\(\int (c e+d e x)^2 (a+b \arctan (c+d x))^3 \, dx\) [15]
Optimal result
Integrand size = 23, antiderivative size = 271 \[
\int (c e+d e x)^2 (a+b \arctan (c+d x))^3 \, dx=a b^2 e^2 x+\frac {b^3 e^2 (c+d x) \arctan (c+d x)}{d}-\frac {b e^2 (a+b \arctan (c+d x))^2}{2 d}-\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d}-\frac {i e^2 (a+b \arctan (c+d x))^3}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^3}{3 d}-\frac {b e^2 (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {b^3 e^2 \log \left (1+(c+d x)^2\right )}{2 d}-\frac {i b^2 e^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d}-\frac {b^3 e^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d}
\]
[Out]
a*b^2*e^2*x+b^3*e^2*(d*x+c)*arctan(d*x+c)/d-1/2*b*e^2*(a+b*arctan(d*x+c))^2/d-1/2*b*e^2*(d*x+c)^2*(a+b*arctan(
d*x+c))^2/d-1/3*I*e^2*(a+b*arctan(d*x+c))^3/d+1/3*e^2*(d*x+c)^3*(a+b*arctan(d*x+c))^3/d-b*e^2*(a+b*arctan(d*x+
c))^2*ln(2/(1+I*(d*x+c)))/d-1/2*b^3*e^2*ln(1+(d*x+c)^2)/d-I*b^2*e^2*(a+b*arctan(d*x+c))*polylog(2,1-2/(1+I*(d*
x+c)))/d-1/2*b^3*e^2*polylog(3,1-2/(1+I*(d*x+c)))/d
Rubi [A] (verified)
Time = 0.30 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of
steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5151, 12, 4946, 5036, 4930,
266, 5004, 5040, 4964, 5114, 6745} \[
\int (c e+d e x)^2 (a+b \arctan (c+d x))^3 \, dx=-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))}{d}-\frac {b e^2 (a+b \arctan (c+d x))^2}{2 d}-\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^3}{3 d}-\frac {i e^2 (a+b \arctan (c+d x))^3}{3 d}-\frac {b e^2 \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^2}{d}+a b^2 e^2 x+\frac {b^3 e^2 (c+d x) \arctan (c+d x)}{d}-\frac {b^3 e^2 \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )}{2 d}-\frac {b^3 e^2 \log \left ((c+d x)^2+1\right )}{2 d}
\]
[In]
Int[(c*e + d*e*x)^2*(a + b*ArcTan[c + d*x])^3,x]
[Out]
a*b^2*e^2*x + (b^3*e^2*(c + d*x)*ArcTan[c + d*x])/d - (b*e^2*(a + b*ArcTan[c + d*x])^2)/(2*d) - (b*e^2*(c + d*
x)^2*(a + b*ArcTan[c + d*x])^2)/(2*d) - ((I/3)*e^2*(a + b*ArcTan[c + d*x])^3)/d + (e^2*(c + d*x)^3*(a + b*ArcT
an[c + d*x])^3)/(3*d) - (b*e^2*(a + b*ArcTan[c + d*x])^2*Log[2/(1 + I*(c + d*x))])/d - (b^3*e^2*Log[1 + (c + d
*x)^2])/(2*d) - (I*b^2*e^2*(a + b*ArcTan[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d - (b^3*e^2*PolyLog[3
, 1 - 2/(1 + I*(c + d*x))])/(2*d)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 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 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 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 5114
Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[b*p*(I/2), Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 -
u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - 2
*(I/(I - c*x)))^2, 0]
Rule 5151
Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[(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]
Rule 6745
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
; !FalseQ[w]] /; FreeQ[n, x]
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int e^2 x^2 (a+b \arctan (x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int x^2 (a+b \arctan (x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^3}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x^3 (a+b \arctan (x))^2}{1+x^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^3}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int x (a+b \arctan (x))^2 \, dx,x,c+d x\right )}{d}+\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x (a+b \arctan (x))^2}{1+x^2} \, dx,x,c+d x\right )}{d} \\ & = -\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d}-\frac {i e^2 (a+b \arctan (c+d x))^3}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^3}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2}{i-x} \, dx,x,c+d x\right )}{d}+\frac {\left (b^2 e^2\right ) \text {Subst}\left (\int \frac {x^2 (a+b \arctan (x))}{1+x^2} \, dx,x,c+d x\right )}{d} \\ & = -\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d}-\frac {i e^2 (a+b \arctan (c+d x))^3}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^3}{3 d}-\frac {b e^2 (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d}+\frac {\left (b^2 e^2\right ) \text {Subst}(\int (a+b \arctan (x)) \, dx,x,c+d x)}{d}-\frac {\left (b^2 e^2\right ) \text {Subst}\left (\int \frac {a+b \arctan (x)}{1+x^2} \, dx,x,c+d x\right )}{d}+\frac {\left (2 b^2 e^2\right ) \text {Subst}\left (\int \frac {(a+b \arctan (x)) \log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d} \\ & = a b^2 e^2 x-\frac {b e^2 (a+b \arctan (c+d x))^2}{2 d}-\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d}-\frac {i e^2 (a+b \arctan (c+d x))^3}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^3}{3 d}-\frac {b e^2 (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {i b^2 e^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d}+\frac {\left (i b^3 e^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d}+\frac {\left (b^3 e^2\right ) \text {Subst}(\int \arctan (x) \, dx,x,c+d x)}{d} \\ & = a b^2 e^2 x+\frac {b^3 e^2 (c+d x) \arctan (c+d x)}{d}-\frac {b e^2 (a+b \arctan (c+d x))^2}{2 d}-\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d}-\frac {i e^2 (a+b \arctan (c+d x))^3}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^3}{3 d}-\frac {b e^2 (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {i b^2 e^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d}-\frac {b^3 e^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d}-\frac {\left (b^3 e^2\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{d} \\ & = a b^2 e^2 x+\frac {b^3 e^2 (c+d x) \arctan (c+d x)}{d}-\frac {b e^2 (a+b \arctan (c+d x))^2}{2 d}-\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d}-\frac {i e^2 (a+b \arctan (c+d x))^3}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^3}{3 d}-\frac {b e^2 (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {b^3 e^2 \log \left (1+(c+d x)^2\right )}{2 d}-\frac {i b^2 e^2 (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d}-\frac {b^3 e^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.58 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.29
\[
\int (c e+d e x)^2 (a+b \arctan (c+d x))^3 \, dx=\frac {e^2 \left (-3 a^2 b (c+d x)^2+2 a^3 (c+d x)^3+6 a^2 b (c+d x)^3 \arctan (c+d x)+3 a^2 b \log \left (1+(c+d x)^2\right )+6 a b^2 \left (c+d x-\arctan (c+d x)-(c+d x)^2 \arctan (c+d x)+i \arctan (c+d x)^2+(c+d x)^3 \arctan (c+d x)^2-2 \arctan (c+d x) \log \left (1+e^{2 i \arctan (c+d x)}\right )+i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )+b^3 \left (6 (c+d x) \arctan (c+d x)-3 \left (1+(c+d x)^2\right ) \arctan (c+d x)^2+2 i \arctan (c+d x)^3-2 (c+d x) \arctan (c+d x)^3+2 (c+d x) \left (1+(c+d x)^2\right ) \arctan (c+d x)^3-6 \arctan (c+d x)^2 \log \left (1+e^{2 i \arctan (c+d x)}\right )+6 \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )+6 i \arctan (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )-3 \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )\right )\right )}{6 d}
\]
[In]
Integrate[(c*e + d*e*x)^2*(a + b*ArcTan[c + d*x])^3,x]
[Out]
(e^2*(-3*a^2*b*(c + d*x)^2 + 2*a^3*(c + d*x)^3 + 6*a^2*b*(c + d*x)^3*ArcTan[c + d*x] + 3*a^2*b*Log[1 + (c + d*
x)^2] + 6*a*b^2*(c + d*x - ArcTan[c + d*x] - (c + d*x)^2*ArcTan[c + d*x] + I*ArcTan[c + d*x]^2 + (c + d*x)^3*A
rcTan[c + d*x]^2 - 2*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + I*PolyLog[2, -E^((2*I)*ArcTan[c + d*
x])]) + b^3*(6*(c + d*x)*ArcTan[c + d*x] - 3*(1 + (c + d*x)^2)*ArcTan[c + d*x]^2 + (2*I)*ArcTan[c + d*x]^3 - 2
*(c + d*x)*ArcTan[c + d*x]^3 + 2*(c + d*x)*(1 + (c + d*x)^2)*ArcTan[c + d*x]^3 - 6*ArcTan[c + d*x]^2*Log[1 + E
^((2*I)*ArcTan[c + d*x])] + 6*Log[1/Sqrt[1 + (c + d*x)^2]] + (6*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I)*ArcTan
[c + d*x])] - 3*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])))/(6*d)
Maple [C] (warning: unable to verify)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.18 (sec) , antiderivative size = 1249, normalized size of antiderivative =
4.61
| | |
| method | result | size |
| | |
| derivativedivides |
\(\text {Expression too large to display}\) |
\(1249\) |
| default |
\(\text {Expression too large to display}\) |
\(1249\) |
| parts |
\(\text {Expression too large to display}\) |
\(1257\) |
| | |
|
|
|
|
[In]
int((d*e*x+c*e)^2*(a+b*arctan(d*x+c))^3,x,method=_RETURNVERBOSE)
[Out]
1/d*(1/3*e^2*a^3*(d*x+c)^3+e^2*b^3*(1/3*(d*x+c)^3*arctan(d*x+c)^3-1/2*(d*x+c)^2*arctan(d*x+c)^2+1/2*arctan(d*x
+c)^2*ln(1+(d*x+c)^2)-arctan(d*x+c)^2*ln((1+I*(d*x+c))/(1+(d*x+c)^2)^(1/2))+I*arctan(d*x+c)*polylog(2,-(1+I*(d
*x+c))^2/(1+(d*x+c)^2))-1/2*polylog(3,-(1+I*(d*x+c))^2/(1+(d*x+c)^2))+1/12*I*arctan(d*x+c)*(3*csgn(I*(1+I*(d*x
+c))/(1+(d*x+c)^2)^(1/2))^2*csgn(I*(1+I*(d*x+c))^2/(1+(d*x+c)^2))*Pi*arctan(d*x+c)-6*csgn(I*(1+I*(d*x+c))/(1+(
d*x+c)^2)^(1/2))*csgn(I*(1+I*(d*x+c))^2/(1+(d*x+c)^2))^2*Pi*arctan(d*x+c)+3*csgn(I*(1+I*(d*x+c))^2/(1+(d*x+c)^
2))^3*Pi*arctan(d*x+c)-3*csgn(I*(1+I*(d*x+c))^2/(1+(d*x+c)^2))*csgn(I*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(1+(1+I*(d
*x+c))^2/(1+(d*x+c)^2))^2)^2*Pi*arctan(d*x+c)+3*csgn(I*(1+I*(d*x+c))^2/(1+(d*x+c)^2))*csgn(I*(1+I*(d*x+c))^2/(
1+(d*x+c)^2)/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2))^2)*csgn(I/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2))^2)*Pi*arctan(d*x+c)
-3*csgn(I*(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))^2*csgn(I*(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2))^2)*Pi*arctan(d*x+c)+6*
csgn(I*(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))*csgn(I*(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2))^2)^2*Pi*arctan(d*x+c)-3*csg
n(I*(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2))^2)^3*Pi*arctan(d*x+c)+3*csgn(I*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(1+(1+I*(d*
x+c))^2/(1+(d*x+c)^2))^2)^3*Pi*arctan(d*x+c)-3*csgn(I*(1+I*(d*x+c))^2/(1+(d*x+c)^2)/(1+(1+I*(d*x+c))^2/(1+(d*x
+c)^2))^2)^2*csgn(I/(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2))^2)*Pi*arctan(d*x+c)+4*arctan(d*x+c)^2+12*I*ln(2)*arctan(
d*x+c)+6*I*arctan(d*x+c)-12-12*I*(d*x+c))+ln(1+(1+I*(d*x+c))^2/(1+(d*x+c)^2)))+3*e^2*a*b^2*(1/3*(d*x+c)^3*arct
an(d*x+c)^2-1/3*(d*x+c)^2*arctan(d*x+c)+1/3*arctan(d*x+c)*ln(1+(d*x+c)^2)+1/3*d*x+1/3*c-1/3*arctan(d*x+c)+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))))+3*e
^2*a^2*b*(1/3*(d*x+c)^3*arctan(d*x+c)-1/6*(d*x+c)^2+1/6*ln(1+(d*x+c)^2)))
Fricas [F]
\[
\int (c e+d e x)^2 (a+b \arctan (c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x }
\]
[In]
integrate((d*e*x+c*e)^2*(a+b*arctan(d*x+c))^3,x, algorithm="fricas")
[Out]
integral(a^3*d^2*e^2*x^2 + 2*a^3*c*d*e^2*x + a^3*c^2*e^2 + (b^3*d^2*e^2*x^2 + 2*b^3*c*d*e^2*x + b^3*c^2*e^2)*a
rctan(d*x + c)^3 + 3*(a*b^2*d^2*e^2*x^2 + 2*a*b^2*c*d*e^2*x + a*b^2*c^2*e^2)*arctan(d*x + c)^2 + 3*(a^2*b*d^2*
e^2*x^2 + 2*a^2*b*c*d*e^2*x + a^2*b*c^2*e^2)*arctan(d*x + c), x)
Sympy [F]
\[
\int (c e+d e x)^2 (a+b \arctan (c+d x))^3 \, dx=e^{2} \left (\int a^{3} c^{2}\, dx + \int a^{3} d^{2} x^{2}\, dx + \int b^{3} c^{2} \operatorname {atan}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} c^{2} \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b c^{2} \operatorname {atan}{\left (c + d x \right )}\, dx + \int 2 a^{3} c d x\, dx + \int b^{3} d^{2} x^{2} \operatorname {atan}^{3}{\left (c + d x \right )}\, dx + \int 3 a b^{2} d^{2} x^{2} \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 3 a^{2} b d^{2} x^{2} \operatorname {atan}{\left (c + d x \right )}\, dx + \int 2 b^{3} c d x \operatorname {atan}^{3}{\left (c + d x \right )}\, dx + \int 6 a b^{2} c d x \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 6 a^{2} b c d x \operatorname {atan}{\left (c + d x \right )}\, dx\right )
\]
[In]
integrate((d*e*x+c*e)**2*(a+b*atan(d*x+c))**3,x)
[Out]
e**2*(Integral(a**3*c**2, x) + Integral(a**3*d**2*x**2, x) + Integral(b**3*c**2*atan(c + d*x)**3, x) + Integra
l(3*a*b**2*c**2*atan(c + d*x)**2, x) + Integral(3*a**2*b*c**2*atan(c + d*x), x) + Integral(2*a**3*c*d*x, x) +
Integral(b**3*d**2*x**2*atan(c + d*x)**3, x) + Integral(3*a*b**2*d**2*x**2*atan(c + d*x)**2, x) + Integral(3*a
**2*b*d**2*x**2*atan(c + d*x), x) + Integral(2*b**3*c*d*x*atan(c + d*x)**3, x) + Integral(6*a*b**2*c*d*x*atan(
c + d*x)**2, x) + Integral(6*a**2*b*c*d*x*atan(c + d*x), x))
Maxima [F]
\[
\int (c e+d e x)^2 (a+b \arctan (c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x }
\]
[In]
integrate((d*e*x+c*e)^2*(a+b*arctan(d*x+c))^3,x, algorithm="maxima")
[Out]
7/8*b^3*c^4*e^2*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d + 3*a*b^2*c^4*e^2*arctan(d*x + c)^2*arctan((d^2*x
+ c*d)/d)/d - (3*arctan(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^4*e^2 - 7/
32*(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^4*e^2 + 1/3*a^3*d^2*e^2*x^3 + 7/8*b^3*c^2*e^2*arctan(d*x + c)^3*arctan((d^2*x + c*d
)/d)/d + 28*b^3*d^4*e^2*integrate(1/32*x^4*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*d^4*e^2
*integrate(1/32*x^4*arctan(d*x + c)*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) + 96*
a*b^2*d^4*e^2*integrate(1/32*x^4*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 112*b^3*c*d^3*e^2*integ
rate(1/32*x^3*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 4*b^3*d^4*e^2*integrate(1/32*x^4*arctan(d*
x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^3*c*d^3*e^2*integrate(1/32*x^
3*arctan(d*x + c)*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) + 384*a*b^2*c*d^3*e^2*i
ntegrate(1/32*x^3*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 168*b^3*c^2*d^2*e^2*integrate(1/32*x^2
*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 16*b^3*c*d^3*e^2*integrate(1/32*x^3*arctan(d*x + c)*log
(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 18*b^3*c^2*d^2*e^2*integrate(1/32*x^2*arctan
(d*x + c)*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) + 576*a*b^2*c^2*d^2*e^2*integra
te(1/32*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 112*b^3*c^3*d*e^2*integrate(1/32*x*arctan(d*
x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 24*b^3*c^2*d^2*e^2*integrate(1/32*x^2*arctan(d*x + c)*log(d^2*x^2
+ 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^3*c^3*d*e^2*integrate(1/32*x*arctan(d*x + c)*lo
g(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 384*a*b^2*c^3*d*e^2*integrate(1/32*x*arct
an(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^3*c^3*d*e^2*integrate(1/32*x*arctan(d*x + c)*log(d^2*x^
2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*c^4*e^2*integrate(1/32*arctan(d*x + c)*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) + a^3*c*d*e^2*x^2 + 3*a*b^2*c^2*e^2*arctan(d*x
+ c)^2*arctan((d^2*x + c*d)/d)/d - 4*b^3*d^3*e^2*integrate(1/32*x^3*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2
+ 1), x) + b^3*d^3*e^2*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
) - 12*b^3*c*d^2*e^2*integrate(1/32*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*c*d^2*e^2*
integrate(1/32*x^2*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) - 12*b^3*c^2*d*e^2*int
egrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*c^2*d*e^2*integrate(1/32*x*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) - (3*arctan(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^2*e^2 - 7/32*(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^2*e^2 + 3*(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*c*d*e^2 + 1/2*
(2*x^3*arctan(d*x + c) - d*((d*x^2 - 4*c*x)/d^3 - 2*(c^3 - 3*c)*arctan((d^2*x + c*d)/d)/d^4 + (3*c^2 - 1)*log(
d^2*x^2 + 2*c*d*x + c^2 + 1)/d^4))*a^2*b*d^2*e^2 + a^3*c^2*e^2*x + 28*b^3*d^2*e^2*integrate(1/32*x^2*arctan(d*
x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*d^2*e^2*integrate(1/32*x^2*arctan(d*x + c)*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) + 96*a*b^2*d^2*e^2*integrate(1/32*x^2*arctan(d*x + c)^2/(
d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*c*d*e^2*integrate(1/32*x*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2
+ 1), x) + 6*b^3*c*d*e^2*integrate(1/32*x*arctan(d*x + c)*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) + 192*a*b^2*c*d*e^2*integrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*
b^3*c^2*e^2*integrate(1/32*arctan(d*x + c)*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
) + 3/2*(2*(d*x + c)*arctan(d*x + c) - log((d*x + c)^2 + 1))*a^2*b*c^2*e^2/d + 1/24*(b^3*d^2*e^2*x^3 + 3*b^3*c
*d*e^2*x^2 + 3*b^3*c^2*e^2*x)*arctan(d*x + c)^3 - 1/32*(b^3*d^2*e^2*x^3 + 3*b^3*c*d*e^2*x^2 + 3*b^3*c^2*e^2*x)
*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2
Giac [F]
\[
\int (c e+d e x)^2 (a+b \arctan (c+d x))^3 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x }
\]
[In]
integrate((d*e*x+c*e)^2*(a+b*arctan(d*x+c))^3,x, algorithm="giac")
[Out]
sage0*x
Mupad [F(-1)]
Timed out. \[
\int (c e+d e x)^2 (a+b \arctan (c+d x))^3 \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3 \,d x
\]
[In]
int((c*e + d*e*x)^2*(a + b*atan(c + d*x))^3,x)
[Out]
int((c*e + d*e*x)^2*(a + b*atan(c + d*x))^3, x)