\(\int \frac {(d+e x^2)^2 (a+b \arctan (c x))^2}{x^2} \, dx\) [1259]

Optimal result

Integrand size = 23, antiderivative size = 343 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^2} \, dx=\frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \arctan (c x)}{3 c^3}-\frac {b e^2 x^2 (a+b \arctan (c x))}{3 c}-i c d^2 (a+b \arctan (c x))^2+\frac {2 i d e (a+b \arctan (c x))^2}{c}-\frac {i e^2 (a+b \arctan (c x))^2}{3 c^3}-\frac {d^2 (a+b \arctan (c x))^2}{x}+2 d e x (a+b \arctan (c x))^2+\frac {1}{3} e^2 x^3 (a+b \arctan (c x))^2+\frac {4 b d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+2 b c d^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )+\frac {2 i b^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3} \]

[Out]

1/3*b^2*e^2*x/c^2-1/3*b^2*e^2*arctan(c*x)/c^3-1/3*b*e^2*x^2*(a+b*arctan(c*x))/c-I*c*d^2*(a+b*arctan(c*x))^2+2*
I*d*e*(a+b*arctan(c*x))^2/c-1/3*I*e^2*(a+b*arctan(c*x))^2/c^3-d^2*(a+b*arctan(c*x))^2/x+2*d*e*x*(a+b*arctan(c*
x))^2+1/3*e^2*x^3*(a+b*arctan(c*x))^2+4*b*d*e*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c-2/3*b*e^2*(a+b*arctan(c*x))*
ln(2/(1+I*c*x))/c^3+2*b*c*d^2*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x))-I*b^2*c*d^2*polylog(2,-1+2/(1-I*c*x))+2*I*b^
2*d*e*polylog(2,1-2/(1+I*c*x))/c-1/3*I*b^2*e^2*polylog(2,1-2/(1+I*c*x))/c^3

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {5100, 4930, 5040, 4964, 2449, 2352, 4946, 5044, 4988, 2497, 5036, 327, 209} \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^2} \, dx=-\frac {i e^2 (a+b \arctan (c x))^2}{3 c^3}-\frac {2 b e^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{3 c^3}-i c d^2 (a+b \arctan (c x))^2-\frac {d^2 (a+b \arctan (c x))^2}{x}+2 b c d^2 \log \left (2-\frac {2}{1-i c x}\right ) (a+b \arctan (c x))+2 d e x (a+b \arctan (c x))^2+\frac {2 i d e (a+b \arctan (c x))^2}{c}+\frac {4 b d e \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {1}{3} e^2 x^3 (a+b \arctan (c x))^2-\frac {b e^2 x^2 (a+b \arctan (c x))}{3 c}-\frac {b^2 e^2 \arctan (c x)}{3 c^3}-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{3 c^3}+\frac {b^2 e^2 x}{3 c^2}-i b^2 c d^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right )+\frac {2 i b^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c} \]

[In]

Int[((d + e*x^2)^2*(a + b*ArcTan[c*x])^2)/x^2,x]

[Out]

(b^2*e^2*x)/(3*c^2) - (b^2*e^2*ArcTan[c*x])/(3*c^3) - (b*e^2*x^2*(a + b*ArcTan[c*x]))/(3*c) - I*c*d^2*(a + b*A
rcTan[c*x])^2 + ((2*I)*d*e*(a + b*ArcTan[c*x])^2)/c - ((I/3)*e^2*(a + b*ArcTan[c*x])^2)/c^3 - (d^2*(a + b*ArcT
an[c*x])^2)/x + 2*d*e*x*(a + b*ArcTan[c*x])^2 + (e^2*x^3*(a + b*ArcTan[c*x])^2)/3 + (4*b*d*e*(a + b*ArcTan[c*x
])*Log[2/(1 + I*c*x)])/c - (2*b*e^2*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(3*c^3) + 2*b*c*d^2*(a + b*ArcTan[
c*x])*Log[2 - 2/(1 - I*c*x)] - I*b^2*c*d^2*PolyLog[2, -1 + 2/(1 - I*c*x)] + ((2*I)*b^2*d*e*PolyLog[2, 1 - 2/(1
+ I*c*x)])/c - ((I/3)*b^2*e^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^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 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 5100

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

Rubi steps \begin{align*} \text {integral}& = \int \left (2 d e (a+b \arctan (c x))^2+\frac {d^2 (a+b \arctan (c x))^2}{x^2}+e^2 x^2 (a+b \arctan (c x))^2\right ) \, dx \\ & = d^2 \int \frac {(a+b \arctan (c x))^2}{x^2} \, dx+(2 d e) \int (a+b \arctan (c x))^2 \, dx+e^2 \int x^2 (a+b \arctan (c x))^2 \, dx \\ & = -\frac {d^2 (a+b \arctan (c x))^2}{x}+2 d e x (a+b \arctan (c x))^2+\frac {1}{3} e^2 x^3 (a+b \arctan (c x))^2+\left (2 b c d^2\right ) \int \frac {a+b \arctan (c x)}{x \left (1+c^2 x^2\right )} \, dx-(4 b c d e) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx-\frac {1}{3} \left (2 b c e^2\right ) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = -i c d^2 (a+b \arctan (c x))^2+\frac {2 i d e (a+b \arctan (c x))^2}{c}-\frac {d^2 (a+b \arctan (c x))^2}{x}+2 d e x (a+b \arctan (c x))^2+\frac {1}{3} e^2 x^3 (a+b \arctan (c x))^2+\left (2 i b c d^2\right ) \int \frac {a+b \arctan (c x)}{x (i+c x)} \, dx+(4 b d e) \int \frac {a+b \arctan (c x)}{i-c x} \, dx-\frac {\left (2 b e^2\right ) \int x (a+b \arctan (c x)) \, dx}{3 c}+\frac {\left (2 b e^2\right ) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 c} \\ & = -\frac {b e^2 x^2 (a+b \arctan (c x))}{3 c}-i c d^2 (a+b \arctan (c x))^2+\frac {2 i d e (a+b \arctan (c x))^2}{c}-\frac {i e^2 (a+b \arctan (c x))^2}{3 c^3}-\frac {d^2 (a+b \arctan (c x))^2}{x}+2 d e x (a+b \arctan (c x))^2+\frac {1}{3} e^2 x^3 (a+b \arctan (c x))^2+\frac {4 b d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}+2 b c d^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )-\left (2 b^2 c^2 d^2\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx-\left (4 b^2 d e\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\frac {1}{3} \left (b^2 e^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx-\frac {\left (2 b e^2\right ) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{3 c^2} \\ & = \frac {b^2 e^2 x}{3 c^2}-\frac {b e^2 x^2 (a+b \arctan (c x))}{3 c}-i c d^2 (a+b \arctan (c x))^2+\frac {2 i d e (a+b \arctan (c x))^2}{c}-\frac {i e^2 (a+b \arctan (c x))^2}{3 c^3}-\frac {d^2 (a+b \arctan (c x))^2}{x}+2 d e x (a+b \arctan (c x))^2+\frac {1}{3} e^2 x^3 (a+b \arctan (c x))^2+\frac {4 b d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+2 b c d^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )+\frac {\left (4 i b^2 d e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c}-\frac {\left (b^2 e^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 c^2}+\frac {\left (2 b^2 e^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2} \\ & = \frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \arctan (c x)}{3 c^3}-\frac {b e^2 x^2 (a+b \arctan (c x))}{3 c}-i c d^2 (a+b \arctan (c x))^2+\frac {2 i d e (a+b \arctan (c x))^2}{c}-\frac {i e^2 (a+b \arctan (c x))^2}{3 c^3}-\frac {d^2 (a+b \arctan (c x))^2}{x}+2 d e x (a+b \arctan (c x))^2+\frac {1}{3} e^2 x^3 (a+b \arctan (c x))^2+\frac {4 b d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+2 b c d^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )+\frac {2 i b^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}-\frac {\left (2 i b^2 e^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^3} \\ & = \frac {b^2 e^2 x}{3 c^2}-\frac {b^2 e^2 \arctan (c x)}{3 c^3}-\frac {b e^2 x^2 (a+b \arctan (c x))}{3 c}-i c d^2 (a+b \arctan (c x))^2+\frac {2 i d e (a+b \arctan (c x))^2}{c}-\frac {i e^2 (a+b \arctan (c x))^2}{3 c^3}-\frac {d^2 (a+b \arctan (c x))^2}{x}+2 d e x (a+b \arctan (c x))^2+\frac {1}{3} e^2 x^3 (a+b \arctan (c x))^2+\frac {4 b d e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {2 b e^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{3 c^3}+2 b c d^2 (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )+\frac {2 i b^2 d e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{3 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))^2}{x^2} \, dx=\frac {1}{3} \left (-\frac {3 a^2 d^2}{x}+6 a^2 d e x+a^2 e^2 x^3+\frac {6 a b d e \left (2 c x \arctan (c x)-\log \left (1+c^2 x^2\right )\right )}{c}+\frac {a b e^2 \left (-c^2 x^2+2 c^3 x^3 \arctan (c x)+\log \left (1+c^2 x^2\right )\right )}{c^3}-\frac {3 a b d^2 \left (2 \arctan (c x)+c x \left (-2 \log (c x)+\log \left (1+c^2 x^2\right )\right )\right )}{x}+\frac {6 b^2 d e \left (\arctan (c x) \left ((-i+c x) \arctan (c x)+2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{c}+\frac {b^2 e^2 \left (c x+\left (i+c^3 x^3\right ) \arctan (c x)^2-\arctan (c x) \left (1+c^2 x^2+2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )\right )}{c^3}+3 b^2 c d^2 \left (\arctan (c x) \left (\left (-i-\frac {1}{c x}\right ) \arctan (c x)+2 \log \left (1-e^{2 i \arctan (c x)}\right )\right )-i \operatorname {PolyLog}\left (2,e^{2 i \arctan (c x)}\right )\right )\right ) \]

[In]

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

[Out]

((-3*a^2*d^2)/x + 6*a^2*d*e*x + a^2*e^2*x^3 + (6*a*b*d*e*(2*c*x*ArcTan[c*x] - Log[1 + c^2*x^2]))/c + (a*b*e^2*
(-(c^2*x^2) + 2*c^3*x^3*ArcTan[c*x] + Log[1 + c^2*x^2]))/c^3 - (3*a*b*d^2*(2*ArcTan[c*x] + c*x*(-2*Log[c*x] +
Log[1 + c^2*x^2])))/x + (6*b^2*d*e*(ArcTan[c*x]*((-I + c*x)*ArcTan[c*x] + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) -
I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c + (b^2*e^2*(c*x + (I + c^3*x^3)*ArcTan[c*x]^2 - ArcTan[c*x]*(1 + c^2*
x^2 + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) + I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]))/c^3 + 3*b^2*c*d^2*(ArcTan[c*x
]*((-I - 1/(c*x))*ArcTan[c*x] + 2*Log[1 - E^((2*I)*ArcTan[c*x])]) - I*PolyLog[2, E^((2*I)*ArcTan[c*x])]))/3

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.53

method	result	size
derivativedivides	\(c \left (\frac {a^{2} \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {b^{2} \left (2 \arctan \left (c x \right )^{2} c^{3} x d e +\frac {\arctan \left (c x \right )^{2} e^{2} c^{3} x^{3}}{3}-\frac {\arctan \left (c x \right )^{2} c^{3} d^{2}}{x}-\frac {\arctan \left (c x \right ) e^{2} c^{2} x^{2}}{3}+2 \arctan \left (c x \right ) c^{4} d^{2} \ln \left (c x \right )-\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{4} d^{2}-2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d e +\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{2}}{3}+\frac {e^{2} \left (c x -\arctan \left (c x \right )\right )}{3}+\frac {\left (3 c^{4} d^{2}+6 c^{2} d e -e^{2}\right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{3}-2 c^{4} d^{2} \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )\right )}{c^{4}}+\frac {2 a b \left (2 \arctan \left (c x \right ) c^{3} d e x +\frac {\arctan \left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {\arctan \left (c x \right ) c^{3} d^{2}}{x}-\frac {e^{2} c^{2} x^{2}}{6}-\frac {\left (3 c^{4} d^{2}+6 c^{2} d e -e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{6}+c^{4} d^{2} \ln \left (c x \right )\right )}{c^{4}}\right )\)	\(524\)
default	\(c \left (\frac {a^{2} \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {b^{2} \left (2 \arctan \left (c x \right )^{2} c^{3} x d e +\frac {\arctan \left (c x \right )^{2} e^{2} c^{3} x^{3}}{3}-\frac {\arctan \left (c x \right )^{2} c^{3} d^{2}}{x}-\frac {\arctan \left (c x \right ) e^{2} c^{2} x^{2}}{3}+2 \arctan \left (c x \right ) c^{4} d^{2} \ln \left (c x \right )-\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{4} d^{2}-2 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d e +\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{2}}{3}+\frac {e^{2} \left (c x -\arctan \left (c x \right )\right )}{3}+\frac {\left (3 c^{4} d^{2}+6 c^{2} d e -e^{2}\right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{3}-2 c^{4} d^{2} \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )\right )}{c^{4}}+\frac {2 a b \left (2 \arctan \left (c x \right ) c^{3} d e x +\frac {\arctan \left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {\arctan \left (c x \right ) c^{3} d^{2}}{x}-\frac {e^{2} c^{2} x^{2}}{6}-\frac {\left (3 c^{4} d^{2}+6 c^{2} d e -e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{6}+c^{4} d^{2} \ln \left (c x \right )\right )}{c^{4}}\right )\)	\(524\)
parts	\(a^{2} \left (\frac {e^{2} x^{3}}{3}+2 d e x -\frac {d^{2}}{x}\right )+b^{2} c \left (\frac {\arctan \left (c x \right )^{2} e^{2} x^{3}}{3 c}+\frac {2 \arctan \left (c x \right )^{2} x d e}{c}-\frac {\arctan \left (c x \right )^{2} d^{2}}{c x}-\frac {2 \left (\frac {\arctan \left (c x \right ) e^{2} c^{2} x^{2}}{2}-3 \arctan \left (c x \right ) c^{4} d^{2} \ln \left (c x \right )+\frac {3 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{4} d^{2}}{2}+3 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) c^{2} d e -\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) e^{2}}{2}-\frac {e^{2} \left (c x -\arctan \left (c x \right )\right )}{2}-\frac {\left (3 c^{4} d^{2}+6 c^{2} d e -e^{2}\right ) \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{2}-\frac {3 i c^{4} d^{2} \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {3 i c^{4} d^{2} \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {3 i c^{4} d^{2} \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {3 i c^{4} d^{2} \operatorname {dilog}\left (-i c x +1\right )}{2}\right )}{3 c^{4}}\right )+2 a b c \left (\frac {\arctan \left (c x \right ) e^{2} x^{3}}{3 c}+\frac {2 \arctan \left (c x \right ) x d e}{c}-\frac {\arctan \left (c x \right ) d^{2}}{c x}-\frac {\frac {e^{2} c^{2} x^{2}}{2}-3 c^{4} d^{2} \ln \left (c x \right )+\frac {\left (3 c^{4} d^{2}+6 c^{2} d e -e^{2}\right ) \ln \left (c^{2} x^{2}+1\right )}{2}}{3 c^{4}}\right )\)	\(534\)

[In]

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

[Out]

c*(a^2/c^4*(2*c^3*d*e*x+1/3*e^2*c^3*x^3-c^3*d^2/x)+b^2/c^4*(2*arctan(c*x)^2*c^3*x*d*e+1/3*arctan(c*x)^2*e^2*c^
3*x^3-arctan(c*x)^2*c^3*d^2/x-1/3*arctan(c*x)*e^2*c^2*x^2+2*arctan(c*x)*c^4*d^2*ln(c*x)-arctan(c*x)*ln(c^2*x^2
+1)*c^4*d^2-2*arctan(c*x)*ln(c^2*x^2+1)*c^2*d*e+1/3*arctan(c*x)*ln(c^2*x^2+1)*e^2+1/3*e^2*(c*x-arctan(c*x))+1/
3*(3*c^4*d^2+6*c^2*d*e-e^2)*(-1/2*I*(ln(c*x-I)*ln(c^2*x^2+1)-1/2*ln(c*x-I)^2-dilog(-1/2*I*(I+c*x))-ln(c*x-I)*l
n(-1/2*I*(I+c*x)))+1/2*I*(ln(I+c*x)*ln(c^2*x^2+1)-1/2*ln(I+c*x)^2-dilog(1/2*I*(c*x-I))-ln(I+c*x)*ln(1/2*I*(c*x
-I))))-2*c^4*d^2*(-1/2*I*ln(c*x)*ln(1+I*c*x)+1/2*I*ln(c*x)*ln(1-I*c*x)-1/2*I*dilog(1+I*c*x)+1/2*I*dilog(1-I*c*
x)))+2*a*b/c^4*(2*arctan(c*x)*c^3*d*e*x+1/3*arctan(c*x)*e^2*c^3*x^3-arctan(c*x)*c^3*d^2/x-1/6*e^2*c^2*x^2-1/6*
(3*c^4*d^2+6*c^2*d*e-e^2)*ln(c^2*x^2+1)+c^4*d^2*ln(c*x)))

Fricas [F]

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

[In]

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

[Out]

integral((a^2*e^2*x^4 + 2*a^2*d*e*x^2 + a^2*d^2 + (b^2*e^2*x^4 + 2*b^2*d*e*x^2 + b^2*d^2)*arctan(c*x)^2 + 2*(a
*b*e^2*x^4 + 2*a*b*d*e*x^2 + a*b*d^2)*arctan(c*x))/x^2, x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/3*a^2*e^2*x^3 - (c*(log(c^2*x^2 + 1) - log(x^2)) + 2*arctan(c*x)/x)*a*b*d^2 + 1/3*(2*x^3*arctan(c*x) - c*(x^
2/c^2 - log(c^2*x^2 + 1)/c^4))*a*b*e^2 + 2*a^2*d*e*x + 2*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a*b*d*e/c - a^
2*d^2/x + 1/48*(4*(b^2*e^2*x^4 + 6*b^2*d*e*x^2 - 3*b^2*d^2)*arctan(c*x)^2 - (b^2*e^2*x^4 + 6*b^2*d*e*x^2 - 3*b
^2*d^2)*log(c^2*x^2 + 1)^2 + 12*(b^2*c*d^2*arctan(c*x)^3 + 144*b^2*c^2*e^2*integrate(1/48*x^6*arctan(c*x)^2/(c
^2*x^4 + x^2), x) + 12*b^2*c^2*e^2*integrate(1/48*x^6*log(c^2*x^2 + 1)^2/(c^2*x^4 + x^2), x) + 16*b^2*c^2*e^2*
integrate(1/48*x^6*log(c^2*x^2 + 1)/(c^2*x^4 + x^2), x) + 288*b^2*c^2*d*e*integrate(1/48*x^4*arctan(c*x)^2/(c^
2*x^4 + x^2), x) + 24*b^2*c^2*d*e*integrate(1/48*x^4*log(c^2*x^2 + 1)^2/(c^2*x^4 + x^2), x) + 96*b^2*c^2*d*e*i
ntegrate(1/48*x^4*log(c^2*x^2 + 1)/(c^2*x^4 + x^2), x) + 12*b^2*c^2*d^2*integrate(1/48*x^2*log(c^2*x^2 + 1)^2/
(c^2*x^4 + x^2), x) - 48*b^2*c^2*d^2*integrate(1/48*x^2*log(c^2*x^2 + 1)/(c^2*x^4 + x^2), x) + 2*b^2*d*e*arcta
n(c*x)^3/c - 32*b^2*c*e^2*integrate(1/48*x^5*arctan(c*x)/(c^2*x^4 + x^2), x) - 192*b^2*c*d*e*integrate(1/48*x^
3*arctan(c*x)/(c^2*x^4 + x^2), x) + 96*b^2*c*d^2*integrate(1/48*x*arctan(c*x)/(c^2*x^4 + x^2), x) + 144*b^2*e^
2*integrate(1/48*x^4*arctan(c*x)^2/(c^2*x^4 + x^2), x) + 12*b^2*e^2*integrate(1/48*x^4*log(c^2*x^2 + 1)^2/(c^2
*x^4 + x^2), x) + 24*b^2*d*e*integrate(1/48*x^2*log(c^2*x^2 + 1)^2/(c^2*x^4 + x^2), x) + 144*b^2*d^2*integrate
(1/48*arctan(c*x)^2/(c^2*x^4 + x^2), x) + 12*b^2*d^2*integrate(1/48*log(c^2*x^2 + 1)^2/(c^2*x^4 + x^2), x))*x)
/x

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

int(((a + b*atan(c*x))^2*(d + e*x^2)^2)/x^2,x)

[Out]

int(((a + b*atan(c*x))^2*(d + e*x^2)^2)/x^2, x)