Integrand size = 21, antiderivative size = 164 \[ \int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx=-\frac {3 i b e (a+b \arctan (c+d x))^2}{2 d}-\frac {3 b e (c+d x) (a+b \arctan (c+d x))^2}{2 d}+\frac {e (a+b \arctan (c+d x))^3}{2 d}+\frac {e (c+d x)^2 (a+b \arctan (c+d x))^3}{2 d}-\frac {3 b^2 e (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {3 i b^3 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d} \] Output:
-3/2*I*b*e*(a+b*arctan(d*x+c))^2/d-3/2*b*e*(d*x+c)*(a+b*arctan(d*x+c))^2/d +1/2*e*(a+b*arctan(d*x+c))^3/d+1/2*e*(d*x+c)^2*(a+b*arctan(d*x+c))^3/d-3*b ^2*e*(a+b*arctan(d*x+c))*ln(2/(1+I*(d*x+c)))/d-3/2*I*b^3*e*polylog(2,1-2/( 1+I*(d*x+c)))/d
Time = 0.16 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.20 \[ \int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx=\frac {e \left (3 b^2 (-i+c+d x) (-b+a (i+c+d x)) \arctan (c+d x)^2+b^3 \left (1+c^2+2 c d x+d^2 x^2\right ) \arctan (c+d x)^3+3 b \arctan (c+d x) \left (a \left (-2 b (c+d x)+a \left (1+c^2+2 c d x+d^2 x^2\right )\right )-2 b^2 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )+a \left (a (c+d x) (-3 b+a c+a d x)-6 b^2 \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )\right )+3 i b^3 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )}{2 d} \] Input:
Integrate[(c*e + d*e*x)*(a + b*ArcTan[c + d*x])^3,x]
Output:
(e*(3*b^2*(-I + c + d*x)*(-b + a*(I + c + d*x))*ArcTan[c + d*x]^2 + b^3*(1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTan[c + d*x]^3 + 3*b*ArcTan[c + d*x]*(a*(-2 *b*(c + d*x) + a*(1 + c^2 + 2*c*d*x + d^2*x^2)) - 2*b^2*Log[1 + E^((2*I)*A rcTan[c + d*x])]) + a*(a*(c + d*x)*(-3*b + a*c + a*d*x) - 6*b^2*Log[1/Sqrt [1 + (c + d*x)^2]]) + (3*I)*b^3*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]))/( 2*d)
Time = 0.88 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.92, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5566, 27, 5361, 5451, 5345, 5419, 5455, 5379, 2849, 2752}
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.
| \(\displaystyle \int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 5566 |
| \(\displaystyle \frac {\int e (c+d x) (a+b \arctan (c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int (c+d x) (a+b \arctan (c+d x))^3d(c+d x)}{d}\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arctan (c+d x))^3-\frac {3}{2} b \int \frac {(c+d x)^2 (a+b \arctan (c+d x))^2}{(c+d x)^2+1}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arctan (c+d x))^3-\frac {3}{2} b \left (\int (a+b \arctan (c+d x))^2d(c+d x)-\int \frac {(a+b \arctan (c+d x))^2}{(c+d x)^2+1}d(c+d x)\right )\right )}{d}\) |
| \(\Big \downarrow \) 5345 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arctan (c+d x))^3-\frac {3}{2} b \left (-2 b \int \frac {(c+d x) (a+b \arctan (c+d x))}{(c+d x)^2+1}d(c+d x)-\int \frac {(a+b \arctan (c+d x))^2}{(c+d x)^2+1}d(c+d x)+(c+d x) (a+b \arctan (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arctan (c+d x))^3-\frac {3}{2} b \left (-2 b \int \frac {(c+d x) (a+b \arctan (c+d x))}{(c+d x)^2+1}d(c+d x)-\frac {(a+b \arctan (c+d x))^3}{3 b}+(c+d x) (a+b \arctan (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arctan (c+d x))^3-\frac {3}{2} b \left (-2 b \left (-\int \frac {a+b \arctan (c+d x)}{-c-d x+i}d(c+d x)-\frac {i (a+b \arctan (c+d x))^2}{2 b}\right )-\frac {(a+b \arctan (c+d x))^3}{3 b}+(c+d x) (a+b \arctan (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arctan (c+d x))^3-\frac {3}{2} b \left (-2 b \left (b \int \frac {\log \left (\frac {2}{i (c+d x)+1}\right )}{(c+d x)^2+1}d(c+d x)-\frac {i (a+b \arctan (c+d x))^2}{2 b}-\log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))\right )-\frac {(a+b \arctan (c+d x))^3}{3 b}+(c+d x) (a+b \arctan (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arctan (c+d x))^3-\frac {3}{2} b \left (-2 b \left (-i b \int \frac {\log \left (\frac {2}{i (c+d x)+1}\right )}{1-\frac {2}{i (c+d x)+1}}d\frac {1}{i (c+d x)+1}-\frac {i (a+b \arctan (c+d x))^2}{2 b}-\log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))\right )-\frac {(a+b \arctan (c+d x))^3}{3 b}+(c+d x) (a+b \arctan (c+d x))^2\right )\right )}{d}\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \arctan (c+d x))^3-\frac {3}{2} b \left (-2 b \left (-\frac {i (a+b \arctan (c+d x))^2}{2 b}-\log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))-\frac {1}{2} i b \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )\right )-\frac {(a+b \arctan (c+d x))^3}{3 b}+(c+d x) (a+b \arctan (c+d x))^2\right )\right )}{d}\) |
Input:
Int[(c*e + d*e*x)*(a + b*ArcTan[c + d*x])^3,x]
Output:
(e*(((c + d*x)^2*(a + b*ArcTan[c + d*x])^3)/2 - (3*b*((c + d*x)*(a + b*Arc Tan[c + d*x])^2 - (a + b*ArcTan[c + d*x])^3/(3*b) - 2*b*(((-1/2*I)*(a + b* ArcTan[c + d*x])^2)/b - (a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x))] - (I/2)*b*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])))/2))/d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-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]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[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])
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_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[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 ]
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[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] )^p, x], x] - Simp[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]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si mp[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]
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. 320 vs. \(2 (150 ) = 300\).
Time = 0.65 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.96
| method | result | size |
| derivativedivides | \(\frac {\frac {e \,a^{3} \left (d x +c \right )^{2}}{2}+e \,b^{3} \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )^{3}}{2}+\frac {\arctan \left (d x +c \right )^{3}}{2}-\frac {3 \left (d x +c \right ) \arctan \left (d x +c \right )^{2}}{2}+\frac {3 \arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\frac {3 i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{4}-\frac {3 i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{4}\right )+3 e a \,b^{2} \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )^{2}}{2}+\frac {\arctan \left (d x +c \right )^{2}}{2}-\left (d x +c \right ) \arctan \left (d x +c \right )+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )+3 e \,a^{2} b \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}+\frac {\arctan \left (d x +c \right )}{2}\right )}{d}\) | \(321\) |
| default | \(\frac {\frac {e \,a^{3} \left (d x +c \right )^{2}}{2}+e \,b^{3} \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )^{3}}{2}+\frac {\arctan \left (d x +c \right )^{3}}{2}-\frac {3 \left (d x +c \right ) \arctan \left (d x +c \right )^{2}}{2}+\frac {3 \arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\frac {3 i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{4}-\frac {3 i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{4}\right )+3 e a \,b^{2} \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )^{2}}{2}+\frac {\arctan \left (d x +c \right )^{2}}{2}-\left (d x +c \right ) \arctan \left (d x +c \right )+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )+3 e \,a^{2} b \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}+\frac {\arctan \left (d x +c \right )}{2}\right )}{d}\) | \(321\) |
| parts | \(e \,a^{3} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e \,b^{3} \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )^{3}}{2}+\frac {\arctan \left (d x +c \right )^{3}}{2}-\frac {3 \left (d x +c \right ) \arctan \left (d x +c \right )^{2}}{2}+\frac {3 \arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\frac {3 i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{4}-\frac {3 i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{4}\right )}{d}+\frac {3 e a \,b^{2} \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )^{2}}{2}+\frac {\arctan \left (d x +c \right )^{2}}{2}-\left (d x +c \right ) \arctan \left (d x +c \right )+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {3 e \,a^{2} b \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}+\frac {\arctan \left (d x +c \right )}{2}\right )}{d}\) | \(328\) |
| risch | \(\text {Expression too large to display}\) | \(1413\) |
Input:
int((d*e*x+c*e)*(a+b*arctan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/2*e*a^3*(d*x+c)^2+e*b^3*(1/2*(d*x+c)^2*arctan(d*x+c)^3+1/2*arctan(d *x+c)^3-3/2*(d*x+c)*arctan(d*x+c)^2+3/2*arctan(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))-l n(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*e*a *b^2*(1/2*(d*x+c)^2*arctan(d*x+c)^2+1/2*arctan(d*x+c)^2-(d*x+c)*arctan(d*x +c)+1/2*ln(1+(d*x+c)^2))+3*e*a^2*b*(1/2*(d*x+c)^2*arctan(d*x+c)-1/2*d*x-1/ 2*c+1/2*arctan(d*x+c)))
\[ \int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx=\int { {\left (d e x + c e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:
integrate((d*e*x+c*e)*(a+b*arctan(d*x+c))^3,x, algorithm="fricas")
Output:
integral(a^3*d*e*x + a^3*c*e + (b^3*d*e*x + b^3*c*e)*arctan(d*x + c)^3 + 3 *(a*b^2*d*e*x + a*b^2*c*e)*arctan(d*x + c)^2 + 3*(a^2*b*d*e*x + a^2*b*c*e) *arctan(d*x + c), x)
\[ \int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx=e \left (\int a^{3} c\, dx + \int a^{3} d x\, dx + \int b^{3} c \operatorname {atan}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} c \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b c \operatorname {atan}{\left (c + d x \right )}\, dx + \int b^{3} d x \operatorname {atan}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} d x \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b d x \operatorname {atan}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate((d*e*x+c*e)*(a+b*atan(d*x+c))**3,x)
Output:
e*(Integral(a**3*c, x) + Integral(a**3*d*x, x) + Integral(b**3*c*atan(c + d*x)**3, x) + Integral(3*a*b**2*c*atan(c + d*x)**2, x) + Integral(3*a**2*b *c*atan(c + d*x), x) + Integral(b**3*d*x*atan(c + d*x)**3, x) + Integral(3 *a*b**2*d*x*atan(c + d*x)**2, x) + Integral(3*a**2*b*d*x*atan(c + d*x), x) )
\[ \int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx=\int { {\left (d e x + c e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:
integrate((d*e*x+c*e)*(a+b*arctan(d*x+c))^3,x, algorithm="maxima")
Output:
1/2*a^3*d*e*x^2 + 3/2*(x^2*arctan(d*x + c) - d*(x/d^2 + (c^2 - 1)*arctan(( d^2*x + c*d)/d)/d^3 - c*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d^3))*a^2*b*d*e + a^3*c*e*x + 3/2*(2*(d*x + c)*arctan(d*x + c) - log((d*x + c)^2 + 1))*a^2* b*c*e/d + 1/32*(8*(b^3*d^2*e*x^2 + 2*b^3*c*d*e*x + (b^3*c^2 + b^3)*e)*arct an(d*x + c)^3 + 12*(a*b^2*d^2*e*x^2 + (2*a*b^2*c - b^3)*d*e*x)*arctan(d*x + c)^2 - 3*(a*b^2*d^2*e*x^2 + (2*a*b^2*c - b^3)*d*e*x)*log(d^2*x^2 + 2*c*d *x + c^2 + 1)^2 + 4*(4*b^3*c^3*e*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d) /d + 18*a*b^2*c^3*e*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d - 6*(3*arc tan(d*x + c)*arctan((d^2*x + c*d)/d)^2/d - arctan((d^2*x + c*d)/d)^3/d)*a* b^2*c^3*e - (6*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)^2/d - 4*arctan(d* x + c)*arctan((d^2*x + c*d)/d)^3/d + arctan((d^2*x + c*d)/d)^4/d)*b^3*c^3* e - 3*b^3*c^2*e*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d + 4*b^3*c*e*ar ctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d + 128*b^3*d^3*e*integrate(1/32*x ^3*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 576*a*b^2*d^3*e*i ntegrate(1/32*x^3*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 38 4*b^3*c*d^2*e*integrate(1/32*x^2*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^ 2 + 1), x) + 48*a*b^2*d^3*e*integrate(1/32*x^3*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 1728*a*b^2*c*d^2*e*integrate(1 /32*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 384*b^3*c^2* d*e*integrate(1/32*x*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x...
\[ \int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx=\int { {\left (d e x + c e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \] Input:
integrate((d*e*x+c*e)*(a+b*arctan(d*x+c))^3,x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3 \,d x \] Input:
int((c*e + d*e*x)*(a + b*atan(c + d*x))^3,x)
Output:
int((c*e + d*e*x)*(a + b*atan(c + d*x))^3, x)
\[ \int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx=\frac {e \left (\mathit {atan} \left (d x +c \right )^{3} b^{3} c^{2}+2 \mathit {atan} \left (d x +c \right )^{3} b^{3} c d x +\mathit {atan} \left (d x +c \right )^{3} b^{3} d^{2} x^{2}+\mathit {atan} \left (d x +c \right )^{3} b^{3}+3 \mathit {atan} \left (d x +c \right )^{2} a \,b^{2} c^{2}+6 \mathit {atan} \left (d x +c \right )^{2} a \,b^{2} c d x +3 \mathit {atan} \left (d x +c \right )^{2} a \,b^{2} d^{2} x^{2}+3 \mathit {atan} \left (d x +c \right )^{2} a \,b^{2}-3 \mathit {atan} \left (d x +c \right )^{2} b^{3} d x +3 \mathit {atan} \left (d x +c \right ) a^{2} b \,c^{2}+6 \mathit {atan} \left (d x +c \right ) a^{2} b c d x +3 \mathit {atan} \left (d x +c \right ) a^{2} b \,d^{2} x^{2}+3 \mathit {atan} \left (d x +c \right ) a^{2} b -6 \mathit {atan} \left (d x +c \right ) a \,b^{2} c -6 \mathit {atan} \left (d x +c \right ) a \,b^{2} d x +6 \left (\int \frac {\mathit {atan} \left (d x +c \right ) x}{d^{2} x^{2}+2 c d x +c^{2}+1}d x \right ) b^{3} d^{2}+3 \,\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) a \,b^{2}+2 a^{3} c d x +a^{3} d^{2} x^{2}-3 a^{2} b d x \right )}{2 d} \] Input:
int((d*e*x+c*e)*(a+b*atan(d*x+c))^3,x)
Output:
(e*(atan(c + d*x)**3*b**3*c**2 + 2*atan(c + d*x)**3*b**3*c*d*x + atan(c + d*x)**3*b**3*d**2*x**2 + atan(c + d*x)**3*b**3 + 3*atan(c + d*x)**2*a*b**2 *c**2 + 6*atan(c + d*x)**2*a*b**2*c*d*x + 3*atan(c + d*x)**2*a*b**2*d**2*x **2 + 3*atan(c + d*x)**2*a*b**2 - 3*atan(c + d*x)**2*b**3*d*x + 3*atan(c + d*x)*a**2*b*c**2 + 6*atan(c + d*x)*a**2*b*c*d*x + 3*atan(c + d*x)*a**2*b* d**2*x**2 + 3*atan(c + d*x)*a**2*b - 6*atan(c + d*x)*a*b**2*c - 6*atan(c + d*x)*a*b**2*d*x + 6*int((atan(c + d*x)*x)/(c**2 + 2*c*d*x + d**2*x**2 + 1 ),x)*b**3*d**2 + 3*log(c**2 + 2*c*d*x + d**2*x**2 + 1)*a*b**2 + 2*a**3*c*d *x + a**3*d**2*x**2 - 3*a**2*b*d*x))/(2*d)