\(\int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx\) [8]

Optimal result

Integrand size = 23, antiderivative size = 183 \[ \int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx=\frac {1}{3} b^2 e^2 x-\frac {b^2 e^2 \arctan (c+d x)}{3 d}-\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))}{3 d}-\frac {i e^2 (a+b \arctan (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^2}{3 d}-\frac {2 b e^2 (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{3 d}-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{3 d} \]

[Out]

1/3*b^2*e^2*x-1/3*b^2*e^2*arctan(d*x+c)/d-1/3*b*e^2*(d*x+c)^2*(a+b*arctan(d*x+c))/d-1/3*I*e^2*(a+b*arctan(d*x+
c))^2/d+1/3*e^2*(d*x+c)^3*(a+b*arctan(d*x+c))^2/d-2/3*b*e^2*(a+b*arctan(d*x+c))*ln(2/(1+I*(d*x+c)))/d-1/3*I*b^
2*e^2*polylog(2,1-2/(1+I*(d*x+c)))/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5151, 12, 4946, 5036, 327, 209, 5040, 4964, 2449, 2352} \[ \int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx=\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^2}{3 d}-\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))}{3 d}-\frac {i e^2 (a+b \arctan (c+d x))^2}{3 d}-\frac {2 b e^2 \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))}{3 d}-\frac {b^2 e^2 \arctan (c+d x)}{3 d}-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{3 d}+\frac {1}{3} b^2 e^2 x \]

[In]

Int[(c*e + d*e*x)^2*(a + b*ArcTan[c + d*x])^2,x]

[Out]

(b^2*e^2*x)/3 - (b^2*e^2*ArcTan[c + d*x])/(3*d) - (b*e^2*(c + d*x)^2*(a + b*ArcTan[c + d*x]))/(3*d) - ((I/3)*e
^2*(a + b*ArcTan[c + d*x])^2)/d + (e^2*(c + d*x)^3*(a + b*ArcTan[c + d*x])^2)/(3*d) - (2*b*e^2*(a + b*ArcTan[c
+ d*x])*Log[2/(1 + I*(c + d*x))])/(3*d) - ((I/3)*b^2*e^2*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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, 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 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 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^2 x^2 (a+b \arctan (x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int x^2 (a+b \arctan (x))^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^2}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {x^3 (a+b \arctan (x))}{1+x^2} \, dx,x,c+d x\right )}{3 d} \\ & = \frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^2}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}(\int x (a+b \arctan (x)) \, dx,x,c+d x)}{3 d}+\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {x (a+b \arctan (x))}{1+x^2} \, dx,x,c+d x\right )}{3 d} \\ & = -\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))}{3 d}-\frac {i e^2 (a+b \arctan (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^2}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {a+b \arctan (x)}{i-x} \, dx,x,c+d x\right )}{3 d}+\frac {\left (b^2 e^2\right ) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,c+d x\right )}{3 d} \\ & = \frac {1}{3} b^2 e^2 x-\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))}{3 d}-\frac {i e^2 (a+b \arctan (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^2}{3 d}-\frac {2 b e^2 (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{3 d}-\frac {\left (b^2 e^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,c+d x\right )}{3 d}+\frac {\left (2 b^2 e^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{3 d} \\ & = \frac {1}{3} b^2 e^2 x-\frac {b^2 e^2 \arctan (c+d x)}{3 d}-\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))}{3 d}-\frac {i e^2 (a+b \arctan (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^2}{3 d}-\frac {2 b e^2 (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{3 d}-\frac {\left (2 i b^2 e^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{3 d} \\ & = \frac {1}{3} b^2 e^2 x-\frac {b^2 e^2 \arctan (c+d x)}{3 d}-\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))}{3 d}-\frac {i e^2 (a+b \arctan (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^2}{3 d}-\frac {2 b e^2 (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{3 d}-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.89 \[ \int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx=\frac {e^2 \left (a^2 (c+d x)^3+a b \left (-(c+d x)^2+2 (c+d x)^3 \arctan (c+d x)+\log \left (1+(c+d x)^2\right )\right )+b^2 \left (c+d x-\arctan (c+d x)-(c+d x)^2 \arctan (c+d x)+i \arctan (c+d x)^2+(c+d x)^3 \arctan (c+d x)^2-2 \arctan (c+d x) \log \left (1+e^{2 i \arctan (c+d x)}\right )+i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )\right )}{3 d} \]

[In]

Integrate[(c*e + d*e*x)^2*(a + b*ArcTan[c + d*x])^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.51

method	result	size
derivativedivides	\(\frac {\frac {e^{2} a^{2} \left (d x +c \right )^{3}}{3}+b^{2} e^{2} \left (\frac {\left (d x +c \right )^{3} \arctan \left (d x +c \right )^{2}}{3}-\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{3}+\frac {\arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{3}+\frac {d x}{3}+\frac {c}{3}-\frac {\arctan \left (d x +c \right )}{3}+\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{6}\right )+2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \arctan \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2}}{6}+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{6}\right )}{d}\)	\(276\)
default	\(\frac {\frac {e^{2} a^{2} \left (d x +c \right )^{3}}{3}+b^{2} e^{2} \left (\frac {\left (d x +c \right )^{3} \arctan \left (d x +c \right )^{2}}{3}-\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{3}+\frac {\arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{3}+\frac {d x}{3}+\frac {c}{3}-\frac {\arctan \left (d x +c \right )}{3}+\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{6}\right )+2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \arctan \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2}}{6}+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{6}\right )}{d}\)	\(276\)
parts	\(\frac {e^{2} a^{2} \left (d x +c \right )^{3}}{3 d}+\frac {b^{2} e^{2} \left (\frac {\left (d x +c \right )^{3} \arctan \left (d x +c \right )^{2}}{3}-\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{3}+\frac {\arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{3}+\frac {d x}{3}+\frac {c}{3}-\frac {\arctan \left (d x +c \right )}{3}+\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{6}\right )}{d}+\frac {2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \arctan \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2}}{6}+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{6}\right )}{d}\)	\(281\)
risch	\(\text {Expression too large to display}\)	\(1383\)

[In]

int((d*e*x+c*e)^2*(a+b*arctan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/3*e^2*a^2*(d*x+c)^3+b^2*e^2*(1/3*(d*x+c)^3*arctan(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))))+2*e^2*a*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]

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

[In]

integrate((d*e*x+c*e)^2*(a+b*arctan(d*x+c))^2,x, algorithm="fricas")

[Out]

integral(a^2*d^2*e^2*x^2 + 2*a^2*c*d*e^2*x + a^2*c^2*e^2 + (b^2*d^2*e^2*x^2 + 2*b^2*c*d*e^2*x + b^2*c^2*e^2)*a
rctan(d*x + c)^2 + 2*(a*b*d^2*e^2*x^2 + 2*a*b*c*d*e^2*x + a*b*c^2*e^2)*arctan(d*x + c), x)

Sympy [F]

\[ \int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx=e^{2} \left (\int a^{2} c^{2}\, dx + \int a^{2} d^{2} x^{2}\, dx + \int b^{2} c^{2} \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{2} \operatorname {atan}{\left (c + d x \right )}\, dx + \int 2 a^{2} c d x\, dx + \int b^{2} d^{2} x^{2} \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{2} x^{2} \operatorname {atan}{\left (c + d x \right )}\, dx + \int 2 b^{2} c d x \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 4 a b c d x \operatorname {atan}{\left (c + d x \right )}\, dx\right ) \]

[In]

integrate((d*e*x+c*e)**2*(a+b*atan(d*x+c))**2,x)

[Out]

e**2*(Integral(a**2*c**2, x) + Integral(a**2*d**2*x**2, x) + Integral(b**2*c**2*atan(c + d*x)**2, x) + Integra
l(2*a*b*c**2*atan(c + d*x), x) + Integral(2*a**2*c*d*x, x) + Integral(b**2*d**2*x**2*atan(c + d*x)**2, x) + In
tegral(2*a*b*d**2*x**2*atan(c + d*x), x) + Integral(2*b**2*c*d*x*atan(c + d*x)**2, x) + Integral(4*a*b*c*d*x*a
tan(c + d*x), x))

Maxima [F]

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

[In]

integrate((d*e*x+c*e)^2*(a+b*arctan(d*x+c))^2,x, algorithm="maxima")

[Out]

3/4*b^2*c^4*e^2*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d - 1/4*(3*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^2
/d - arctan((d^2*x + c*d)/d)^3/d)*b^2*c^4*e^2 + 1/3*a^2*d^2*e^2*x^3 + 36*b^2*d^4*e^2*integrate(1/48*x^4*arctan
(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^2*d^4*e^2*integrate(1/48*x^4*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) + 144*b^2*c*d^3*e^2*integrate(1/48*x^3*arctan(d*x + c)^2/(d^2*x^2 + 2*
c*d*x + c^2 + 1), x) + 4*b^2*d^4*e^2*integrate(1/48*x^4*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^2*c*d^3*e^2*integrate(1/48*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) + 216*b^2*c^2*d^2*e^2*integrate(1/48*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 16*b^
2*c*d^3*e^2*integrate(1/48*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) + 18*b^2*c^2
*d^2*e^2*integrate(1/48*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) + 144*b^2*c^3
*d*e^2*integrate(1/48*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 24*b^2*c^2*d^2*e^2*integrate(1/4
8*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) + 12*b^2*c^3*d*e^2*integrate(1/48*x*l
og(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^2*c^3*d*e^2*integrate(1/48*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) + 3*b^2*c^4*e^2*integrate(1/48*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^2*c*d*e^2*x^2 + 3/4*b^2*c^2*e^2*arctan(d*x + c)^2*arcta
n((d^2*x + c*d)/d)/d - 8*b^2*d^3*e^2*integrate(1/48*x^3*arctan(d*x + c)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) - 24
*b^2*c*d^2*e^2*integrate(1/48*x^2*arctan(d*x + c)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) - 24*b^2*c^2*d*e^2*integra
te(1/48*x*arctan(d*x + c)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) - 1/4*(3*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^2
/d - arctan((d^2*x + c*d)/d)^3/d)*b^2*c^2*e^2 + 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*b*c*d*e^2 + 1/3*(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*
b*d^2*e^2 + a^2*c^2*e^2*x + 36*b^2*d^2*e^2*integrate(1/48*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1),
x) + 3*b^2*d^2*e^2*integrate(1/48*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) +
72*b^2*c*d*e^2*integrate(1/48*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^2*c*d*e^2*integrate(
1/48*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*b^2*c^2*e^2*integrate(1/48*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) + (2*(d*x + c)*arctan(d*x + c) - log((d*x +
c)^2 + 1))*a*b*c^2*e^2/d + 1/12*(b^2*d^2*e^2*x^3 + 3*b^2*c*d*e^2*x^2 + 3*b^2*c^2*e^2*x)*arctan(d*x + c)^2 - 1/
48*(b^2*d^2*e^2*x^3 + 3*b^2*c*d*e^2*x^2 + 3*b^2*c^2*e^2*x)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2

Giac [F]

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

[In]

integrate((d*e*x+c*e)^2*(a+b*arctan(d*x+c))^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int((c*e + d*e*x)^2*(a + b*atan(c + d*x))^2,x)

[Out]

int((c*e + d*e*x)^2*(a + b*atan(c + d*x))^2, x)