Integrand size = 23, antiderivative size = 180 \[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=-\frac {3 i b (a+b \arctan (c+d x))^2}{2 d e^3}-\frac {3 b (a+b \arctan (c+d x))^2}{2 d e^3 (c+d x)}-\frac {(a+b \arctan (c+d x))^3}{2 d e^3}-\frac {(a+b \arctan (c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 (a+b \arctan (c+d x)) \log \left (2-\frac {2}{1-i (c+d x)}\right )}{d e^3}-\frac {3 i b^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i (c+d x)}\right )}{2 d e^3} \]
-3/2*I*b*(a+b*arctan(d*x+c))^2/d/e^3-3/2*b*(a+b*arctan(d*x+c))^2/d/e^3/(d* x+c)-1/2*(a+b*arctan(d*x+c))^3/d/e^3-1/2*(a+b*arctan(d*x+c))^3/d/e^3/(d*x+ c)^2+3*b^2*(a+b*arctan(d*x+c))*ln(2-2/(1-I*(d*x+c)))/d/e^3-3/2*I*b^3*polyl og(2,-1+2/(1-I*(d*x+c)))/d/e^3
Time = 0.33 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=-\frac {a^3+b^3 \left (1+c^2+2 c d x+d^2 x^2\right ) \arctan (c+d x)^3+3 a^2 b \left (c+d x+\left (1+(c+d x)^2\right ) \arctan (c+d x)\right )+3 a b^2 \left (2 (c+d x) \arctan (c+d x)+\left (1+(c+d x)^2\right ) \arctan (c+d x)^2-2 (c+d x)^2 \log \left (\frac {c+d x}{\sqrt {1+(c+d x)^2}}\right )\right )+3 b^3 (c+d x) \left (\arctan (c+d x)^2-2 (c+d x) \arctan (c+d x) \log \left (1-e^{2 i \arctan (c+d x)}\right )+i (c+d x) \left (\arctan (c+d x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arctan (c+d x)}\right )\right )\right )}{2 d e^3 (c+d x)^2} \]
-1/2*(a^3 + b^3*(1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTan[c + d*x]^3 + 3*a^2*b* (c + d*x + (1 + (c + d*x)^2)*ArcTan[c + d*x]) + 3*a*b^2*(2*(c + d*x)*ArcTa n[c + d*x] + (1 + (c + d*x)^2)*ArcTan[c + d*x]^2 - 2*(c + d*x)^2*Log[(c + d*x)/Sqrt[1 + (c + d*x)^2]]) + 3*b^3*(c + d*x)*(ArcTan[c + d*x]^2 - 2*(c + d*x)*ArcTan[c + d*x]*Log[1 - E^((2*I)*ArcTan[c + d*x])] + I*(c + d*x)*(Ar cTan[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcTan[c + d*x])])))/(d*e^3*(c + d*x )^2)
Time = 0.86 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.91, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5566, 27, 5361, 5453, 5361, 5419, 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.
| \(\displaystyle \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx\) |
| \(\Big \downarrow \) 5566 |
| \(\displaystyle \frac {\int \frac {(a+b \arctan (c+d x))^3}{e^3 (c+d x)^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \arctan (c+d x))^3}{(c+d x)^3}d(c+d x)}{d e^3}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {\frac {3}{2} b \int \frac {(a+b \arctan (c+d x))^2}{(c+d x)^2 \left ((c+d x)^2+1\right )}d(c+d x)-\frac {(a+b \arctan (c+d x))^3}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {\frac {3}{2} b \left (\int \frac {(a+b \arctan (c+d x))^2}{(c+d x)^2}d(c+d x)-\int \frac {(a+b \arctan (c+d x))^2}{(c+d x)^2+1}d(c+d x)\right )-\frac {(a+b \arctan (c+d x))^3}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {\frac {3}{2} b \left (2 b \int \frac {a+b \arctan (c+d x)}{(c+d x) \left ((c+d x)^2+1\right )}d(c+d x)-\int \frac {(a+b \arctan (c+d x))^2}{(c+d x)^2+1}d(c+d x)-\frac {(a+b \arctan (c+d x))^2}{c+d x}\right )-\frac {(a+b \arctan (c+d x))^3}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {\frac {3}{2} b \left (2 b \int \frac {a+b \arctan (c+d x)}{(c+d x) \left ((c+d x)^2+1\right )}d(c+d x)-\frac {(a+b \arctan (c+d x))^3}{3 b}-\frac {(a+b \arctan (c+d x))^2}{c+d x}\right )-\frac {(a+b \arctan (c+d x))^3}{2 (c+d x)^2}}{d e^3}\) |
| \(\Big \downarrow \) 5459 |
| \(\displaystyle \frac {-\frac {(a+b \arctan (c+d x))^3}{2 (c+d x)^2}+\frac {3}{2} b \left (2 b \left (i \int \frac {a+b \arctan (c+d x)}{(c+d x) (c+d x+i)}d(c+d x)-\frac {i (a+b \arctan (c+d x))^2}{2 b}\right )-\frac {(a+b \arctan (c+d x))^3}{3 b}-\frac {(a+b \arctan (c+d x))^2}{c+d x}\right )}{d e^3}\) |
| \(\Big \downarrow \) 5403 |
| \(\displaystyle \frac {-\frac {(a+b \arctan (c+d x))^3}{2 (c+d x)^2}+\frac {3}{2} b \left (2 b \left (i \left (i b \int \frac {\log \left (2-\frac {2}{1-i (c+d x)}\right )}{(c+d x)^2+1}d(c+d x)-i \log \left (2-\frac {2}{1-i (c+d x)}\right ) (a+b \arctan (c+d x))\right )-\frac {i (a+b \arctan (c+d x))^2}{2 b}\right )-\frac {(a+b \arctan (c+d x))^3}{3 b}-\frac {(a+b \arctan (c+d x))^2}{c+d x}\right )}{d e^3}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {-\frac {(a+b \arctan (c+d x))^3}{2 (c+d x)^2}+\frac {3}{2} b \left (2 b \left (i \left (-i \log \left (2-\frac {2}{1-i (c+d x)}\right ) (a+b \arctan (c+d x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{1-i (c+d x)}-1\right )\right )-\frac {i (a+b \arctan (c+d x))^2}{2 b}\right )-\frac {(a+b \arctan (c+d x))^3}{3 b}-\frac {(a+b \arctan (c+d x))^2}{c+d x}\right )}{d e^3}\) |
(-1/2*(a + b*ArcTan[c + d*x])^3/(c + d*x)^2 + (3*b*(-((a + b*ArcTan[c + d* x])^2/(c + d*x)) - (a + b*ArcTan[c + d*x])^3/(3*b) + 2*b*(((-1/2*I)*(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))))/2)/(d*e^3)
3.1.19.3.1 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[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_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> 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]
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. 422 vs. \(2 (166 ) = 332\).
Time = 1.42 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.35
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{3} \left (-\frac {\arctan \left (d x +c \right )^{3}}{2 \left (d x +c \right )^{2}}-\frac {3 \arctan \left (d x +c \right )^{2}}{2 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )^{3}}{2}+3 \ln \left (d x +c \right ) \arctan \left (d x +c \right )-\frac {3 \arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{4}+\frac {3 i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{4}+\frac {3 i \ln \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )\right )}{2}-\frac {3 i \ln \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (d x +c \right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (d x +c \right )\right )}{2}\right )}{e^{3}}+\frac {3 a \,b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\arctan \left (d x +c \right )}{d x +c}-\frac {\arctan \left (d x +c \right )^{2}}{2}+\ln \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{e^{3}}+\frac {3 a^{2} b \left (-\frac {\arctan \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {1}{2 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )}{2}\right )}{e^{3}}}{d}\) | \(423\) |
| default | \(\frac {-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{3} \left (-\frac {\arctan \left (d x +c \right )^{3}}{2 \left (d x +c \right )^{2}}-\frac {3 \arctan \left (d x +c \right )^{2}}{2 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )^{3}}{2}+3 \ln \left (d x +c \right ) \arctan \left (d x +c \right )-\frac {3 \arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{4}+\frac {3 i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{4}+\frac {3 i \ln \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )\right )}{2}-\frac {3 i \ln \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (d x +c \right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (d x +c \right )\right )}{2}\right )}{e^{3}}+\frac {3 a \,b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\arctan \left (d x +c \right )}{d x +c}-\frac {\arctan \left (d x +c \right )^{2}}{2}+\ln \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{e^{3}}+\frac {3 a^{2} b \left (-\frac {\arctan \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {1}{2 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )}{2}\right )}{e^{3}}}{d}\) | \(423\) |
| parts | \(-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2} d}+\frac {b^{3} \left (-\frac {\arctan \left (d x +c \right )^{3}}{2 \left (d x +c \right )^{2}}-\frac {3 \arctan \left (d x +c \right )^{2}}{2 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )^{3}}{2}+3 \ln \left (d x +c \right ) \arctan \left (d x +c \right )-\frac {3 \arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{4}+\frac {3 i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{4}+\frac {3 i \ln \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )\right )}{2}-\frac {3 i \ln \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (d x +c \right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (d x +c \right )\right )}{2}\right )}{e^{3} d}+\frac {3 a^{2} b \left (-\frac {\arctan \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {1}{2 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )}{2}\right )}{e^{3} d}+\frac {3 a \,b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\arctan \left (d x +c \right )}{d x +c}-\frac {\arctan \left (d x +c \right )^{2}}{2}+\ln \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{e^{3} d}\) | \(431\) |
1/d*(-1/2*a^3/e^3/(d*x+c)^2+b^3/e^3*(-1/2/(d*x+c)^2*arctan(d*x+c)^3-3/2/(d *x+c)*arctan(d*x+c)^2-1/2*arctan(d*x+c)^3+3*ln(d*x+c)*arctan(d*x+c)-3/2*ar ctan(d*x+c)*ln(1+(d*x+c)^2)-3/4*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)))+3/4*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)))+3/2*I*ln(d*x+c)*ln(1+I*(d*x+c))-3/2*I*ln(d*x+c)* ln(1-I*(d*x+c))+3/2*I*dilog(1+I*(d*x+c))-3/2*I*dilog(1-I*(d*x+c)))+3*a*b^2 /e^3*(-1/2/(d*x+c)^2*arctan(d*x+c)^2-1/(d*x+c)*arctan(d*x+c)-1/2*arctan(d* x+c)^2+ln(d*x+c)-1/2*ln(1+(d*x+c)^2))+3*a^2*b/e^3*(-1/2/(d*x+c)^2*arctan(d *x+c)-1/2/(d*x+c)-1/2*arctan(d*x+c)))
\[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \arctan \left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{3}} \,d x } \]
integral((b^3*arctan(d*x + c)^3 + 3*a*b^2*arctan(d*x + c)^2 + 3*a^2*b*arct an(d*x + c) + a^3)/(d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3*e^ 3), x)
\[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=\frac {\int \frac {a^{3}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{3} \operatorname {atan}^{3}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 a b^{2} \operatorname {atan}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 a^{2} b \operatorname {atan}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]
(Integral(a**3/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integ ral(b**3*atan(c + d*x)**3/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(3*a*b**2*atan(c + d*x)**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x** 2 + d**3*x**3), x) + Integral(3*a**2*b*atan(c + d*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x))/e**3
\[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \arctan \left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{3}} \,d x } \]
-3/2*(d*(1/(d^3*e^3*x + c*d^2*e^3) + arctan((d^2*x + c*d)/d)/(d^2*e^3)) + arctan(d*x + c)/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3))*a^2*b - 3/2*(2* d*(1/(d^3*e^3*x + c*d^2*e^3) + arctan((d^2*x + c*d)/d)/(d^2*e^3))*arctan(d *x + c) - (arctan(d*x + c)^2 - log(d^2*x^2 + 2*c*d*x + c^2 + 1) + 2*log(d* x + c))/(d*e^3))*a*b^2 - 3/2*a*b^2*arctan(d*x + c)^2/(d^3*e^3*x^2 + 2*c*d^ 2*e^3*x + c^2*d*e^3) - 1/32*(8*(d^2*x^2 + 2*c*d*x + c^2 + 1)*arctan(d*x + c)^3 + 12*(d*x + c)*arctan(d*x + c)^2 - 3*(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 - 32*(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)*integrate(1/32 *(16*(d^2*x^2 + 2*c*d*x + c^2 + 1)*arctan(d*x + c)^3 + 12*(d^3*x^3 + 3*c*d ^2*x^2 + c^3 + (3*c^2 + 1)*d*x + c)*arctan(d*x + c)^2 + 3*(d^3*x^3 + 3*c*d ^2*x^2 + c^3 + (3*c^2 + 1)*d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 2 4*(d^2*x^2 + 2*c*d*x + c^2)*arctan(d*x + c) - 12*(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*log(d^2*x^2 + 2*c*d*x + c^2 + 1))/(d^5*e^3*x^5 + 5*c*d^4* e^3*x^4 + (10*c^2 + 1)*d^3*e^3*x^3 + (10*c^3 + 3*c)*d^2*e^3*x^2 + (5*c^4 + 3*c^2)*d*e^3*x + (c^5 + c^3)*e^3), x))*b^3/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 1/2*a^3/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)
Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]