{210 a^2}+\frac {17 c^2 x^3}{630}+\frac {1}{105} a^2 c^2 x^5+\frac {c^2 \arctan (a x)}{210 a^3}-\frac {8 c^2 x^2 \arctan (a x)}{105 a}-\frac {9}{70} a c^2 x^4 \arctan (a x)-\frac {1}{21} a^3 c^2 x^6 \arctan (a x)-\frac {8 i c^2 \arctan (a x)^2}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^2+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^2+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^2-\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {8 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{105 a^3} \]

[Out]

-1/210*c^2*x/a^2+17/630*c^2*x^3+1/105*a^2*c^2*x^5+1/210*c^2*arctan(a*x)/a^3-8/105*c^2*x^2*arctan(a*x)/a-9/70*a
*c^2*x^4*arctan(a*x)-1/21*a^3*c^2*x^6*arctan(a*x)-8/105*I*c^2*arctan(a*x)^2/a^3+1/3*c^2*x^3*arctan(a*x)^2+2/5*
a^2*c^2*x^5*arctan(a*x)^2+1/7*a^4*c^2*x^7*arctan(a*x)^2-16/105*c^2*arctan(a*x)*ln(2/(1+I*a*x))/a^3-8/105*I*c^2
*polylog(2,1-2/(1+I*a*x))/a^3

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 44, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5068, 4946, 5036, 327, 209, 5040, 4964, 2449, 2352, 308} \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^2-\frac {1}{21} a^3 c^2 x^6 \arctan (a x)-\frac {8 i c^2 \arctan (a x)^2}{105 a^3}+\frac {c^2 \arctan (a x)}{210 a^3}-\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {8 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{105 a^3}+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^2+\frac {1}{105} a^2 c^2 x^5-\frac {c^2 x}{210 a^2}-\frac {9}{70} a c^2 x^4 \arctan (a x)+\frac {1}{3} c^2 x^3 \arctan (a x)^2-\frac {8 c^2 x^2 \arctan (a x)}{105 a}+\frac {17 c^2 x^3}{630} \]

[In]

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

[Out]

-1/210*(c^2*x)/a^2 + (17*c^2*x^3)/630 + (a^2*c^2*x^5)/105 + (c^2*ArcTan[a*x])/(210*a^3) - (8*c^2*x^2*ArcTan[a*
x])/(105*a) - (9*a*c^2*x^4*ArcTan[a*x])/70 - (a^3*c^2*x^6*ArcTan[a*x])/21 - (((8*I)/105)*c^2*ArcTan[a*x]^2)/a^
3 + (c^2*x^3*ArcTan[a*x]^2)/3 + (2*a^2*c^2*x^5*ArcTan[a*x]^2)/5 + (a^4*c^2*x^7*ArcTan[a*x]^2)/7 - (16*c^2*ArcT
an[a*x]*Log[2/(1 + I*a*x)])/(105*a^3) - (((8*I)/105)*c^2*PolyLog[2, 1 - 2/(1 + I*a*x)])/a^3

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 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 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 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 (c^2 x^2 \arctan (a x)^2+2 a^2 c^2 x^4 \arctan (a x)^2+a^4 c^2 x^6 \arctan (a x)^2\right ) \, dx \\ & = c^2 \int x^2 \arctan (a x)^2 \, dx+\left (2 a^2 c^2\right ) \int x^4 \arctan (a x)^2 \, dx+\left (a^4 c^2\right ) \int x^6 \arctan (a x)^2 \, dx \\ & = \frac {1}{3} c^2 x^3 \arctan (a x)^2+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^2+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^2-\frac {1}{3} \left (2 a c^2\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (4 a^3 c^2\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{7} \left (2 a^5 c^2\right ) \int \frac {x^7 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {1}{3} c^2 x^3 \arctan (a x)^2+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^2+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^2-\frac {\left (2 c^2\right ) \int x \arctan (a x) \, dx}{3 a}+\frac {\left (2 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{3 a}-\frac {1}{5} \left (4 a c^2\right ) \int x^3 \arctan (a x) \, dx+\frac {1}{5} \left (4 a c^2\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{7} \left (2 a^3 c^2\right ) \int x^5 \arctan (a x) \, dx+\frac {1}{7} \left (2 a^3 c^2\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = -\frac {c^2 x^2 \arctan (a x)}{3 a}-\frac {1}{5} a c^2 x^4 \arctan (a x)-\frac {1}{21} a^3 c^2 x^6 \arctan (a x)-\frac {i c^2 \arctan (a x)^2}{3 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^2+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^2+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^2+\frac {1}{3} c^2 \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {\left (2 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{3 a^2}+\frac {\left (4 c^2\right ) \int x \arctan (a x) \, dx}{5 a}-\frac {\left (4 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{7} \left (2 a c^2\right ) \int x^3 \arctan (a x) \, dx-\frac {1}{7} \left (2 a c^2\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^2 c^2\right ) \int \frac {x^4}{1+a^2 x^2} \, dx+\frac {1}{21} \left (a^4 c^2\right ) \int \frac {x^6}{1+a^2 x^2} \, dx \\ & = \frac {c^2 x}{3 a^2}+\frac {c^2 x^2 \arctan (a x)}{15 a}-\frac {9}{70} a c^2 x^4 \arctan (a x)-\frac {1}{21} a^3 c^2 x^6 \arctan (a x)+\frac {i c^2 \arctan (a x)^2}{15 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^2+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^2+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^2-\frac {2 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {1}{5} \left (2 c^2\right ) \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{3 a^2}+\frac {\left (2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}+\frac {\left (4 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{5 a^2}-\frac {\left (2 c^2\right ) \int x \arctan (a x) \, dx}{7 a}+\frac {\left (2 c^2\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{7 a}-\frac {1}{14} \left (a^2 c^2\right ) \int \frac {x^4}{1+a^2 x^2} \, dx+\frac {1}{5} \left (a^2 c^2\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 {1}{21} \left (a^4 c^2\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = -\frac {23 c^2 x}{105 a^2}+\frac {16 c^2 x^3}{315}+\frac {1}{105} a^2 c^2 x^5-\frac {c^2 \arctan (a x)}{3 a^3}-\frac {8 c^2 x^2 \arctan (a x)}{105 a}-\frac {9}{70} a c^2 x^4 \arctan (a x)-\frac {1}{21} a^3 c^2 x^6 \arctan (a x)-\frac {8 i c^2 \arctan (a x)^2}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^2+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^2+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^2+\frac {2 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^3}+\frac {1}{7} c^2 \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {\left (2 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a^3}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{21 a^2}+\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (2 c^2\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{7 a^2}+\frac {\left (2 c^2\right ) \int \frac {1}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (4 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}-\frac {1}{14} \left (a^2 c^2\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 {c^2 x}{210 a^2}+\frac {17 c^2 x^3}{630}+\frac {1}{105} a^2 c^2 x^5+\frac {23 c^2 \arctan (a x)}{105 a^3}-\frac {8 c^2 x^2 \arctan (a x)}{105 a}-\frac {9}{70} a c^2 x^4 \arctan (a x)-\frac {1}{21} a^3 c^2 x^6 \arctan (a x)-\frac {8 i c^2 \arctan (a x)^2}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^2+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^2+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^2-\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{3 a^3}+\frac {\left (4 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^3}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{14 a^2}-\frac {c^2 \int \frac {1}{1+a^2 x^2} \, dx}{7 a^2}+\frac {\left (2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2} \\ & = -\frac {c^2 x}{210 a^2}+\frac {17 c^2 x^3}{630}+\frac {1}{105} a^2 c^2 x^5+\frac {c^2 \arctan (a x)}{210 a^3}-\frac {8 c^2 x^2 \arctan (a x)}{105 a}-\frac {9}{70} a c^2 x^4 \arctan (a x)-\frac {1}{21} a^3 c^2 x^6 \arctan (a x)-\frac {8 i c^2 \arctan (a x)^2}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^2+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^2+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^2-\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{105 a^3}+\frac {i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{15 a^3}-\frac {\left (2 i c^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{7 a^3} \\ & = -\frac {c^2 x}{210 a^2}+\frac {17 c^2 x^3}{630}+\frac {1}{105} a^2 c^2 x^5+\frac {c^2 \arctan (a x)}{210 a^3}-\frac {8 c^2 x^2 \arctan (a x)}{105 a}-\frac {9}{70} a c^2 x^4 \arctan (a x)-\frac {1}{21} a^3 c^2 x^6 \arctan (a x)-\frac {8 i c^2 \arctan (a x)^2}{105 a^3}+\frac {1}{3} c^2 x^3 \arctan (a x)^2+\frac {2}{5} a^2 c^2 x^5 \arctan (a x)^2+\frac {1}{7} a^4 c^2 x^7 \arctan (a x)^2-\frac {16 c^2 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{105 a^3}-\frac {8 i c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{105 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.59 \[ \int x^2 \left (c+a^2 c x^2\right )^2 \arctan (a x)^2 \, dx=\frac {c^2 \left (a x \left (-3+17 a^2 x^2+6 a^4 x^4\right )+6 \left (8 i+35 a^3 x^3+42 a^5 x^5+15 a^7 x^7\right ) \arctan (a x)^2-3 \arctan (a x) \left (-1+16 a^2 x^2+27 a^4 x^4+10 a^6 x^6+32 \log \left (1+e^{2 i \arctan (a x)}\right )\right )+48 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{630 a^3} \]

[In]

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

[Out]

(c^2*(a*x*(-3 + 17*a^2*x^2 + 6*a^4*x^4) + 6*(8*I + 35*a^3*x^3 + 42*a^5*x^5 + 15*a^7*x^7)*ArcTan[a*x]^2 - 3*Arc
Tan[a*x]*(-1 + 16*a^2*x^2 + 27*a^4*x^4 + 10*a^6*x^6 + 32*Log[1 + E^((2*I)*ArcTan[a*x])]) + (48*I)*PolyLog[2, -
E^((2*I)*ArcTan[a*x])]))/(630*a^3)

Maple [A] (verified)

Time = 2.31 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.19

method	result	size
parts	\(\frac {a^{4} c^{2} x^{7} \arctan \left (a x \right )^{2}}{7}+\frac {2 a^{2} c^{2} x^{5} \arctan \left (a x \right )^{2}}{5}+\frac {c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}-\frac {2 c^{2} \left (\frac {5 a^{3} \arctan \left (a x \right ) x^{6}}{2}+\frac {27 a \arctan \left (a x \right ) x^{4}}{4}+\frac {4 \arctan \left (a x \right ) x^{2}}{a}-\frac {4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{a^{3}}-\frac {2 a^{5} x^{5}+\frac {17 a^{3} x^{3}}{3}-a x +\arctan \left (a x \right )+8 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )-8 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{4 a^{3}}\right )}{105}\)	\(267\)
derivativedivides	\(\frac {\frac {c^{2} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {2 c^{2} \arctan \left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {a^{3} c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}-\frac {2 c^{2} \left (\frac {5 a^{6} \arctan \left (a x \right ) x^{6}}{2}+\frac {27 \arctan \left (a x \right ) a^{4} x^{4}}{4}+4 a^{2} \arctan \left (a x \right ) x^{2}-4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{5} x^{5}}{2}-\frac {17 a^{3} x^{3}}{12}+\frac {a x}{4}-\frac {\arctan \left (a x \right )}{4}-2 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+2 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{105}}{a^{3}}\)	\(269\)
default	\(\frac {\frac {c^{2} \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {2 c^{2} \arctan \left (a x \right )^{2} a^{5} x^{5}}{5}+\frac {a^{3} c^{2} x^{3} \arctan \left (a x \right )^{2}}{3}-\frac {2 c^{2} \left (\frac {5 a^{6} \arctan \left (a x \right ) x^{6}}{2}+\frac {27 \arctan \left (a x \right ) a^{4} x^{4}}{4}+4 a^{2} \arctan \left (a x \right ) x^{2}-4 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {a^{5} x^{5}}{2}-\frac {17 a^{3} x^{3}}{12}+\frac {a x}{4}-\frac {\arctan \left (a x \right )}{4}-2 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+2 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )\right )}{105}}{a^{3}}\)	\(269\)
risch	\(-\frac {c^{2} a^{2} \ln \left (i a x +1\right )^{2} x^{5}}{10}-\frac {c^{2} a^{4} \ln \left (i a x +1\right )^{2} x^{7}}{28}-\frac {c^{2} a^{4} \ln \left (-i a x +1\right )^{2} x^{7}}{28}-\frac {c^{2} a^{2} \ln \left (-i a x +1\right )^{2} x^{5}}{10}-\frac {c^{2} x}{210 a^{2}}+\frac {a^{2} c^{2} x^{5}}{105}+\frac {c^{2} \arctan \left (a x \right )}{210 a^{3}}+\frac {17 c^{2} x^{3}}{630}-\frac {177151 i c^{2}}{2315250 a^{3}}-\frac {c^{2} \ln \left (i a x +1\right )^{2} x^{3}}{12}-\frac {c^{2} \ln \left (-i a x +1\right )^{2} x^{3}}{12}+\frac {c^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{6}-\frac {8 i c^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{105 a^{3}}-\frac {2 i c^{2} \ln \left (i a x +1\right )^{2}}{105 a^{3}}+\frac {2 i c^{2} \ln \left (-i a x +1\right )^{2}}{105 a^{3}}+\frac {8 i c^{2} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{105 a^{3}}-\frac {8 i c^{2} \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{105 a^{3}}+\frac {c^{2} a^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{5}}{5}+\frac {c^{2} a^{4} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{7}}{14}+\frac {9 i c^{2} a \ln \left (i a x +1\right ) x^{4}}{140}+\frac {4 i c^{2} \ln \left (i a x +1\right ) x^{2}}{105 a}-\frac {i c^{2} a^{3} \ln \left (-i a x +1\right ) x^{6}}{42}-\frac {9 i c^{2} a \ln \left (-i a x +1\right ) x^{4}}{140}-\frac {4 i c^{2} \ln \left (-i a x +1\right ) x^{2}}{105 a}+\frac {i c^{2} a^{3} \ln \left (i a x +1\right ) x^{6}}{42}-\frac {4 i c^{2} \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{105 a^{3}}\)	\(495\)

[In]

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

[Out]

1/7*a^4*c^2*x^7*arctan(a*x)^2+2/5*a^2*c^2*x^5*arctan(a*x)^2+1/3*c^2*x^3*arctan(a*x)^2-2/105*c^2*(5/2*a^3*arcta
n(a*x)*x^6+27/4*a*arctan(a*x)*x^4+4/a*arctan(a*x)*x^2-4/a^3*arctan(a*x)*ln(a^2*x^2+1)-1/4/a^3*(2*a^5*x^5+17/3*
a^3*x^3-a*x+arctan(a*x)+8*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)-8*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]

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

[In]

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

[Out]

integral((a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2)*arctan(a*x)^2, x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/420*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 + 35*c^2*x^3)*arctan(a*x)^2 - 1/1680*(15*a^4*c^2*x^7 + 42*a^2*c^2*x^5 +
35*c^2*x^3)*log(a^2*x^2 + 1)^2 + integrate(1/1680*(1260*(a^6*c^2*x^8 + 3*a^4*c^2*x^6 + 3*a^2*c^2*x^4 + c^2*x^
2)*arctan(a*x)^2 + 105*(a^6*c^2*x^8 + 3*a^4*c^2*x^6 + 3*a^2*c^2*x^4 + c^2*x^2)*log(a^2*x^2 + 1)^2 - 8*(15*a^5*
c^2*x^7 + 42*a^3*c^2*x^5 + 35*a*c^2*x^3)*arctan(a*x) + 4*(15*a^6*c^2*x^8 + 42*a^4*c^2*x^6 + 35*a^2*c^2*x^4)*lo
g(a^2*x^2 + 1))/(a^2*x^2 + 1), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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