Integrand size = 23, antiderivative size = 194 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^4} \, dx=-\frac {b^2}{3 d e^4 (c+d x)}-\frac {b^2 \arctan (c+d x)}{3 d e^4}-\frac {b (a+b \arctan (c+d x))}{3 d e^4 (c+d x)^2}+\frac {i (a+b \arctan (c+d x))^2}{3 d e^4}-\frac {(a+b \arctan (c+d x))^2}{3 d e^4 (c+d x)^3}-\frac {2 b (a+b \arctan (c+d x)) \log \left (2-\frac {2}{1-i (c+d x)}\right )}{3 d e^4}+\frac {i b^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i (c+d x)}\right )}{3 d e^4} \]
-1/3*b^2/d/e^4/(d*x+c)-1/3*b^2*arctan(d*x+c)/d/e^4-1/3*b*(a+b*arctan(d*x+c ))/d/e^4/(d*x+c)^2+1/3*I*(a+b*arctan(d*x+c))^2/d/e^4-1/3*(a+b*arctan(d*x+c ))^2/d/e^4/(d*x+c)^3-2/3*b*(a+b*arctan(d*x+c))*ln(2-2/(1-I*(d*x+c)))/d/e^4 +1/3*I*b^2*polylog(2,-1+2/(1-I*(d*x+c)))/d/e^4
Time = 0.66 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^4} \, dx=-\frac {a b+\frac {a^2}{(c+d x)^3}+\frac {a b}{(c+d x)^2}+\frac {b^2}{c+d x}+b^2 \left (-i+\frac {1}{(c+d x)^3}\right ) \arctan (c+d x)^2+b \arctan (c+d x) \left (b+\frac {2 a}{(c+d x)^3}+\frac {b}{(c+d x)^2}+2 b \log \left (1-e^{2 i \arctan (c+d x)}\right )\right )+2 a b \log \left (\frac {c+d x}{\sqrt {1+(c+d x)^2}}\right )-i b^2 \operatorname {PolyLog}\left (2,e^{2 i \arctan (c+d x)}\right )}{3 d e^4} \]
-1/3*(a*b + a^2/(c + d*x)^3 + (a*b)/(c + d*x)^2 + b^2/(c + d*x) + b^2*(-I + (c + d*x)^(-3))*ArcTan[c + d*x]^2 + b*ArcTan[c + d*x]*(b + (2*a)/(c + d* x)^3 + b/(c + d*x)^2 + 2*b*Log[1 - E^((2*I)*ArcTan[c + d*x])]) + 2*a*b*Log [(c + d*x)/Sqrt[1 + (c + d*x)^2]] - I*b^2*PolyLog[2, E^((2*I)*ArcTan[c + d *x])])/(d*e^4)
Time = 0.72 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.84, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5566, 27, 5361, 5453, 5361, 264, 216, 5459, 5403, 2897}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^4} \, dx\) |
| \(\Big \downarrow \) 5566 |
| \(\displaystyle \frac {\int \frac {(a+b \arctan (c+d x))^2}{e^4 (c+d x)^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \arctan (c+d x))^2}{(c+d x)^4}d(c+d x)}{d e^4}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {\frac {2}{3} b \int \frac {a+b \arctan (c+d x)}{(c+d x)^3 \left ((c+d x)^2+1\right )}d(c+d x)-\frac {(a+b \arctan (c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {\frac {2}{3} b \left (\int \frac {a+b \arctan (c+d x)}{(c+d x)^3}d(c+d x)-\int \frac {a+b \arctan (c+d x)}{(c+d x) \left ((c+d x)^2+1\right )}d(c+d x)\right )-\frac {(a+b \arctan (c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {\frac {2}{3} b \left (-\int \frac {a+b \arctan (c+d x)}{(c+d x) \left ((c+d x)^2+1\right )}d(c+d x)+\frac {1}{2} b \int \frac {1}{(c+d x)^2 \left ((c+d x)^2+1\right )}d(c+d x)-\frac {a+b \arctan (c+d x)}{2 (c+d x)^2}\right )-\frac {(a+b \arctan (c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\frac {2}{3} b \left (-\int \frac {a+b \arctan (c+d x)}{(c+d x) \left ((c+d x)^2+1\right )}d(c+d x)+\frac {1}{2} b \left (-\int \frac {1}{(c+d x)^2+1}d(c+d x)-\frac {1}{c+d x}\right )-\frac {a+b \arctan (c+d x)}{2 (c+d x)^2}\right )-\frac {(a+b \arctan (c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {2}{3} b \left (-\int \frac {a+b \arctan (c+d x)}{(c+d x) \left ((c+d x)^2+1\right )}d(c+d x)-\frac {a+b \arctan (c+d x)}{2 (c+d x)^2}+\frac {1}{2} b \left (-\arctan (c+d x)-\frac {1}{c+d x}\right )\right )-\frac {(a+b \arctan (c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle \frac {-\frac {(a+b \arctan (c+d x))^2}{3 (c+d x)^3}+\frac {2}{3} b \left (-i \int \frac {a+b \arctan (c+d x)}{(c+d x) (c+d x+i)}d(c+d x)+\frac {i (a+b \arctan (c+d x))^2}{2 b}-\frac {a+b \arctan (c+d x)}{2 (c+d x)^2}+\frac {1}{2} b \left (-\arctan (c+d x)-\frac {1}{c+d x}\right )\right )}{d e^4}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {-\frac {(a+b \arctan (c+d x))^2}{3 (c+d x)^3}+\frac {2}{3} b \left (-i \left (i b \int \frac {\log \left (2-\frac {2}{1-i (c+d x)}\right )}{(c+d x)^2+1}d(c+d x)-i \log \left (2-\frac {2}{1-i (c+d x)}\right ) (a+b \arctan (c+d x))\right )+\frac {i (a+b \arctan (c+d x))^2}{2 b}-\frac {a+b \arctan (c+d x)}{2 (c+d x)^2}+\frac {1}{2} b \left (-\arctan (c+d x)-\frac {1}{c+d x}\right )\right )}{d e^4}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {-\frac {(a+b \arctan (c+d x))^2}{3 (c+d x)^3}+\frac {2}{3} b \left (-i \left (-i \log \left (2-\frac {2}{1-i (c+d x)}\right ) (a+b \arctan (c+d x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{1-i (c+d x)}-1\right )\right )+\frac {i (a+b \arctan (c+d x))^2}{2 b}-\frac {a+b \arctan (c+d x)}{2 (c+d x)^2}+\frac {1}{2} b \left (-\arctan (c+d x)-\frac {1}{c+d x}\right )\right )}{d e^4}\) |
(-1/3*(a + b*ArcTan[c + d*x])^2/(c + d*x)^3 + (2*b*((b*(-(c + d*x)^(-1) - ArcTan[c + d*x]))/2 - (a + b*ArcTan[c + d*x])/(2*(c + d*x)^2) + ((I/2)*(a + b*ArcTan[c + d*x])^2)/b - I*((-I)*(a + b*ArcTan[c + d*x])*Log[2 - 2/(1 - I*(c + d*x))] - (b*PolyLog[2, -1 + 2/(1 - I*(c + d*x))])/2)))/3)/(d*e^4)
3.1.13.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
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] - Simp[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]
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] - Si mp[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]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/(d + e*x^2) ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*d*(p + 1))), x] + Si mp[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]
Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[1/d Subst[Int[(f*(x/d))^m*(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (176 ) = 352\).
Time = 2.56 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.90
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {\arctan \left (d x +c \right )}{3 \left (d x +c \right )^{2}}-\frac {2 \ln \left (d x +c \right ) \arctan \left (d x +c \right )}{3}+\frac {\arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{3}+\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{6}-\frac {1}{3 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )}{3}-\frac {i \ln \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )\right )}{3}+\frac {i \ln \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (d x +c \right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (d x +c \right )\right )}{3}\right )}{e^{4}}+\frac {2 a b \left (-\frac {\arctan \left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {1}{6 \left (d x +c \right )^{2}}-\frac {\ln \left (d x +c \right )}{3}+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{6}\right )}{e^{4}}}{d}\) | \(368\) |
| default | \(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {\arctan \left (d x +c \right )}{3 \left (d x +c \right )^{2}}-\frac {2 \ln \left (d x +c \right ) \arctan \left (d x +c \right )}{3}+\frac {\arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{3}+\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{6}-\frac {1}{3 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )}{3}-\frac {i \ln \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )\right )}{3}+\frac {i \ln \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (d x +c \right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (d x +c \right )\right )}{3}\right )}{e^{4}}+\frac {2 a b \left (-\frac {\arctan \left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {1}{6 \left (d x +c \right )^{2}}-\frac {\ln \left (d x +c \right )}{3}+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{6}\right )}{e^{4}}}{d}\) | \(368\) |
| parts | \(-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3} d}+\frac {b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {\arctan \left (d x +c \right )}{3 \left (d x +c \right )^{2}}-\frac {2 \ln \left (d x +c \right ) \arctan \left (d x +c \right )}{3}+\frac {\arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{3}+\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{6}-\frac {1}{3 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )}{3}-\frac {i \ln \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )\right )}{3}+\frac {i \ln \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (d x +c \right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (d x +c \right )\right )}{3}\right )}{e^{4} d}+\frac {2 a b \left (-\frac {\arctan \left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {1}{6 \left (d x +c \right )^{2}}-\frac {\ln \left (d x +c \right )}{3}+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{6}\right )}{e^{4} d}\) | \(373\) |
1/d*(-1/3*a^2/e^4/(d*x+c)^3+b^2/e^4*(-1/3/(d*x+c)^3*arctan(d*x+c)^2-1/3/(d *x+c)^2*arctan(d*x+c)-2/3*ln(d*x+c)*arctan(d*x+c)+1/3*arctan(d*x+c)*ln(1+( d*x+c)^2)+1/6*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)))-1/6*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)))-1/3/(d*x+c)-1/3*arctan(d*x+c)-1/3*I*ln(d*x+c)*ln(1+I*(d*x+c))+1/3* I*ln(d*x+c)*ln(1-I*(d*x+c))-1/3*I*dilog(1+I*(d*x+c))+1/3*I*dilog(1-I*(d*x+ c)))+2*a*b/e^4*(-1/3/(d*x+c)^3*arctan(d*x+c)-1/6/(d*x+c)^2-1/3*ln(d*x+c)+1 /6*ln(1+(d*x+c)^2)))
\[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \arctan \left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
integral((b^2*arctan(d*x + c)^2 + 2*a*b*arctan(d*x + c) + a^2)/(d^4*e^4*x^ 4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4*x + c^4*e^4), x)
\[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{2}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{2} \operatorname {atan}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {2 a b \operatorname {atan}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
(Integral(a**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d** 4*x**4), x) + Integral(b**2*atan(c + d*x)**2/(c**4 + 4*c**3*d*x + 6*c**2*d **2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(2*a*b*atan(c + d*x)/( c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x))/e** 4
\[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \arctan \left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
-1/3*(d*(1/(d^4*e^4*x^2 + 2*c*d^3*e^4*x + c^2*d^2*e^4) - log(d^2*x^2 + 2*c *d*x + c^2 + 1)/(d^2*e^4) + 2*log(d*x + c)/(d^2*e^4)) + 2*arctan(d*x + c)/ (d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4))*a*b - 1/48* (4*arctan(d*x + c)^2 - 48*(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4)*integrate(1/48*(36*(d^2*x^2 + 2*c*d*x + c^2 + 1)*arctan(d*x + c)^2 + 3*(d^2*x^2 + 2*c*d*x + c^2 + 1)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^ 2 + 8*(d*x + c)*arctan(d*x + c) - 4*(d^2*x^2 + 2*c*d*x + c^2)*log(d^2*x^2 + 2*c*d*x + c^2 + 1))/(d^6*e^4*x^6 + 6*c*d^5*e^4*x^5 + (15*c^2 + 1)*d^4*e^ 4*x^4 + 4*(5*c^3 + c)*d^3*e^4*x^3 + 3*(5*c^4 + 2*c^2)*d^2*e^4*x^2 + 2*(3*c ^5 + 2*c^3)*d*e^4*x + (c^6 + c^4)*e^4), x) - log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2)*b^2/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) - 1/3*a^2/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4)
Timed out. \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]