\(\int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx\) [19]

Optimal result

Integrand size = 23, antiderivative size = 180 \[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=-\frac {3 i b (a+b \arctan (c+d x))^2}{2 d e^3}-\frac {3 b (a+b \arctan (c+d x))^2}{2 d e^3 (c+d x)}-\frac {(a+b \arctan (c+d x))^3}{2 d e^3}-\frac {(a+b \arctan (c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 (a+b \arctan (c+d x)) \log \left (2-\frac {2}{1-i (c+d x)}\right )}{d e^3}-\frac {3 i b^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i (c+d x)}\right )}{2 d e^3} \]

[Out]

-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*pol
ylog(2,-1+2/(1-I*(d*x+c)))/d/e^3

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5151, 12, 4946, 5038, 5044, 4988, 2497, 5004} \[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=\frac {3 b^2 \log \left (2-\frac {2}{1-i (c+d x)}\right ) (a+b \arctan (c+d x))}{d e^3}-\frac {3 b (a+b \arctan (c+d x))^2}{2 d e^3 (c+d x)}-\frac {3 i b (a+b \arctan (c+d x))^2}{2 d e^3}-\frac {(a+b \arctan (c+d x))^3}{2 d e^3 (c+d x)^2}-\frac {(a+b \arctan (c+d x))^3}{2 d e^3}-\frac {3 i b^3 \operatorname {PolyLog}\left (2,\frac {2}{1-i (c+d x)}-1\right )}{2 d e^3} \]

[In]

Int[(a + b*ArcTan[c + d*x])^3/(c*e + d*e*x)^3,x]

[Out]

(((-3*I)/2)*b*(a + b*ArcTan[c + d*x])^2)/(d*e^3) - (3*b*(a + b*ArcTan[c + d*x])^2)/(2*d*e^3*(c + d*x)) - (a +
b*ArcTan[c + d*x])^3/(2*d*e^3) - (a + b*ArcTan[c + d*x])^3/(2*d*e^3*(c + d*x)^2) + (3*b^2*(a + b*ArcTan[c + d*
x])*Log[2 - 2/(1 - I*(c + d*x))])/(d*e^3) - (((3*I)/2)*b^3*PolyLog[2, -1 + 2/(1 - I*(c + d*x))])/(d*e^3)

Rule 12

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

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 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 4988

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] - Dist[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]

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 5038

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Dist[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]

Rule 5044

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*d*(p + 1))), x] + Dist[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]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \arctan (x))^3}{e^3 x^3} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \arctan (x))^3}{x^3} \, dx,x,c+d x\right )}{d e^3} \\ & = -\frac {(a+b \arctan (c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {(3 b) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2}{x^2 \left (1+x^2\right )} \, dx,x,c+d x\right )}{2 d e^3} \\ & = -\frac {(a+b \arctan (c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {(3 b) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2}{x^2} \, dx,x,c+d x\right )}{2 d e^3}-\frac {(3 b) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2}{1+x^2} \, dx,x,c+d x\right )}{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 {\left (3 b^2\right ) \text {Subst}\left (\int \frac {a+b \arctan (x)}{x \left (1+x^2\right )} \, dx,x,c+d x\right )}{d e^3} \\ & = -\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 {\left (3 i b^2\right ) \text {Subst}\left (\int \frac {a+b \arctan (x)}{x (i+x)} \, dx,x,c+d x\right )}{d e^3} \\ & = -\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 {\left (3 b^3\right ) \text {Subst}\left (\int \frac {\log \left (2-\frac {2}{1-i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d e^3} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=-\frac {a^3+b^3 \left (1+c^2+2 c d x+d^2 x^2\right ) \arctan (c+d x)^3+3 a^2 b \left (c+d x+\left (1+(c+d x)^2\right ) \arctan (c+d x)\right )+3 a b^2 \left (2 (c+d x) \arctan (c+d x)+\left (1+(c+d x)^2\right ) \arctan (c+d x)^2-2 (c+d x)^2 \log \left (\frac {c+d x}{\sqrt {1+(c+d x)^2}}\right )\right )+3 b^3 (c+d x) \left (\arctan (c+d x)^2-2 (c+d x) \arctan (c+d x) \log \left (1-e^{2 i \arctan (c+d x)}\right )+i (c+d x) \left (\arctan (c+d x)^2+\operatorname {PolyLog}\left (2,e^{2 i \arctan (c+d x)}\right )\right )\right )}{2 d e^3 (c+d x)^2} \]

[In]

Integrate[(a + b*ArcTan[c + d*x])^3/(c*e + d*e*x)^3,x]

[Out]

-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)*ArcTan[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)*(ArcTan[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcTan[c + d*x])])))/(d*e^3*(c +
d*x)^2)

Maple [B] (verified)

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	\(-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2} d}+\frac {b^{3} \left (-\frac {\arctan \left (d x +c \right )^{3}}{2 \left (d x +c \right )^{2}}-\frac {3 \arctan \left (d x +c \right )^{2}}{2 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )^{3}}{2}+3 \ln \left (d x +c \right ) \arctan \left (d x +c \right )-\frac {3 \arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{4}+\frac {3 i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{4}+\frac {3 i \ln \left (d x +c \right ) \ln \left (1+i \left (d x +c \right )\right )}{2}-\frac {3 i \ln \left (d x +c \right ) \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (d x +c \right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (d x +c \right )\right )}{2}\right )}{e^{3} d}+\frac {3 a^{2} b \left (-\frac {\arctan \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {1}{2 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )}{2}\right )}{e^{3} d}+\frac {3 a \,b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\arctan \left (d x +c \right )}{d x +c}-\frac {\arctan \left (d x +c \right )^{2}}{2}+\ln \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{e^{3} d}\)	\(431\)

[In]

int((a+b*arctan(d*x+c))^3/(d*e*x+c*e)^3,x,method=_RETURNVERBOSE)

[Out]

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*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))-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*ar
ctan(d*x+c)-1/2/(d*x+c)-1/2*arctan(d*x+c)))

Fricas [F]

\[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \arctan \left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{3}} \,d x } \]

[In]

integrate((a+b*arctan(d*x+c))^3/(d*e*x+c*e)^3,x, algorithm="fricas")

[Out]

integral((b^3*arctan(d*x + c)^3 + 3*a*b^2*arctan(d*x + c)^2 + 3*a^2*b*arctan(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)

Sympy [F]

\[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=\frac {\int \frac {a^{3}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{3} \operatorname {atan}^{3}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 a b^{2} \operatorname {atan}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 a^{2} b \operatorname {atan}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]

[In]

integrate((a+b*atan(d*x+c))**3/(d*e*x+c*e)**3,x)

[Out]

(Integral(a**3/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(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

Maxima [F]

\[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \arctan \left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{3}} \,d x } \]

[In]

integrate((a+b*arctan(d*x+c))^3/(d*e*x+c*e)^3,x, algorithm="maxima")

[Out]

-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 + 24*(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)

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=\text {Timed out} \]

[In]

integrate((a+b*arctan(d*x+c))^3/(d*e*x+c*e)^3,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c+d x))^3}{(c e+d e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]

[In]

int((a + b*atan(c + d*x))^3/(c*e + d*e*x)^3,x)

[Out]

int((a + b*atan(c + d*x))^3/(c*e + d*e*x)^3, x)