\(\int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx\) [16]

Optimal result

Integrand size = 21, antiderivative size = 164 \[ \int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx=-\frac {3 i b e (a+b \arctan (c+d x))^2}{2 d}-\frac {3 b e (c+d x) (a+b \arctan (c+d x))^2}{2 d}+\frac {e (a+b \arctan (c+d x))^3}{2 d}+\frac {e (c+d x)^2 (a+b \arctan (c+d x))^3}{2 d}-\frac {3 b^2 e (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d}-\frac {3 i b^3 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5151, 12, 4946, 5036, 4930, 5040, 4964, 2449, 2352, 5004} \[ \int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx=-\frac {3 b^2 e \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))}{d}-\frac {3 i b e (a+b \arctan (c+d x))^2}{2 d}-\frac {3 b e (c+d x) (a+b \arctan (c+d x))^2}{2 d}+\frac {e (c+d x)^2 (a+b \arctan (c+d x))^3}{2 d}+\frac {e (a+b \arctan (c+d x))^3}{2 d}-\frac {3 i b^3 e \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{2 d} \]

[In]

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

[Out]

(((-3*I)/2)*b*e*(a + b*ArcTan[c + d*x])^2)/d - (3*b*e*(c + d*x)*(a + b*ArcTan[c + d*x])^2)/(2*d) + (e*(a + b*A
rcTan[c + d*x])^3)/(2*d) + (e*(c + d*x)^2*(a + b*ArcTan[c + d*x])^3)/(2*d) - (3*b^2*e*(a + b*ArcTan[c + d*x])*
Log[2/(1 + I*(c + d*x))])/d - (((3*I)/2)*b^3*e*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-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]

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

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.20 \[ \int (c e+d e x) (a+b \arctan (c+d x))^3 \, dx=\frac {e \left (3 b^2 (-i+c+d x) (-b+a (i+c+d x)) \arctan (c+d x)^2+b^3 \left (1+c^2+2 c d x+d^2 x^2\right ) \arctan (c+d x)^3+3 b \arctan (c+d x) \left (a \left (-2 b (c+d x)+a \left (1+c^2+2 c d x+d^2 x^2\right )\right )-2 b^2 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )+a \left (a (c+d x) (-3 b+a c+a d x)-6 b^2 \log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )\right )+3 i b^3 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )}{2 d} \]

[In]

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

[Out]

(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)
*ArcTan[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*Po
lyLog[2, -E^((2*I)*ArcTan[c + d*x])]))/(2*d)

Maple [B] (verified)

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 = 0.55 (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	\(\text {Expression too large to display}\)	\(1413\)

[In]

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

[Out]

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))
-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*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)))

Fricas [F]

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

[In]

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

[Out]

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)

Sympy [F]

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

[In]

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

[Out]

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*at
an(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))

Maxima [F]

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

[In]

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

[Out]

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)*arctan(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*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^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*arctan(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*integrate(1/32*x^3*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1
), x) + 384*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*arc
tan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 96*a*b^2*d^3*e*integrate(1/32*x^3*log(d^2*x^2 + 2*c*d*x + c
^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 144*a*b^2*c*d^2*e*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) + 1728*a*b^2*c^2*d*e*integrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c
*d*x + c^2 + 1), x) + 288*a*b^2*c*d^2*e*integrate(1/32*x^2*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x
+ c^2 + 1), x) + 144*a*b^2*c^2*d*e*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) + 192*a*b^2*c^2*d*e*integrate(1/32*x*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1
), x) + 48*a*b^2*c^3*e*integrate(1/32*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)*b^3*c^2*e + 18*a*b^2*c*e*arctan(d
*x + c)^2*arctan((d^2*x + c*d)/d)/d - 96*b^3*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) - 24*b^3*d^2*e*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) - 192*a*b^2*d^2*e*integrate(1/32*x^2*arctan(d*x + c)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) - 192*b^3*c*d*e*in
tegrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) - 96*b^3*d^2*e*integrate(1/32*x^2*log(d^2*x
^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) - 48*b^3*c*d*e*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) - 384*a*b^2*c*d*e*integrate(1/32*x*arctan(d*x + c)/(d^2*x^2 +
2*c*d*x + c^2 + 1), x) - 96*b^3*c*d*e*integrate(1/32*x*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x +
c^2 + 1), x) - 24*b^3*c^2*e*integrate(1/32*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
) - 6*(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*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*e - 3*b^3*e*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d + 128*b^3*d*e*integrate(1/32*x*arctan(d*x +
c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 576*a*b^2*d*e*integrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x +
c^2 + 1), x) + 48*a*b^2*d*e*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) + 192*b^3*d*e*integrate(1/32*x*arctan(d*x + c)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 48*a*b^2*c*e*integrate
(1/32*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)*b^3*e - 24*b^3*e*integrate(1/32*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))*d)/d

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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