\(\int \frac {(a+b \arctan (c+d x))^2}{(e+f x)^2} \, dx\) [35]

Optimal result

Integrand size = 20, antiderivative size = 568 \[ \int \frac {(a+b \arctan (c+d x))^2}{(e+f x)^2} \, dx=\frac {2 a b d (d e-c f) \arctan (c+d x)}{f \left (f^2+(d e-c f)^2\right )}+\frac {i b^2 d \arctan (c+d x)^2}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^2 d (d e-c f) \arctan (c+d x)^2}{f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )}-\frac {(a+b \arctan (c+d x))^2}{f (e+f x)}+\frac {2 a b d \log (e+f x)}{f^2+(d e-c f)^2}-\frac {2 b^2 d \arctan (c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {2 b^2 d \arctan (c+d x) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {2 b^2 d \arctan (c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {a b d \log \left (1+(c+d x)^2\right )}{f^2+(d e-c f)^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.250, Rules used = {5153, 2007, 719, 31, 648, 632, 210, 642, 6873, 5165, 720, 649, 209, 266, 6820, 12, 6857, 4966, 2449, 2352, 2497, 5104, 5004, 5040, 4964} \[ \int \frac {(a+b \arctan (c+d x))^2}{(e+f x)^2} \, dx=\frac {2 a b d \arctan (c+d x) (d e-c f)}{f \left ((d e-c f)^2+f^2\right )}-\frac {(a+b \arctan (c+d x))^2}{f (e+f x)}+\frac {2 a b d \log (e+f x)}{(d e-c f)^2+f^2}-\frac {a b d \log \left ((c+d x)^2+1\right )}{(d e-c f)^2+f^2}+\frac {i b^2 d \arctan (c+d x)^2}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {b^2 d \arctan (c+d x)^2 (d e-c f)}{f \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}-\frac {2 b^2 d \arctan (c+d x) \log \left (\frac {2}{1-i (c+d x)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {2 b^2 d \arctan (c+d x) \log \left (\frac {2 d (e+f x)}{(1-i (c+d x)) (d e+(-c+i) f)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {2 b^2 d \arctan (c+d x) \log \left (\frac {2}{1+i (c+d x)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}-\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+(i-c) f) (1-i (c+d x))}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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 210

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

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

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 719

Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 - b*d*e + a*e^2
), Int[1/(d + e*x), x], x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x]
/; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 720

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[e^2/(c*d^2 + a*e^2), Int[1/(d + e*x), x],
x] + Dist[1/(c*d^2 + a*e^2), Int[(c*d - c*e*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a
*e^2, 0]

Rule 2007

Int[(u_)^(m_.)*(v_)^(p_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^p, x] /; FreeQ[{m, p}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, 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 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 4966

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

Rule 5004

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

Rule 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 5104

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcTan[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]

Rule 5153

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

Rule 5165

Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_
)^2)^(q_.), x_Symbol] :> Dist[1/d, Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(C/d^2 + (C/d^2)*x^2)^q*(a + b*ArcTan
[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0]
&& EqQ[2*c*C - B*d, 0]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

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

Mathematica [A] (verified)

Time = 5.05 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b \arctan (c+d x))^2}{(e+f x)^2} \, dx=\frac {-\frac {a^2}{f}+\frac {2 a b \left (-\left (\left (-c d e+f+c^2 f-d^2 e x+c d f x\right ) \arctan (c+d x)\right )+d (e+f x) \log \left (\frac {d (e+f x)}{\sqrt {1+(c+d x)^2}}\right )\right )}{d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}+\frac {b^2 d (e+f x) \left (-\frac {e^{i \arctan \left (\frac {d e-c f}{f}\right )} \arctan (c+d x)^2}{f \sqrt {1+\frac {(d e-c f)^2}{f^2}}}+\frac {(c+d x) \arctan (c+d x)^2}{d (e+f x)}-\frac {(d e-c f) \left (-i \left (\pi -2 \arctan \left (\frac {d e-c f}{f}\right )\right ) \arctan (c+d x)-\pi \log \left (1+e^{-2 i \arctan (c+d x)}\right )-2 \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right ) \log \left (1-e^{2 i \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )}\right )+\pi \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )+2 \arctan \left (\frac {d e-c f}{f}\right ) \log \left (\sin \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \left (\arctan \left (\frac {d e-c f}{f}\right )+\arctan (c+d x)\right )}\right )\right )}{f^2 \left (1+\frac {(d e-c f)^2}{f^2}\right )}\right )}{d e-c f}}{e+f x} \]

[In]

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

[Out]

(-(a^2/f) + (2*a*b*(-((-(c*d*e) + f + c^2*f - d^2*e*x + c*d*f*x)*ArcTan[c + d*x]) + d*(e + f*x)*Log[(d*(e + f*
x))/Sqrt[1 + (c + d*x)^2]]))/(d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2) + (b^2*d*(e + f*x)*(-((E^(I*ArcTan[(d*e - c
*f)/f])*ArcTan[c + d*x]^2)/(f*Sqrt[1 + (d*e - c*f)^2/f^2])) + ((c + d*x)*ArcTan[c + d*x]^2)/(d*(e + f*x)) - ((
d*e - c*f)*((-I)*(Pi - 2*ArcTan[(d*e - c*f)/f])*ArcTan[c + d*x] - Pi*Log[1 + E^((-2*I)*ArcTan[c + d*x])] - 2*(
ArcTan[(d*e - c*f)/f] + ArcTan[c + d*x])*Log[1 - E^((2*I)*(ArcTan[(d*e - c*f)/f] + ArcTan[c + d*x]))] + Pi*Log
[1/Sqrt[1 + (c + d*x)^2]] + 2*ArcTan[(d*e - c*f)/f]*Log[Sin[ArcTan[(d*e - c*f)/f] + ArcTan[c + d*x]]] + I*Poly
Log[2, E^((2*I)*(ArcTan[(d*e - c*f)/f] + ArcTan[c + d*x]))]))/(f^2*(1 + (d*e - c*f)^2/f^2))))/(d*e - c*f))/(e
+ f*x)

Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 776, normalized size of antiderivative = 1.37

method	result	size
parts	\(-\frac {a^{2}}{\left (f x +e \right ) f}+\frac {b^{2} \left (-\frac {d^{2} \arctan \left (d x +c \right )^{2}}{\left (f \left (d x +c \right )-c f +d e \right ) f}+\frac {2 d^{2} \left (\frac {\arctan \left (d x +c \right ) f \ln \left (f \left (d x +c \right )-c f +d e \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {\arctan \left (d x +c \right ) f \ln \left (1+\left (d x +c \right )^{2}\right )}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}-\frac {\arctan \left (d x +c \right )^{2} c f}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {\arctan \left (d x +c \right )^{2} d e}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {f^{2} \left (-\frac {i \ln \left (f \left (d x +c \right )-c f +d e \right ) \left (\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )-\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )-\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )\right )}{2 f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {f \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {\left (c f -d e \right ) \arctan \left (d x +c \right )^{2}}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}\right )}{f}\right )}{d}+\frac {2 a b \left (-\frac {d^{2} \arctan \left (d x +c \right )}{\left (f \left (d x +c \right )-c f +d e \right ) f}+\frac {d^{2} \left (\frac {f \ln \left (f \left (d x +c \right )-c f +d e \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}+\frac {-\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (-c f +d e \right ) \arctan \left (d x +c \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}\right )}{f}\right )}{d}\)	\(776\)
derivativedivides	\(\frac {\frac {a^{2} d^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+b^{2} d^{2} \left (\frac {\arctan \left (d x +c \right )^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}-\frac {2 \left (\frac {\arctan \left (d x +c \right ) f \ln \left (1+\left (d x +c \right )^{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {\arctan \left (d x +c \right )^{2} c f}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {\arctan \left (d x +c \right )^{2} d e}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {\arctan \left (d x +c \right ) f \ln \left (c f -d e -f \left (d x +c \right )\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {\left (c f -d e \right ) \arctan \left (d x +c \right )^{2}}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}-\frac {f \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}-\frac {f^{2} \left (-\frac {i \ln \left (c f -d e -f \left (d x +c \right )\right ) \left (\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}\right )}{f}\right )+2 a b \,d^{2} \left (\frac {\arctan \left (d x +c \right )}{\left (c f -d e -f \left (d x +c \right )\right ) f}-\frac {\frac {\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c f -d e \right ) \arctan \left (d x +c \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {f \ln \left (c f -d e -f \left (d x +c \right )\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}}{f}\right )}{d}\)	\(787\)
default	\(\frac {\frac {a^{2} d^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+b^{2} d^{2} \left (\frac {\arctan \left (d x +c \right )^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}-\frac {2 \left (\frac {\arctan \left (d x +c \right ) f \ln \left (1+\left (d x +c \right )^{2}\right )}{2 c^{2} f^{2}-4 c d e f +2 d^{2} e^{2}+2 f^{2}}+\frac {\arctan \left (d x +c \right )^{2} c f}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {\arctan \left (d x +c \right )^{2} d e}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {\arctan \left (d x +c \right ) f \ln \left (c f -d e -f \left (d x +c \right )\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {\left (c f -d e \right ) \arctan \left (d x +c \right )^{2}}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}-\frac {f \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )}-\frac {f^{2} \left (-\frac {i \ln \left (c f -d e -f \left (d x +c \right )\right ) \left (\ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}-\frac {i \left (\operatorname {dilog}\left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )-\operatorname {dilog}\left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )\right )}{2 f}\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}\right )}{f}\right )+2 a b \,d^{2} \left (\frac {\arctan \left (d x +c \right )}{\left (c f -d e -f \left (d x +c \right )\right ) f}-\frac {\frac {\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c f -d e \right ) \arctan \left (d x +c \right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}-\frac {f \ln \left (c f -d e -f \left (d x +c \right )\right )}{c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}}}{f}\right )}{d}\)	\(787\)

[In]

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

[Out]

-a^2/(f*x+e)/f+b^2/d*(-d^2/(f*(d*x+c)-c*f+d*e)/f*arctan(d*x+c)^2+2*d^2/f*(arctan(d*x+c)*f/(c^2*f^2-2*c*d*e*f+d
^2*e^2+f^2)*ln(f*(d*x+c)-c*f+d*e)-1/2*arctan(d*x+c)/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*f*ln(1+(d*x+c)^2)-1/(c^2*f
^2-2*c*d*e*f+d^2*e^2+f^2)*arctan(d*x+c)^2*c*f+1/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*arctan(d*x+c)^2*d*e-f^2/(c^2*f
^2-2*c*d*e*f+d^2*e^2+f^2)*(-1/2*I*ln(f*(d*x+c)-c*f+d*e)*(ln((I*f-f*(d*x+c))/(d*e+I*f-c*f))-ln((I*f+f*(d*x+c))/
(c*f-d*e+I*f)))/f-1/2*I*(dilog((I*f-f*(d*x+c))/(d*e+I*f-c*f))-dilog((I*f+f*(d*x+c))/(c*f-d*e+I*f)))/f)+1/2*f/(
c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*(-1/2*I*(ln(d*x+c-I)*ln(1+(d*x+c)^2)-1/2*ln(d*x+c-I)^2-dilog(-1/2*I*(d*x+c+I))-
ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I)))+1/2*I*(ln(d*x+c+I)*ln(1+(d*x+c)^2)-1/2*ln(d*x+c+I)^2-dilog(1/2*I*(d*x+c-I))-
ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))))+1/2/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*(c*f-d*e)*arctan(d*x+c)^2))+2*a*b/d*(-d^
2/(f*(d*x+c)-c*f+d*e)/f*arctan(d*x+c)+d^2/f*(1/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)*f*ln(f*(d*x+c)-c*f+d*e)+1/(c^2*
f^2-2*c*d*e*f+d^2*e^2+f^2)*(-1/2*f*ln(1+(d*x+c)^2)+(-c*f+d*e)*arctan(d*x+c))))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

(d*(2*(d^2*e - c*d*f)*arctan((d^2*x + c*d)/d)/((d^2*e^2*f - 2*c*d*e*f^2 + (c^2 + 1)*f^3)*d) - log(d^2*x^2 + 2*
c*d*x + c^2 + 1)/(d^2*e^2 - 2*c*d*e*f + (c^2 + 1)*f^2) + 2*log(f*x + e)/(d^2*e^2 - 2*c*d*e*f + (c^2 + 1)*f^2))
- 2*arctan(d*x + c)/(f^2*x + e*f))*a*b - 1/16*(4*arctan(d*x + c)^2 - 16*(f^2*x + e*f)*integrate(1/16*(12*(d^2
*f*x^2 + 2*c*d*f*x + (c^2 + 1)*f)*arctan(d*x + c)^2 + (d^2*f*x^2 + 2*c*d*f*x + (c^2 + 1)*f)*log(d^2*x^2 + 2*c*
d*x + c^2 + 1)^2 + 8*(d*f*x + d*e)*arctan(d*x + c) - 4*(d^2*f*x^2 + c*d*e + (d^2*e + c*d*f)*x)*log(d^2*x^2 + 2
*c*d*x + c^2 + 1))/(d^2*f^3*x^4 + (c^2 + 1)*e^2*f + 2*(d^2*e*f^2 + c*d*f^3)*x^3 + (d^2*e^2*f + 4*c*d*e*f^2 + (
c^2 + 1)*f^3)*x^2 + 2*(c*d*e^2*f + (c^2 + 1)*e*f^2)*x), x) - log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2)*b^2/(f^2*x +
e*f) - a^2/(f^2*x + e*f)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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